Geometric dynamics of Vlasov kinetic theory and its moments

Geometric d ynamics of Vlaso v kinetic theor y and its moments Cesar e T r onci A thesis presen ted for the degree of Do ctor of Philosoph y of the Universit y of London and the Diploma of Imp erial College Departmen t of Mathematics Imp erial College London April 2008 “L a natur e est un temple o` u de vivants pil liers L aissent p arfois sortir de c onfuses p ar oles; L’homme y p asse ` a tr avers des forˆ ets de symb oles Qui l’observent ave c des r e gar ds familiers. ” (C. Baudelaire, Corr esp ondenc es , Les ﬂeurs du mal) 2 Abstract The Vlasov equation of kinetic theory is introduced and the Hamiltonian structure of its momen ts is presen ted. Then w e focus on the geodesic evolution of the Vlaso v momen ts. As a ﬁrst step, these momen t equations generalize the Camassa-Holm equation to its m ulti- comp onen t version. Subsequently , adding electrostatic forces to the geodesic momen t equa- tions relates them to the Benney equations and to the equations for b eam dynamics in particle accelerators. Next, we develop a kinetic theory for self assembly in nano-particles. Darcy’s law is in tro duced as a general principle for aggregation dynamics in friction dominated systems (at diﬀeren t scales). Then, a kinetic equation is in tro duced for the dissipativ e motion of isotropic nano-particles. The zeroth-momen t dynamics of this equation reco vers the classical Darcy’s la w at the macroscopic level. A kinetic-theory description for oriented nano-particles is also presen ted. A t the macroscopic level, the zeroth moments of this kinetic equation reco ver the magnetization dynamics of the Landau-Lifshitz-Gilb ert equation. The moment equations exhibit the sp ontaneous emergence of singular solutions (clumpons) that ﬁnally merge in one singularit y . This b eha viour represents aggregation and alignmen t of oriented nano-particles. Finally , the Smolucho wski description is deriv ed from the dissipative Vlasov equation for anisotropic in teractions. V arious levels of approximate Smolucho wsky descriptions are prop osed as special cases of the general treatmen t. As a result, the macroscopic momen tum emerges as an additional dynamical v ariable that in general cannot b e neglected. I declare that the material presented in this thesis is my own work and any material which is not my own has been ackno wledged. Signed: Cesar e T r onci Date: April 2008 3 Con ten ts Preface 11 0 Outline: motiv ations, results and p ersp ectiv es 14 0.1 Mathematical background of kinetic theory . . . . . . . . . . . . . . . . . . . 14 0.2 Geometry of Vlasov momen ts: state of the art . . . . . . . . . . . . . . . . . . 17 0.3 Motiv ations for the present w ork . . . . . . . . . . . . . . . . . . . . . . . . . 20 0.4 Some op en questions and results in this work . . . . . . . . . . . . . . . . . . 21 0.5 A new mo del for oriented nano-particles . . . . . . . . . . . . . . . . . . . . . 24 0.6 P ersp ectives for future w ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 Singular solutions in contin uum dynamics 27 1.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2 Basic concepts in geometric mechanics . . . . . . . . . . . . . . . . . . . . . . 28 1.3 Euler equation and vortex ﬁlamen ts . . . . . . . . . . . . . . . . . . . . . . . 32 1.4 The Camassa-Holm and EPDiﬀ equations . . . . . . . . . . . . . . . . . . . . 36 1.5 Darcy’s law for aggregation dynamics . . . . . . . . . . . . . . . . . . . . . . 39 1.6 The Vlasov equation in kinetic theory . . . . . . . . . . . . . . . . . . . . . . 43 2 Dynamics of kinetic moments 48 2.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2 Momen t Lie-P oisson dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2.1 Review of the one dimensional case . . . . . . . . . . . . . . . . . . . . 49 2.2.2 Multidimensional treatment I: bac kground . . . . . . . . . . . . . . . . 52 2.2.3 Multidimensional treatment II: a new result . . . . . . . . . . . . . . . 54 2.3 New v ariational principles for moment dynamics . . . . . . . . . . . . . . . . 57 4 CONTENTS 5 2.3.1 Hamilton-P oincar´ e hierarc hy . . . . . . . . . . . . . . . . . . . . . . . 57 2.3.2 Euler-P oincar´ e hierarc hy . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.4 Some results on moments and cotangen t lifts . . . . . . . . . . . . . . . . . . 61 2.4.1 Bac kground on Lagrangian v ariables . . . . . . . . . . . . . . . . . . . 62 2.4.2 Characteristic equations and related results . . . . . . . . . . . . . . . 63 2.4.3 Momen ts and semidirect pro ducts . . . . . . . . . . . . . . . . . . . . 66 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3 Geo desic ﬂo w on the moments: a new problem 69 3.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Applications of the moments and quadratic terms . . . . . . . . . . . . . . . . 70 3.2.1 The Benney equations and particle b eams: a new result . . . . . . . . 70 3.2.2 The wak e-ﬁeld mo del and some sp ecializations . . . . . . . . . . . . . 71 3.2.3 The Maxwell-Vlaso v system . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.4 The EPDiﬀ equation and singular solutions . . . . . . . . . . . . . . . 72 3.3 A new geo desic ﬂow and its singular solutions . . . . . . . . . . . . . . . . . . 73 3.3.1 F ormulation of the problem: quadratic Hamiltonians . . . . . . . . . . 73 3.3.2 A ﬁrst result: the geo desic Vlasov equation (EPSymp) . . . . . . . . . 75 3.3.3 The nature of singular geo desic solutions . . . . . . . . . . . . . . . . 77 3.3.4 Some results on the dynamics of singular solutions . . . . . . . . . . . 79 3.3.5 Connections with the cold plasma solution . . . . . . . . . . . . . . . . 81 3.3.6 A result on truncations: the CH-2 equation . . . . . . . . . . . . . . . 83 3.3.7 Extending EPSymp to anisotropic interactions . . . . . . . . . . . . . 86 3.4 Op en questions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 GOP theory and geometric dissipation 91 4.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Theory of geometric order parameter equations . . . . . . . . . . . . . . . . . 93 4.2.1 Bac kground: geometric structure of Darcy’s law . . . . . . . . . . . . 93 4.2.2 GOP equations: a result on singular solutions . . . . . . . . . . . . . . 95 4.2.3 More background: m ulti-scale v ariations . . . . . . . . . . . . . . . . . 98 4.2.4 Prop erties of the diamond op eration . . . . . . . . . . . . . . . . . . . 99 4.2.5 Energy dissipation in GOP theory . . . . . . . . . . . . . . . . . . . . 100 4.2.6 A general principle for geometric dissipation . . . . . . . . . . . . . . . 103 CONTENTS 6 4.3 Review of scalar GOP equations . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4 New GOP equations for one-forms and tw o-forms . . . . . . . . . . . . . . . . 105 4.4.1 Results on singular solutions . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.2 Exact diﬀerential forms . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.4.3 Singular solutions for exact forms and their p otentials . . . . . . . . . 108 4.5 Applications to vortex dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.5.1 A new GOP equation for ﬂuid vorticit y . . . . . . . . . . . . . . . . . 109 4.5.2 Results in tw o dimensions: point v ortices and steady ﬂows . . . . . . . 111 4.5.3 More results in three dimensions . . . . . . . . . . . . . . . . . . . . . 113 4.6 Tw o more examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6.1 Dissipativ e EPDiﬀ equation . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6.2 Dissipativ e Vlaso v dynamics . . . . . . . . . . . . . . . . . . . . . . . 117 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5 Geometric dissipation for kinetic equations 120 5.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.1.1 History of double-brack et dissipation . . . . . . . . . . . . . . . . . . . 121 5.1.2 The origins: selectiv e decay h yp othesis . . . . . . . . . . . . . . . . . . 122 5.2 Double brack et structure for kinetic equations . . . . . . . . . . . . . . . . . . 123 5.2.1 Bac kground review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.2 A new multiscale dissipativ e kinetic equation . . . . . . . . . . . . . . 124 5.3 Prop erties and consequences of the mo del . . . . . . . . . . . . . . . . . . . . 126 5.3.1 GOP theory and double brack et dissipation: background . . . . . . . . 126 5.3.2 A ﬁrst consequence: conserv ation of en tropy . . . . . . . . . . . . . . . 128 5.3.3 A result on the single-particle solution . . . . . . . . . . . . . . . . . . 131 5.4 Geometric dissipation for kinetic moments . . . . . . . . . . . . . . . . . . . . 132 5.4.1 Review of the moment brac ket . . . . . . . . . . . . . . . . . . . . . . 132 5.4.2 A multiscale dissipativ e moment hierarc hy . . . . . . . . . . . . . . . . 132 5.5 Prop erties of the dissipative momen t hierarch y . . . . . . . . . . . . . . . . . 134 5.5.1 A ﬁrst result: reco vering Darcy’s la w . . . . . . . . . . . . . . . . . . . 134 5.5.2 A new dissipative ﬂuid model and its prop erties . . . . . . . . . . . . 135 5.6 F urther generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.6.1 A double brack et structure for the b -equation. . . . . . . . . . . . . . . 137 5.6.2 A GOP equation for the moments . . . . . . . . . . . . . . . . . . . . 139 CONTENTS 7 5.7 Discussion and op en questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6 Anisotropic in teractions: a new model 142 6.1 In tro duction and bac kground . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.1.1 Geometric mo dels of dissipation in physical systems . . . . . . . . . . 142 6.1.2 Goal and present approac h . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2 Geometric dissipation for anisotropic interactions . . . . . . . . . . . . . . . 144 6.2.1 A dissipative v ersion of the GHK-Vlasov equation . . . . . . . . . . . 144 6.2.2 Dissipativ e momen t dynamics: a new anisotropic mo del . . . . . . . . 145 6.2.3 A ﬁrst prop erty: singular solutions . . . . . . . . . . . . . . . . . . . . 148 6.3 An application: rod-like particles on the line . . . . . . . . . . . . . . . . . . . 150 6.3.1 Momen t equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.3.2 More results: emergence and interaction of singularities . . . . . . . . 150 6.3.3 Higher dimensional treatment . . . . . . . . . . . . . . . . . . . . . . . 155 6.4 A higher order of approximation . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.4.1 Momen t dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.4.2 Cold plasma formulation and momen t closure . . . . . . . . . . . . . . 162 6.4.3 Pro of of Theorem 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.5 Smoluc howski approac h to momen t dynamics . . . . . . . . . . . . . . . . . . 166 6.5.1 A new GOP-Smolucho wski equation . . . . . . . . . . . . . . . . . . . 167 6.5.2 Systematic deriv ation of moment equations . . . . . . . . . . . . . . . 168 6.5.3 A cold plasma-like closure . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.5.4 Some results on sp ecializations and truncations . . . . . . . . . . . . . 171 6.5.5 A divergence form for the momen t equations . . . . . . . . . . . . . . 172 6.6 Summary and outlo ok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7 Conclusions and p ersp ectives 176 7.1 The Schouten concomitan t and moment dynamics . . . . . . . . . . . . . . . 176 7.2 Geo desic momen t equations and EPSymp . . . . . . . . . . . . . . . . . . . . 178 7.3 Geometric dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4 Dissipativ e equation for ﬂuid vorticit y . . . . . . . . . . . . . . . . . . . . . . 180 7.5 Geometric dissipation in kinetic theory . . . . . . . . . . . . . . . . . . . . . . 181 7.6 Double brack et equations for the moments . . . . . . . . . . . . . . . . . . . . 182 7.7 Anisotropic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 CONTENTS 8 7.8 The Smolucho wski approac h . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.9 F uture ob jectiv es in geometric moment dynamics . . . . . . . . . . . . . . . . 185 Bibliograph y 188 List of Figures 1.1 P eakons emerging from Gaussian initial conditions for diﬀerent Gaussian widths w . Figure from [HoSt03]. . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.2 Singularities emerging from a Gaussian initial condition. It is shown how these singularities merge together after their formation. This ﬁgure is taken from [HoPu2005, HoPu2006]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.1 L eft: Emerging singularities. Plots of the smo othed densit y ¯ ρ = H ∗ ρ and orien tation m = H ∗ m (three components), where the smo othing k ernel is the Helmholtzian e −| x | . The ﬁgures sho w ho w the singular solution emerges from a Gaussian initial condition for the energy in (6.12). Smo othed quantities are chosen to av oid the necessit y to represen t δ -functions. Right: Orienton formation in a d = 1 dimensional simulation. The color-co de on the cylinder denotes lo cal aver age d density: black is maximum density while white is ρ = 0. Purple lines denote the three-dimensional vector m = H ∗ m . The formation of sharp p eaks in a veraged quan tities corresponds to the formation of δ -functions. (Figures by V. Putk aradze) . . . . . . . . . . . . . . . . . . . 151 6.2 Ev olution of a ﬂat magnetization ﬁeld and a sinusoidally-v arying densit y . Subﬁgure (a) shows the evolution of ¯ ρ = H ∗ ρ for t ∈ [0 . 5 , 1]; (b) shows the ev olution of ¯ m x . The proﬁles of ¯ m y and ¯ m z are similar. At t = 0 . 5, the initial data hav e formed eight equally spaced, iden tical clump ons, corresp onding to the eight density maxima in the initial conﬁguration. By impulsively shifting the clumpon at x = 0 by a small amount, the equilibrium is disrupted and the clumpons merge repeatedly until only one clump on remains. (Figures b y L. ´ O N´ araigh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 9 LIST OF FIGURES 10 6.3 Sp on taneous emergence of clump ed states in t wo dimensions. Random initial conditions break up into dots. The expression for the energy interaction is E = 1 / 2 R H ( x − y ) ( ρ ( x ) ρ ( y ) + m ( x ) · m ( y )), where H ( x − y ) = e −| x − y | . The left plot shows the smoothed density ¯ ρ , while right plot shows the mod- ulus | m | (Figure by V. Putk aradze). . . . . . . . . . . . . . . . . . . . . . . . 157 6.4 An example of t wo oriented ﬁlaments (red and green) attracting eac h other and un winding at the same time. The blue vectors illustrate the vector m at eac h p oin t on the curve. Time scale is arbitrary . (Figure by V. Putk aradze) . 158 Preface This work is the fruit of my research o ver the last three y ears, during my postgraduate studies at Imp erial College London. Besides the fundamental guide of m y sup ervisor Darryl Holm, the collaboration with John Gibb ons and V akhtang Putk aradze has also b een determinan t. The scien tiﬁc matter of this work is the geometric structure of the Vlaso v equation in kinetic theory and the passage from this microscopic description to the macroscopic ﬂuid treatmen t, giv en by the dynamics of kinetic moments. Vlasov moments are very w ell kno wn since the early tw entieth century , when Chapmann and Enskog formulated their closure of the Boltzmann equation [Chapman1960]. The p ow er of the momen t approac h leaded to the imp ortan t theory of ﬂuid mechanics and its kinetic justiﬁcations in physics. In the collisionless Vlasov limit, the moment hierarc hy turns out to conserv e a purely geometric structure inherited by the Vlaso v Lie-Poisson brac ket. The geometric structure of momen t dynamics is kno wn since the late 70’s [KuMa1978, Le1979] and was found surpris- ingly in a v ery diﬀeren t context from kinetic theory , that is the analysis of in tegrable shallo w w ater equations. The relation with kinetic theory w as found few y ears later [Gi1981], but the geometric properties of moment dynamics were not explored further. Even the ﬂuid closure has alwa ys b een considered in terms of cold plasma solution of the Vlasov equation, without considering the mathematical property that this solution is equiv alen t to a trunc ation of the momen t hierarch y to the ﬁrst tw o moments. This prop ert y is apparen tly trivial, although this work shows that this is crucial in some contexts inv olving dissipative dynamics, where the cold plasma solution is not of muc h use. This w ork takes inspiration from the idea that the geometric prop erties of moment dy- namics deserv e further inv estigation. The topics cov ered in this thesis analyze the geometric prop erties of both Hamiltonian and dissipative ﬂows. The ﬁrst part is devoted to exploring the geo desic motion on the moments and the second part formulates the double brack et 11 PREF A CE 12 equations for dissipative momen t dynamics. The main result is the formulation of a mo del for the aggregation of oriented particles, with p ossible applications in nano-sciences. Plan of the work The thesis pro ceeds in the following order. The ﬁrst chapter reviews some bac kground and form ulates the motiv ations b y fo cusing on singular solutions in contin uum theories. The second c hapter analyzes the geometric structure of moment dynamics. It contains one main result, that is the iden tiﬁcation of the moment Lie brack et with the symmetric Sc houten brack et on symmetric tensors, whic h is diﬀer ent from the Kup ershmidt-Manin brac ket in m ulti-index notation [GiHoT r2008]. The third chapter concerns the study of geodesic motion on the moments: it is explained ho w this is equiv alent to the geo desic motion on canonical transformations (EPSymp) and this fact determines the existence of singular solutions, which may reduce to the single- particle dynamics. At the end of c hapter 3, the geo desic motion on the moments is extended to include anisotropic in teractions and this constitutes an in tro duction to the topics co vered in the last chapter. The fourth chapter formulates the geometric dissipative dynamics for geometric order parameters (GOP). This analysis takes inspiration from the geometric structure of Darcy’s la w [HoPu2005, HoPu2006, HoPu2007] and form ulates a geometric dissipation that extends Darcy’s law to any tensor quantit y , instead of only densities. The b ehavior of singular solutions is analyzed extensiv ely . Moreov er the application of this framew ork to the case of the ﬂuid vorticit y leads to the fact that this form of dissipative dynamics em b o dies to the double brack et approac h, whic h w as established in the early 90’s [BlKrMaRa1996]. The ﬁfth c hapter applies the geometric dissipation to the case of the Vlaso v equation and to the Vlasov kinetic moments. The main result of this section is that Darcy’s law follows v ery naturally as the zero-th moment equation of the dissipative moment hierarc hy . The dissipativ e momen t dynamics is also applied to formulate appropriate equations such as the dissipativ e ﬂuid equations, the b -equation and the momen t GOP equation, each allowing singular solutions. The sixth chapter extends the previous dissipativ e treatment to kinetic theory for anisotropic in teractions. The distribution function no w depends on the orientation of the single (nano)- particle and the moment hierarc hy is again obtained. The analogue of Darcy’s law for this case yields tw o equations, one for the mass density and the other for the polarization, recov- PREF A CE 13 ering the Landau-Lifshitz-Gilb ert dissipation term for the magnetization in ferromagnetic media [Gilb ert1955]. It is imp ortan t to notice that the ﬂuid closure of the dissipativ e mo- men t hierarch y is not obtained through the cold plasma solution of the Vlasov equation, rather it is obtained by a pure truncation of the momen t hierarch y to the ﬁrst t wo momen ts. This constitutes a go o d conﬁrmation for the high importance of moment dynamics in deriv- ing macroscopic contin uum mo dels from kinetic treatmen ts. F urther study is dev oted to the Smoluc howski approac h and it is shown ho w this approach presen ts interesting truncations and sp ecializations, despite the complicated equations arising from the whole hierarch y . Ac kno wledgements My greatest ac knowledgmen ts are addressed to my advisor Darryl Holm for trusting my capabilities in this ﬁeld, despite m y previous exp erience in very applied engineering problems. I w ould also lik e to thank him for guiding me through the diﬃcult w orld of scien tiﬁc researc h and to let me appreciate more and more the beauty of geometric models in physics. Thanks, Darryl! Also I am indebted with V akhtang Putk aradze at the Colorado State Universit y for his great ideas and for his excellen t advices ov er the last year. My work with him has b een a great exp erience, which I hop e will contin ue in the next years. Moreov er I would like to thank John Gibbons at Imperial College London for his expert advice, esp ecially on momen t dynamics; this work would ha ve probably never b een p ossible without his exp ertise in the ﬁeld. P articular ackno wledgements also go to Ugo Amaldi at CERN, who ﬁrst recognized my mathematical taste and supp orted my idea of moving to this ﬁeld. He is the p erson who help ed me most at the b eginning of m y scientiﬁc itinerary and I am enormously indebted to him. In addition I feel need to thank Bruce Carlsten, Paul Channell, Rick ey F ahel and Gio- v anni Lap enta at the Los Alamos National Lab oratory for man y helpful discussions and encouragemen ts. Finally I w ould lik e to thank also m y colleagues Matthew Dixon, for his imp ortan t advice, and Andrea Raimondo for his keen observ ations on momen t dynamics. Chapter 0 Outline: motiv ations, results and p ersp ectiv es 0.1 Mathematical bac kground of kinetic theory The imp ortance of kinetic equations in non-equilibrium statistical mechanics is well kno wn and ﬁnds its ro ots in the pioneering work of Maxwell [Ma1873] and Boltzmann [Bo95]. The mathematical foundations of kinetic theory reside in Liouvil le’s the or em , stating that no matter how large the num b er of particles is in a system, they undergo canonical transformations which preserve the v olume elemen t in the global phase space of the system. More mathematically , one deﬁnes a density v ariable ρ ( q i , p i , t ) (with i = 1 , . . . , N ) for the N -particle system. Then one writes the Liouville equation as a characteristic equation on phase space d dt ρ t = 0 along ˙ q i = ∂ H ∂ p i , ˙ p i = − ∂ H ∂ q i where H is the N -particle Hamiltonian. In Eulerian co ordinates one has the Theorem 1 (Liouville’s equation) Given a phase sp ac e density ρ for the N -p article dis- tribution, its evolution is given by the c onservation e quation ∂ ρ ∂ t + { ρ, H } = 0 so that the fol lowing volume form is pr eserve d ρ 0 ( q (0) i , p (0) i ) dΩ 0 = ρ t ( q ( t ) i , p ( t ) i ) dΩ t 14 OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 15 wher e dΩ is the inﬁnitesimal volume element in phase sp ac e. In the search for appro ximate descriptions of this system, one may think to deal with global quantities that integrate out the information on some of the particles. In particular one deﬁnes a n -particle distribution as f n ( z 1 , . . . , z n , t ) := Z ρ ( z 1 , . . . , z N ) d z n +1 . . . d z N where the notation z i = ( q i , p i ) has b een introduced for compactness of notation. These quan tities are called “BBGKY moments” and their equations constitute an inﬁnite hierarc hy of equations known as BBGKY hier ar chy [MaMoW e1984], or “Bogoliub ov-Born-Green- Kirkw o o d-Yvon equations”. This hierarc hy is rather complicated, although Marsden, Mor- rison and W einstein [MaMoW e1984] ha v e shown that it p ossesses a clear geometric structure in terms of canonical transformations that are symmetric with resp ect to their arguments. In particular, this hierarch y is a Lie-Poisson system, i.e. a Hamiltonian system on a Lie group, as explained in chapter 1. Suitable appro ximations on the equation for the single particle distribution f := f 1 lead to the Boltzmann e quation . F or the purp oses of this w ork it suﬃces to write this equation sc hematically as ∂ f ∂ t + { f , H } =  ∂ f ∂ t  coll where H is now the 1-particle Hamiltonian H = p 2 / 2 + V ( q ). The right hand side collects the information on pairwise collisions among particles and its explicit expression requires a discussion that is out of the purp oses of this w ork. Rather it is imp ortant to discuss an imp ortant appro ximation of the Boltzmann equation, the F okker-Planck e quation [F okker-Plank1931, Ri89]. Indeed, the hypothesis of sto chastic dynamics in terms of Brow- nian motion leads to the following fundamen tal equation ∂ f ∂ t + { f , H } = γ ∂ ∂ p  pf + β − 1 ∂ f ∂ p  where a dissip ative drift-diﬀusion term is evidently substituted to the collision term of the Boltzmann equation. This term is peculiar of the microscopic stochastic dynamics expressed b y the Langevin equation ˙ p = − H q − γ p + p 2 γ β − 1 ˙ w ( t ) for the single particle momentum ( ˙ w is a white noise pro cess). This equation is the most common equation in kinetic theory and it is probably the most used in physical applications. In man y con texts it is p ossible to neglect the eﬀects of collisions. Such contexts range from astrophysical topics (cf. e.g. [Ka1991]) to particle beam dynamics (cf. e.g. [V en turini]), OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 16 whic h is the v ery ﬁrst inspiration for this w ork, given some previous exp erience of the author in the ﬁeld of particle accelerators. In more generalit y , the hypothesis of negligible collisions is most commonly used in the physics of plasmas (electrostatic or magnetized). In the case of collisionless dynamics, the resulting equation ∂ f ∂ t + { f , H } = 0 is called Vlasov e quation [Vl1961] and its underlying mathematical structure has b een widely inv estigated ov er the past decades, especially in terms of geometric arguments [W eMo, MaW e81, Ma82, MaW eRaScSp, CeHoHoMa1998]. In particular, Marsden, W einstein and collab orators [W eMo, MaW e81, Ma82, MaW eRaScSp] hav e shown that this equation p os- sesses a Lie-Poisson structure on the whole group of canonical transformations. The explicit expression of the Lie-Poisson brac ket is { F , G } [ f ] = Z Z f ( q , p, t )  δ F δ f , δ G δ f  d q d p where the Lie brack et {· , ·} is now the canonical Poisson brack et. Even when this equation is coupled with the Maxwell equations and particles are acted on b y an electromagnetic ﬁeld ( Maxwel l-Vlasov system ), the geometric structure persists [W eMo, MaW e81, MaW eRaScSp]. This particular result is also due to Cendra and Holm, who show ed in their joint w ork with Ho yle and Marsden [CeHoHoMa1998] how the Maxw ell-Vlasov equation has also a Lagrangian formulation. This Lagrangian approac h w as ﬁrst pioneered by Low in the late 50’s [Lo58]. As a Lie-Poisson system, the Vlasov equation p ossesses the prop ert y of b eing a kind of c o adjoint motion [MaRa99], so that its evolution map coincides with the c o adjoint gr oup action f t = Ad ∗ g − 1 t f 0 with g t ∈ G as explained in chapter 1. This means that the dynamics is purely geometric and it is uniquely determined by the canonical nature of particle dynamics. A particular kind of Vlaso v equation has been prop osed b y Gibb ons, Holm and Kupersh- midt (GHK) [GiHoKu1982, GiHoKu1983] in order to form ulate a kinetic theory for particles immersed in a Y ang-Mills ﬁeld. Without going in to the details, one can refer to it as a colli- sionless kinetic equation that tak es in to accoun t for an extra-degree of freedom of the single particle. In the case of [GiHoKu1982, GiHoKu1983], this w ould b e a color c harge asso ciated with c hromo dynamics. How ever for the present purposes, this can also b e represen ted by OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 17 a spin-like v ariable which is carried by eac h particle in the system. In more generality this equation can be considered as a kinetic equation for particles with anisotr opic inter actions . The GHK-Vlasov e quation considers a distribution function f = f ( q , p, µ, t ) with µ ∈ g ∗ where g ∗ is the dual of some Lie algebra g . The equation is written as ∂ f ∂ t + { f , H } +  µ,  ∂ f ∂ µ , ∂ H ∂ µ  = 0 where [ · , · ] denotes the Lie brac ket and h· , ·i is the pairing. This equation will be determinan t for the results presented in c hapter 6, where a mo del for oriented nano-particles is form ulated. Although the Vlasov equation enjoys many geometric prop erties (Lie-P oisson brack et, coadjoin t motion, advection), these are not shared by the F okker-Planc k equation, whose geometric in terpretation is far from the theory of symmetry groups used in this w ork. Nev- ertheless, Kandrup [Ka1991] and Blo ch and collab orators [BlKrMaRa1996] ha ve formulated a type of dissip ative Vlasov equation, whic h preserves the geometric nature of the Hamil- tonian ﬂo w while dissipating energy . This theory requires the concept of double br acket dissip ation , i.e. the dissipation is modelled b y the subsequen t application of tw o Poisson brac kets and the corresp onding equation b ecomes ∂ f ∂ t + { f , H } = α { f , { f , H }} . This equation represents an interesting p ossibility for in tro ducing geometric dissipation in kinetic equations, but it has nev er b een considered further. A deep er in vestigation of this equation is presented in c hapter 5 and extended in chapter 6. In the case of isotropic interactions, the Vlasov densit y f dep ends on seven v ariables: six phase space co ordinates plus time. This indicates that the Vlasov equation is still a rather complicated equation even when n umerical eﬀorts are inv olved. Thus it is often conv enient to ﬁnd suitable appro ximations in order to discard unnecessary information while keeping the main feature of collisionless multi particle dynamics. T o this purp ose, one introduces the moments of the Vlasov distribution. 0.2 Geometry of Vlaso v momen ts: state of the art The use of moments in kinetic theory w as introduced by Chapmann and Enskog [Chapman1960], who form ulated their closure of the Boltzmann equation yielding the equations of ﬂuid me- c hanics and its kinetic justiﬁcations in ph ysics. This result show ed ho w the use of moments OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 18 is a p ow erful to ol for obtaining consistent reductions or approximations of the microscopic kinetic description. Since that time, the mathematical prop erties of moments hav e been widely inv estigated The geometric prop erties of Vlasov momen ts mainly arose in tw o very diﬀeren t contexts, particle b eam dynamics and shallow water equations. How ev er it is imp ortant to distin- guish b etw een tw o diﬀeren t classes of momen ts: statistic al moments and kinetic moments . Statistic al moments are deﬁned as g n, b n ( t ) = Z p n q b n f ( q , p, t ) d q d p . These quantities ﬁrst arose in the study of particle beam dynamics [Ch83, Ch90, LyOv88] from the observ ation that the beam emittanc e  :=  g 0 , 2 g 2 , 0 − g 2 1 , 1  1 / 2 is a lab oratory pa- rameter, which is also an inv ariant function of the statistical momen ts. In particular, Chan- nell, Holm, Lysenko and Scov el [Ch90, HoLySc1990, LyP a97] were the ﬁrst to consider the Lie-P oisson structure of the moments, whose explicit expression is given in c hapter 1 as { F , G } = ∞ X b m,m, b n,n =0 ∂ F ∂ g b m,m  b m m − b n n  ∂ G ∂ g b n,n g b m + b n − 1 , m + n − 1 . This geometric framew ork allow ed the systematic construction of symplectic moment in- v arian ts in [HoLySc1990], a question that was also pursued b y Dragt and collab orators in [DrNeRa92]. Sp ecial truncations and approximations of the equations for statistical mo- men ts hav e b een studied also by Scov el and W einstein in [ScW e] in 1994. Besides appli- cations in particle b eam ph ysics, the use of statistical moments has also b een proposed in astroph ysical problems by Channell in [Ch95]. Besides statistical moments, another kind of moments w ere known to b e a p ow erful to ol in kinetic theory , since they had been used by Chapman and Enskog to recov er ﬂuid dynamics from the Boltzmann equation. These are the kinetic moments A n ( q , t ) = Z p n f ( q , p, t ) d p . and the follo wing discussion will refer to these quan tities as simply “momen ts”, unless other- wise sp eciﬁed. The geometric prop erties of these momen ts ﬁrst arose in 1981 [Gi1981], when Gibb ons recognized that these Vlaso v moments are equiv alen t to the v ariables introduced b y Benney in 1973 [Be1973], in the context of shallo w w ater w av es. The Hamiltonian struc- ture of these v ariables was found by Kup ershmidt and Manin [KuMa1978]; later Gibb ons recognized ho w this structure is inherited from the Vlasov Lie-P oisson brack et [Gi1981]. OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 19 The relation b et ween moments and the algebra of generating functions was also known to Leb edev [Le1979], although he did not recognize the connection with Vlasov dynamics. The Lie-P oisson structure for the momen ts is also called Kup ershmidt-Manin structur e and is explicitly written as [KuMa1978] { F , G } = Z A m + n − 1  n δ F δ A n ∂ ∂ q δ G δ A m − m δ G δ A m ∂ ∂ q δ F δ A n  d q whose deriv ation will be presen ted in c hapter 2. The main theorem regarding moments is th us the following Theorem 2 (Gibb ons [Gi1981]) The pr o c ess of taking moments of the Vlasov distribu- tion is a Poisson map , that is it takes the Vlasov Lie-Poisson structur e to another Lie- Poisson structur e, which is given by the Kup ershmidt-Manin br acket. Remark 3 It is imp ortant to notic e that, although the Lie-Poisson moment br acket is wel l known, the c o adjoint gr oup action is not ful ly understo o d and this r epr esents an imp ortant op en question c onc erning the ge ometric dynamics of Vlasov moments. Besides their role in the theory of Benney long wa ves [Be1973], the geometric structure of the momen ts has not b een considered as a whole so far. Even in that context, the use of the Vlasov equation turns out to b e more con venien t. Rather the ﬂuid closure of momen t dynamics is very well understoo d and is given by considering only the ﬁrst tw o moments A 0 and A 1 , which coincide with the ﬂuid densit y and momentum resp ectively . The key to understanding the geometric c haracterization of this closure is to consider the c old plasma solution , i.e. a singular Vlasov solution of the form f ( q , p, t ) = ρ ( q , t ) δ ( p − P ( q, t )) . Substituting this expression in to the Vlaso v equation yields the equations for ρ = A 0 and P = A 1 /ρ . Marsden, Ratiu and collab orators [MaW eRaScSp] sho wed how this solution is a momentum map (cf. e.g. [MaRa99]), which is called plasma-to-ﬂuid map . This imp ortan t prop ert y has b een widely used to form ulate h ydro dynamic mo dels from kinetic theory [MaW eRaScSp] and it has b een extended to account for Y ang-Mills ﬁelds in the w ork of Gibb ons, Holm and Kup ershmidt [GiHoKu1982, GiHoKu1983]. Ho wev er these h ydro dynamical mo dels ha ve usually b een derived directly from the Vlasov equation by direct substitution of the cold plasma solution, rather than considering the momen t hierarc hy in its own. The tw o approaches are clearly equiv alent and this apparently trivial p oin t OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 20 b ecomes a key fact in some con texts where the cold plasma is not of m uch use. An example is provided in chapters 5 and 6, where the substitution of the cold plasma solution is evidently a voided as it yields to cumbersome calculations and results that are not completely clear. 0.3 Motiv ations for the present w ork As men tioned abov e, the topic of Vlaso v momen ts is ﬁrst dictated b y the previous scientiﬁc exp erience of the author with particle accelerators. In particular, b eam dynamics issues assume a central role in many questions of accelerator design, esp ecially for high b eam curren ts ( ∼ 1–100mA), and the Vlaso v approac h is a natural step in this matter. The theory of Vlaso v statistical moments arose in this environmen t. How ever, although the theory of Vlaso v statistical moments is completely understo od [HoLySc1990, ScW e], this is not true for kinetic moments. F or example, it is not known a priori what geometric nature these momen ts should hav e. Is there an y c hance that their geometric prop erties could be relev an t to b eam dynamics and plasma physics? These questions provide the ﬁrst motiv ations for approac hing the topic of Vlasov kinetic momen ts. Also, it is presen ted in c hapter 3 how momen t dynamics recov ers the integrable Camassa- Holm (CH) e quation [CaHo1993] and thus it reco vers its singular p e akon solutions : one ma y w onder whether there is an explanation of the CH in tegrable dynamics in terms of mo- men ts. What would b e a suitable formulation of this problem? What is the relation in terms of singular solutions ? The fact that the CH equation is recov ered by moment dynamics is the main motiv ation for seeking possible ge neralizations of this equation in terms of the momen ts. The dynamics of kinetic moments has nev er b een related with singular solutions and blo w–up phenomena in contin uum PDE’s and this constitutes another motiv ation for pursuing this direction. Moreo ver, Blo ch and collab orators hav e shown how the double br acket dissip ation [BlKrMaRa1996, BlBrCr1997] in kinetic theory recov ers a form of dissipative Vlaso v equa- tion, whic h has b een proposed in astrophysics by Kandrup [Ka1991]. This does not recov er the single p article solution . Why? How can this problem b e solv ed? What is the cor- resp onding in terpretation in terms of the moments? The main motiv ation for pursuing this direction is that the double brac k et dissipation provides an in teresting w ay of inserting dis- sipation in collisionless kinetic equations while preserving the geometric structure of the Vlaso v equation. OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 21 As it easy to see, there are many op en questions that make the geometric prop erties of the Vlasov equation and its mom en ts an intriguing ﬁeld of research. The next section tries to classify these op en questions and explains what the contribution of this w ork is. 0.4 Some op en questions and results in this w ork One can try to classify the open questions in three types: purely geometric questions, Hamiltonian ﬂows on the moments and dissipative geometric ﬂows. At this p oint, one attempts to write a table as follows • Purely geometric questions – Is there a ge ometric char acterization of moments? What kind of geometric quan tities are they? V ector ﬁelds? diﬀeren tial forms? generic tensors? – The BBGKY moments and the statistical momen ts are w ell understoo d as mo- men tum maps [MaMoW e1984, HoLySc1990]: are kinetic moments momentum maps to o? If so, what is the underlying symmetry gr oup ? – Statistical moments p ossess a whole family of in v ariant functions [HoLySc1990]: what are the moment invariants for kinetic moments? – The Euler-Poinc ar´ e e quations are the Lagrangian counterpart of a Lie-Poisson system [MaRa99]: what are the Euler-Poincar ´ e equations for the moments? • Hamiltonian ﬂows on the momen ts – How do es the theory of momen t dynamics apply to physical problems, e.g. b e am dynamics ? – Moment dynamics reco v ers the Camassa-Holm e quation [CaHo1993] from the ev olution of the ﬁrst-order moment: why does this happ en? – Quadratic terms in the moments often app ears in applications: what are the prop erties of pur ely quadr atic Hamiltonians ? – Quadratic Hamiltonians deﬁne ge o desic motion on the moments: what is its geometric interpretation in terms of Vlaso v dynamics? – These systems ma y allo w for singular solutions: what kind of solutions are they? ho w are they related with the CH p eak ons? OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 22 – The CH equation is an integrable equation: do es geo desic momen t dynamics reco ver other inte gr able c ases ? • Geometric dissipative ﬂows – Is it p ossible to extend the double brack et dissipation [BlKrMaRa1996] in the Vlaso v equation to allow for the single p article solution ? – How does the double br acket structur e apply to moment dynamics? – What kind of macroscopic moment e quations arise in this context? what is their meaning? – How do es the GHK-Vlasov e quation [GiHoKu1982, GiHoKu1983] transfer to double brack et dynamics? what is the corresp onding moment dynamics? – What do singular solutions represen t in this case? Ho w do they interact? what happ ens in three dimensions? – Smoluchowski moments dep end on b oth p osition and orientation: what are their equations as they arise from double brack et dynamics? Analogously , this section illustrates the accomplishmen ts of this w ork by following the same scheme. • Results on the momen t brac ket – Chapter 2 shows ho w the moments ha ve possess a deep geometric interpretation in terms of symmetric c ovariant tensors [GiHoT r2007] – The momen t Lie brack et has b een identiﬁed with the Schouten symmetric br acket on symmetric con trav ariant tensors [GiHoT r2008], as explained in c hap- ter 2. – Chapter 2 deriv es the Euler-Poinc ar ´ e e quations for the moments and c hap- ter 3 illustrates some integrable examples [GiHoT r05, GiHoT r2007] • Results on Hamiltonian ﬂo ws – Chapter 3 sho ws ho w the Benney momen t equations [Be1973] regulate the dynam- ics of c o asting b e ams in particle accelerators [V enturini] and this fact [GiHoT r2007] determines the nature of the c oher ent structur es observed in the exp erimen ts [KoHaLi2001, CoDaHoMa04]. OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 23 – Chapter 2 presen ts how the Camassa-Holm equation [CaHo1993] app ears from the restriction of moment dynamics to c otangent lifts of diﬀe omorphisms . This type of ﬂo w also provides an in terpretation of the b -e quation [HoSt03] in terms of moment dynamics [GiHoT r2007]. – The ge o desic ﬂow on the moments has b een form ulated as a new problem in c hapter 3. It has been shown how this is equiv alen t to a ge o desic Vlasov e quation , that is a ge o desic motion on the symple ctic gr oup of canonical transformations [GiHoT r05, GiHoT r2007]. – Chapter 3 also shows how the CH peakons ma y be in terpreted in terms of sin- gular solutions for the moments , i.e. the single particle solution [GiHoT r05, GiHoT r2007] – The two-c omp onent CH e quation [ChLiZh2005, Ku2007] has been shown to emerge as a particular sp ecialization of the geo desic moment equations [GiHoT r2007] – The geo desic momen t equations ha ve b een extended to include anisotr opic in- ter actions [GiHoT r2007] • Results on dissipativ e ﬂows – Chapter 4 shows how the existence of singular solutions can b e allo wed for a whole class of dissipative equations, called GOP e quations [HoPu2007]. This is ap- plied to recov er the double brac ket form of the vorticity e quation [HoPuT r2007] in chapter 4 and of the Vlasov e quation [HoPuT r2007-CR] in chapter 5. – Chapter 5 applies the double br acket dissip ation to form ulate dissip ative e quations for the moments [HoPuT r2007-CR, HoPuT r2007-Poisson], whose zero-th order truncation recov ers Dar cy’s law for p orous media – Chapter 6 applies the double brac ket dissipation to the GHK–Vlasov equation [GiHoKu1982, GiHoKu1983] and to moment dynamics. The zero-th order trun- cation constitutes a gener alization of Dar cy’s law to anisotr opic inter ac- tions , recov ering Landau-Lifshitz-Gilb ert dynamics for magnetization in ferro- magnetic media [HoPuT r2007-Poisson, HoPuT r08, HoOnT r07]. – Chapter 6 explains ho w this extension of Darcy’s law admits singular solu- tions ( orientons ) and presents analytical results on their b eha vior [HoPuT r08, HoOnT r07]. OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 24 – Smoluchowski moment dynamics is also derived in chapter 6 and particular sp ecializations are presented [HoPuT r2007-P oisson] There are tw o main mathematical ideas b ehind these results. The ﬁrst is that taking the moments is a Poisson map [Gi1981]: this allo ws to transfer from the microscopic kinetic side to the macroscopic contin uum lev el. In particular, this idea is of central importance when deriving ﬂuid–like mo dels from kinetic equations. The clear example is given by the form ulation of the double brack et for the moments: the dissipative momen t dynamics need not to b e determined by direct integration of the Vlasov equation, but rather they can b e constructed by following purely geometric argumen ts in the theory of double brack et dissipation. The second k ey idea is that con tinuum mo dels may allow for singular solutions . In the present theory of double brac ket, these singular solutions are not allow ed and it is not clear a priori ho w a smo othing pro cess can b e inserted in order to admit the singularities. The inspiration for the solution of this problem comes from the GOP the ory of Holm and Putk aradze [HoPu2007], whic h deriv es a class of dissipative equations through a suit- able v ariational principle. Chapter 4 shows that the w ay the smoothing pro cess enters in this v ariational principle determines whether singular solutions exist in the GOP family of equations [HoPu2007, HoPuT r2007], which also include double brack et equations. 0.5 A new mo del for orien ted nano-particles The main result of this w ork is presented in c hapter 6. This result is the form ulation of a con tinuum mo del that generalizes Darcy’s law to oriented nano-particles, starting from ﬁrst principles in kinetic theory . The starting p oint is the double brack et for of the GHK-Vlasov equation [GiHoKu1982, GiHoKu1983] ∂ f ∂ t =  f ,  µ [ f ] , δ E δ f  1  1 with n f , h o 1 := n f , h o + m · ∂ f ∂ m × ∂ h ∂ m where E is the energy functional and µ [ f ] is a smo othed copy of f , i.e. a conv olution µ [ f ] = K ∗ f with some kernel K [HoPuT r2007-P oisson]. Once this equation is introduced, one pro ceeds by considering the leading-order momen ts ( ρ, M ) = Z (1 , m ) f ( q , p , m , t ) d 3 p d 3 m OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 25 so that ρ ( q , t ) is the mass density and M ( q , t ) is the p olarization . A t this p oint, it suﬃces to calculate the dissipative equations for ρ and M , which turn out to b e [HoPuT r08] ∂ ρ ∂ t = div ρ  µ ρ ∇ δ E δ ρ + µ M · ∇ δ E δ M  ! ∂ M ∂ t = div M ⊗  µ ρ ∇ δ E δ ρ + µ M · ∇ δ E δ M  ! + M × µ M × δ E δ M where the last term in the second equation is the dissipative term for magnetization dynamics in ferr omagnetics . Thus the Landau-Lifshitz-Gilbert dissipation [Gilb ert1955] is derived from ﬁrst principles in kinetic the ory and this mo del can also b e applied to systems of ferromagnetic particles. This mo del allows for singular solutions of the form [HoPuT r08] ρ ( q , t ) = N X i =1 Z δ ( q − Q i ( s, t )) d s M ( q , t ) = N X i =1 Z w M ,i ( s, t ) δ ( q − Q i ( s, t )) d s where s is a co ordinate on a submanifold of R 3 : if s is a one-dimensional coordinate, then one gets an orientation ﬁlament , while in tw o dimensions one has an orientation she et . When the problem is studied in only one spatial dime nsion, then the singular solutions tak e the simpler form [HoPuT r2007-Poisson] ρ ( q , t ) = N X i =1 w ρ,i ( t ) δ ( q − Q i ( t )) M ( q , t ) = N X i =1 w M ,i ( t ) δ ( q − Q i ( t )) and w ρ , w M and Q undergo the following dynamics ˙ w ρ,i = 0 , ˙ w M ,i = w M ,i ×  µ M × δ E δ M  q = Q i ˙ Q i = −  µ ρ ∂ ∂ q δ E δ ρ + µ M · ∂ ∂ q δ E δ M  q = Q i so that these singular solutions represen t the dynamics of N particles. Numerical sim ulations sho w that these solutions ma y form sp ontane ously fr om any initial c onﬁgur ation [HoOnT r07]. The study of pairwise interactions in chapter 6 shows that there is a wide class of possible situations where these particles exhibit clumping and alignmen t phenomena [HoOnT r07]. OUTLINE: MOTIV A TIONS, RESUL TS AND PERSPECTIVES 26 0.6 P ersp ectiv es for future w ork Besides its achiev ements, the present study raises many imp ortant questions concerning v arious topics, from purely geometric matters to singularities in double brack et equations. F or example, the result that momen t dynamics is determined b y the symmetric Schouten brac ket could b e used to identify the symmetry group determining the moment Lie-P oisson structure. This would allo w to deﬁne moments as momen tum maps [MaRa99]. The study of the geo desic moment equations generates several open questions. Chap- ter 3 shows ho w this hierarch y reco vers t wo imp ortan t in tegrable equations, the CH equation [CaHo1993] and its tw o-comp onent version [ChLiZh2005, Ku2007]. Th us one may wonder if there exist other truncations of the moment hierarc hy with remark able b ehavior, suc h as in tegrability . The geo desic Vlasov equation presented in ch apter 3 is very similar in con- struction to the Bloch-Iserles system [BlIsMaRa05] (geodesic ﬂow on Hamiltonian matrices) and it w ould b e in teresting to explore this connection further. Also, the dynamics of singu- lar solutions still deserves further in vestigation, esp ecially in higher dimensions (ﬁlaments and sheets). In the case of the CH equation dual p airs [MaW e83] emerge in the analysis of singular solutions [HoMa2004]: is this p ossible for the t w o-comp onent CH equation? and for other truncations of the moment equations? Similar questions concern the singular solutions of the geo desic moment equations for anisotropic in teractions [GiHoT r2007]. The same questions regarding singular solutions and their b ehavior can b e extended to the double brack et moment equations in chapters 5 and 6. In particular, one would w onder how the clumping and alignment phenomena transfer to the case of ﬁlaments and sheets. An imp ortant question is whether these ﬁlaments emerge sp on taneously in tw o or three dimensions. F urther developmen t is needed also for the geometric structure of the Smolucho wski moment equations. An analysis of their closures and study of singular solutions is required. Later, one can hop e to apply this theory to real problems in volving orien ted particles and ferromagnetic materials in nano-science. Chapter 1 Singular solutions in con tin uum dynamics 1.1 In tro duction The use of geometric concepts in contin uum mo dels has highly increased in the last 40 years and mainly related to physical systems which presen t some contin uous symmetry [MaRa99]. Suc h an approach has provided an imp ortant insight into the mathematics of ﬂuid mo dels and has been successfully used for physical modeling and other applications (turbulence [F oHoTi01], imaging [HoRaT rY o2004], numerics [BuIs99], etc.). It has been sho wn that many imp ortant contin uum systems in physics (ﬂuid dynamics [HoMaRa], plasma physics [HoMaRa], elasticity [SiMaKr88], etc.) follo w a purely geometric ﬂo w, uniquely determined b y their total energy and b y their symmetry prop erties. In partic- ular, many geometric ﬂuid mo dels hav e b een widely studied (LAE- α [HoNiPu06], LANS- α [F oHoTi01], etc.) in the last years. One imp ortan t feature that arises in many contin uum systems is the existence of singular measure-v alued solutions. Probably , the most famous example of singular solution in ﬂuids is the p oint vortex solution for the v orticity equation on the plane. These solutions are delta-lik e solutions that follo w a m ulti-particle dynamics. In three dimensions one extends this concept to vortex ﬁlamen ts or v ortex sheets, for which the v orticity is supp orted on a low er dimensional submanifold (1D or 2D resp ectively) of the Euclidean space R 3 . The dynamics of these 27 CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 28 solutions has been widely inv estigated and is still a source of imp ortant results in b oth ﬂuid dynamics and geometry . The existence of these solutions is a result of the nonlo cal nature of the equation describing the dynamics [MaW e83]. Also, these solutions form an inv ariant manifold and they are not exp ected to b e created by ﬂuid motion. Another important example of ﬂuid mo del admitting singular solutions is the Camassa- Holm (or EPDiﬀ ) equation, whic h is an in tegrable equation describing shallo w water wa ves (b esides its applications in other areas suc h as turbulence and imaging). How ever this equation has one more interesting feature, that is the sp ontane ous emergence of singular solutions from any conﬁned initial conﬁguration. The dynamical v ariable is the ﬂuid v elo city and the nature of the singular solutions goes bac k to the tra jectory of the single ﬂuid particle. F or this particular case, the singular solutions also hav e a soliton b ehavior. Singular solution also arise in plasma physics (magnetic vortex lines, cf. e.g. [Ga06]), ki- netic theory (phase space particle tra jectories, cf. e.g. [GiHoT r05, GiHoT r2007]), and other mo dels for aggregation dynamics in friction dominated systems [HoPu2005, HoPu2006]. The latter are dissipative contin uum mo dels, which in volv e a ﬂuid velocity that is prop ortional to the collective force. In some cases these dissipative mo dels exhibit the sp ontaneous for- mation of singularities that clump together in a ﬁnite time. This b ehavior is dominated b y the dissipation of energy and describ es aggregation of particles. These considerations suggest that the properties of singular solutions in con tinuum mod- els deserve further in vestigation. In particular, this work presents geo desic and dissipativ e ﬂo ws that exhibit the sp ontaneous emergence of singularities. These ﬂows are then related to the kinetic description for m ulti-particle systems. The connection from the microscopic kinetic level to the macroscopic level is provided by the kinetic momen ts. How ever, b efore going in to the details of kinetic theory , this chapter reviews the mathematical properties of the ﬂuid equations allowing for singular solutions. 1.2 Basic concepts in geometric mechanics The basic geometric setting for ﬂuid equations is given b y Lagrangian (or Hamiltonian) systems deﬁned on Lie groups and Lie algebras. (This paragraph uses some of the concepts and the notation introduced in [MaRa99].) When a system is inv ariant with resp ect to the Lie group G ov er which it is deﬁned, then it is p ossible to rewrite its equations on the Lie CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 29 algebra g = T e G (or its dual g ∗ = T ∗ e G ) of that group. F or example, if one takes the (right) in v ariant Hamiltonian H = H ( g , p ) : T ∗ G → R , then one writes H ( g g − 1 , pg − 1 ) = H ( e, µ ) = h ( µ ) , µ ∈ T ∗ e G . so that the Hamiltonian h ( µ ) is deﬁned on the dual Lie algebra g ∗ . Analogously , for a (righ t) in v ariant Lagrangian L = L ( g , ˙ g ) : T G → R one writes L ( g g − 1 , ˙ g g − 1 ) = L ( e, ξ ) = l ( ξ ) , ξ ∈ T e G . The present work will mainly consider symmetric contin uous systems whose equations are already written on the Lie algebra of some Lie group. This theory is called Euler-Poinc ar´ e (or Lie-Poisson ) reduction and is extensively presen ted in [MaRa99]. Lie-P oisson and Euler-Poincar ´ e equations. The starting p oint for the presen t analysis is the Lie-Poisson br acket . Deﬁnition 4 A Hamiltonian system is c al le d Lie-Poisson iﬀ it is deﬁne d on the dual of a Lie algebr a g ∗ and the Poisson br acket is given by { F , G } ( µ ) = ±  µ,  δ F δ µ , δ G δ µ  with µ ∈ g ∗ wher e F , G ar e functionals of µ , the notation δ F /δ µ denotes the functional derivative, [ · , · ] is the Lie br acket and h· , ·i denotes the natur al p airing b etwe en a ve ctor sp ac e and its dual. It is imp ortan t to say that the sign in the brack et dep ends only on whether the system is righ t- or left-inv ariant (plus and minus resp ectively). The following sections will explore v arious examples of Lie-P oisson systems both right and left in v arian t. How ev er this section k eeps the plus sign for right-in v ariant systems. The equations arising from this structure are called Lie-Poisson e quations and are written as ∂ µ ∂ t + ad ∗ δ H δ µ µ = 0 (1.1) where the coadjoint operator ad ∗ is deﬁned as the dual of the Lie brack et  ad ∗ η ν, ξ  :=  ν, ad η ξ  =  ν, [ η , ξ ]  with ν ∈ g ∗ and η , ξ ∈ g . CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 30 If the Hamiltonian is suc h that the Legendre transform is inv ertible ( r e gular Hamilto- nian ), then one can introduce the Lagrangian L ( ξ ) in terms of the Lie algebra v ariable ξ ∈ g µ = δ L δ ξ so that the Lagrangian is written as L ( ξ ) = h µ, ξ i − H ( µ ) and the Euler-Lagrange equations are written in the form ∂ ∂ t δ L δ ξ + ad ∗ ξ δ L δ ξ = 0 whic h are called Euler-Poinc ar ´ e equations. This work will mainly consider inﬁnite dimensional Lie groups acting on some manifold M . The most general example is the group of diﬀeomorphisms Diﬀ( M ), whose Lie algebra X ( M ) consists of all p ossible vector ﬁelds on M . The manifold M will b e R n and the Lie brac ket among the vector ﬁelds X and Y is given b y the Jac obi-Lie br acket [ X , Y ] J L = ( X · ∇ ) Y − ( Y · ∇ ) X . As it happens in ordinary ﬁnite-dimensional classical mechanics, b oth Lie-P oisson and Euler-P oincar´ e equations can b e derived from the follo wing v ariational principles Euler-P oincar´ e: δ Z t 2 t 1 L ( ξ ) dt = 0 Lie-P oisson: δ Z t 2 t 1  h µ, ξ i − H ( µ )  dt = 0 for v ariations of the form δ ξ = ˙ η − [ ξ , η ], where η ( t ) is a curv e that v anishes at the end points η ( t 1 ) = η ( t 2 ) = 0. The second of these v ariational principles is called Hamilton-Poinc ar´ e v ariational principle [CeMaP eRa]. Coadjoin t motion. F rom abov e, one can see that Lie-P oisson (or Euler-Poincar ´ e) dy- namics is a strongly geometric t yp e of dynamics. This p oin t is ev en more eviden t, once one writes the solution of the equations as [MaRa99] µ ( t ) = Ad ∗ g − 1 ( t ) µ (0) (1.2) CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 31 where g ( t ) = exp ( t δ H /δ µ ). The op erator Ad ∗ : G × g ∗ 7→ g ∗ denotes the c o adjoint gr oup action on the Lie algebra g and is deﬁned as the dual of the adjoint group action giv en b y Ad g ξ := d dτ     τ =0 g ◦ e τ ξ ◦ g − 1 ∀ g ∈ G, ξ ∈ g , so that h µ, Ad g ξ i =  Ad ∗ g µ, ξ  . Such a motion is called c o adjoint motion and is said to o ccur on c o adjoint orbits , where the coadjoint orbit O ( µ ) of µ ∈ g ∗ is the subset of g ∗ deﬁned by O ( µ ) := G · µ :=  Ad ∗ g − 1 µ : g ∈ G  In order to see how Lie-Poisson equations (1.1) are reco vered from equation (1.2), one tak es pairing of (1.2) with a Lie algebra element η ∈ g as follows h µ ( t ) , η i = D Ad ∗ g − 1 ( t ) µ (0) , η E =  µ (0) , Ad g − 1 ( t ) η  (1.3) where Ad g − 1 ( t ) η = d dτ     τ =0 e − t δH δµ ◦ e τ η ◦ e t δH δµ No w taking the time deriv ative of (1.3) and ev aluating it at the initial condition t = 0 yields h ˙ µ (0) , η i =  µ (0) , d dt Ad exp( − t δH /δµ ) η    t =0  = −  µ (0) , ad δH δµ η  = −  ad ∗ δH δµ µ (0) , η  where the relation (cf. e.g. [MaRa99]) ad ξ η = d dt Ad exp( t ξ ) η    t =0 has been used. Consequently , a system undergoing coadjoint orbits is a Lie-Poisson system. In particular, if the tra jectory of a Lie-Poisson system starts on O , then it sta ys in O [MaRa99]. This kind of motion explains how the geometry of the Lie group generates the dynamics. Lie deriv ativ e of tensor ﬁelds. An imp ortant op erator which is fundamental for the purp oses of the present w ork is the Lie deriv ative. In order to in tro duce this op eration as an inﬁnitesimal generator, one can fo cus on the action of diﬀeomorphism group on set of tensor ﬁelds deﬁned on some manifold Q . Explicitly , the action Φ of a group elemen t g ∈ Diﬀ (a smo oth in vertible c hange of co ordinates g : q 7→ g ( q ) ) on a tensor ﬁeld T ( q ) is given b y Φ( g , T ) = g ∗ T CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 32 where the notation g ∗ indicates the pull-back op eration [MaRa99]. If one considers a one- parameter subgroup, i.e. a curve g ( t ) := g t on the Diﬀ group (such that g 0 = e , where e is the iden tity), then this action transp orts the tensor T along this curv e, according to g ∗ t T . A Lie algebr a action ξ M on a manifold M is deﬁned by the inﬁnitesimal generator. In particular, if M is the space T ( Q ) of tensor ﬁelds on Q the inﬁnitesimal generator is ev aluated on the tensor T as follows ξ T ( Q ) T := d dt     t =0 Φ( g t , T ) = d dt     t =0 g ∗ t T Ho wev er, an element of a one-parameter subgroup can be expressed in terms of a Lie algebra elemen t ξ through the exp onential map g t = e t ξ so that ξ T ( Q ) T = d dt     t =0 g ∗ t T = d dt     t =0 Φ( e t ξ , T ) = d dt     t =0  e t ξ  ∗ T Since the Lie algebra of the Diﬀ group is the space of vector ﬁelds X , the one-parameter subgroup g t is identiﬁed with the ﬂow of the vector ﬁeld ξ ∈ X . Thus ξ T ( Q ) T is the X -Lie algebra action on the space of tensor ﬁelds T ( Q ). A t this p oin t the deﬁnition of the Lie deriv ativ e is simply Deﬁnition 5 (Lie deriv ativ e of a tensor ﬁeld) The Lie derivative of a tensor ﬁeld T ( q ) on some manifold Q along a ve ctor ﬁeld X ( q ) on the same manifold is deﬁne d as the in- ﬁnitesimal gener ator of the gr oup of diﬀe omorphisms acting on Q £ X T := ξ T ( Q ) T = d dt     t =0  e t X  ∗ T . A particular case is provided by the possibility T = Y ∈ X , since now g ∗ Y =: Ad g − 1 Y and th us £ X Y = d dt     t =0 Ad exp( − tX ) Y =: ad − X Y =: [ X , Y ] J L so that the Lie derivative of two ve ctor ﬁelds is given by the Jac obi-Lie br acket . 1.3 Euler equation and v ortex ﬁlamen ts An imp ortant physical system in the context of geometric dynamics is the Lie-P oisson system for the vorticit y of an ideal Euler ﬂuid. As its primary geometric characteristic, Euler’s CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 33 ﬂuid theory represen ts ﬂuid ﬂow as Hamiltonian geo desic motion on the space of smo oth in vertible maps acting on the ﬂo w domain and p ossessing smooth in verses. These smo oth maps (diﬀeomorphisms) act on the ﬂuid reference conﬁguration so as to mov e the ﬂuid particles around in their container. And their smo oth inv erses recall the initial reference conﬁguration (or label) for the ﬂuid particle curren tly occupying an y giv en position in space. Th us, the motion of all the ﬂuid particles in a container is represented as a time-dep endent curv e in the inﬁnite-dimensional group of diﬀeomorphisms. Moreo ver, this curv e describing the sequen tial actions of the diﬀeomorphisms on the ﬂuid domain is a sp ecial optimal curv e that distills the ﬂuid motion into a single statement. Namely , “A ﬂuid mov es to get out of its own wa y as eﬃciently as p ossible.” Put more mathematically , ﬂuid ﬂow o ccurs along a curv e in the diﬀeomorphism group which is a geo desic with resp ect to the metric on its tangen t space supplied by its kinetic energy . F or incompressible ﬂuids, one restricts to diﬀeomorphisms that preserve the volume elemen t and the ﬂuid is describ ed by its vorticit y , which satisﬁes a Lie-Poisson equation. This section reviews some of the results presented in [MaW e83]. In order to understand the Lie-P oisson structure, one introduces Euler’s vorticit y equation as ω t + curl  ω × v  = 0 . (1.4) where the v orticity is deﬁned in terms of the v elo city as ω = curl v . F ollowing [MaW e83], this equation represen ts the advection equation for an exact tw o-form ω = ω · d S appearing as the vorticit y for incompressible motion along the ﬂuid v elo city v and thus it can b e written in terms of Lie deriv ative £ along the v elo city v ector ﬁeld ω t + £ v ω = 0 . (1.5) The Lie-Poisson brac ket for vorticit y is written on the dual X ∗ vol of the Lie-algebra X vol of v olume-preserving diﬀeomorphisms, which is isomorphic to the set of exact one-forms: ω = dα , where α is a generic one-form. In this case the Jacobi-Lie brack et b etw een tw o v olume-preserving v ector ﬁelds ξ 1 and ξ 2 in R 3 ma y b e written as [ ξ 1 , ξ 2 ] J L = − curl ( ξ 1 × ξ 2 ) . In terms of the vector p otentials for whic h ξ 1 = curl ψ 1 and ξ 2 = curl ψ 2 this brack et b ecomes [ ξ 1 , ξ 2 ] J L = − curl  curl ψ 1 × curl ψ 2  . CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 34 The vector p otentials ψ 1 and ψ 2 are deﬁned up to a gradien t of a scalar function so that one can alw a ys choose a gauge in which div ψ = 0. Pairing the vector ﬁeld given b y the Lie brac ket with a one-form (density) α then yields, after an integration b y parts,  α , [ ξ 1 , ξ 2 ] J L  = −  curl α ,  curl ψ 1 × curl ψ 2   = −  ω ,  ψ 1 , ψ 2   where α is deﬁned up to an exact one-form d f and one introduces the notation  ψ 1 , ψ 2  := curl ψ 1 × curl ψ 2 . The brack et [ · , · ] deﬁnes a Lie algebra structure on the space of vector potentials whose dual space may b e naturally iden tiﬁed with exact tw o-forms ω = curl α . A t this p oint, the expression for the Lie-Poisson brac ket for functionals of v orticity ma y b e introduced as { F , H } =  ω ,  δ F δ ω , δ H δ ω  = Z ω ·  curl δ F δ ω × curl δ H δ ω  d 3 x , where H = 1 2 Z ω · ( − ∆) − 1 ω d 3 x = 1 2 Z | u | 2 d 3 x = 1 2 k u k 2 is the ﬂuid’s kinetic energy expressed in terms of vorticit y . No w, vorticit y dynamics is an example of geo desic motion on a (inﬁnite-dimensional) Lie group [Ar1966] ω t = − ad ∗ δ H/δ ω ω = − curl  ω × curl ( − ∆) − 1 ω  = curl  curl ψ × ω  = − £ curl ψ ω . where the ad ∗ is now deﬁned as the dual of the [ [ · , · ] ] Lie brack et and one has ad ∗ ψ ω = £ curl ψ ω . As stated in section 1.1, this equation allows singular solutions in the form of v ortex ﬁlamen ts, distributions of vorticit y supp orted on a curve Q ( s, t ). These are represented by the following expression ω ( x , t ) = Z ∂ Q ( s, t ) ∂ s δ ( x − Q ( s, t )) d s where s is a curvilinear coordinate and ∂ s Q is the tangent to the curve. The dynamics of Q ( s, t ) is presented in the work by Holm and Stechmann [Ho2003, HoSt04]. These solutions are widely studied in man y areas of ﬂuid dynamics as well as in condensed state theory , within the theory of sup erﬂuids [RaRe]. CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 35 All the arguments ab ov e can b e pro jected down onto the plane to obtain the 2D Euler equation ω t +  ω , ( − ∆) − 1 ω  = 0 , (1.6) where {· , ·} denotes the canonical Poisson brack et in the plane co ordinates x, y . This equa- tion also allows for singular solutions, the p oint v ortex solutions moving on the plane. The expression for a p oint v ortex is easily written as ω ( x, y , t ) = δ ( x − X ( t )) δ ( y − Y ( t )) where the dynamics of X and Y is just ordinary Hamiltonian dynamics for the tw o conjugate v ariables, so that p oint v ortices mov e around as if they were particles. This 2D equation is imp ortan t because it is completely equiv alent to the collisionless Boltzmann equation in kinetic theory and thus pro vides a slight in tro duction to the cen tral topic of this work. Singular solutions are allow ed b ecause of a com bination of the form of the equation and the smoothing of the Lie algebra element ψ = ∆ − 1 ω b y the P oisson kernel ∆ − 1 . Indeed, if one chooses a Hamiltonian that is quadratic with resp ect to the Euclidean norm ( H = 1 2 R | ω | 2 d 3 x ), one readily realizes that singular solutions are forbidden by the dynamics. Consequen tly the presence of a smo oth vector p otential is of central imp ortance in the existence of singular solutions. F or example, one could think to mo dify the equations in order to allow for a diﬀerent regularization of the solution, that is changing the norm ∆ − 1 with another k ernel which, p ossibly introduces a regularization length-scale. An example is provided b y the Euler-alpha mo del [HoNiPu06] that introduces a smo othed velocity u = (1 − α 2 ∆) − 1 v , so that up on deﬁning ω = curl u (regularized vorticit y) and q = curl v (singular vorticit y), the Hamiltonian is given b y H = 1 / 2 Z u · v d 3 x = 1 / 2 Z (1 − α 2 ∆) − 1 v · curl − 1 q d 3 x = 1 / 2 Z (1 − α 2 ∆) − 1 curl − 1 q · curl − 1 q d 3 x = 1 / 2 Z q · curl − 1 (1 − α 2 ∆) − 1 curl − 1 q d 3 x = 1 / 2 Z q · ( − ∆) − 1 (1 − α 2 ∆) − 1 q d 3 x where the last step is justiﬁed by the fact that the integral op erators (1 − α 2 ∆) − 1 and curl − 1 comm ute. In this wa y , the motion is again geo desic with resp ect to the singular v orticity q . The previous arguments show how the dynamics of the Euler ﬂuid is giv en by CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 36 a w ell kno w geometric ﬂow, the ge o desic ﬂow on the gr oup of volume pr eserving diﬀe omorphisms [Ar1966, ArKe98]. The next section sho ws that this kind of ﬂo w plays a cen tral role in the theory of singular solutions for contin uous Hamiltonian dynamics. This idea relates the formation of singular solutions to geo desic motion on diﬀerent inﬁnite-dimensional Lie groups, like the group of diﬀeomorphisms or the group of canonical transformations (symplectomorphisms). The latter will b e a central topic in this work. 1.4 The Camassa-Holm and EPDiﬀ equations The Euler equation admits v ortex ﬁlamen ts and these solutions are related to a geo desic ﬂo w on an inﬁnite-dimensional Lie group. How ever for the Euler equations, the singular solutions are an inv ariant submanifold, that is they do not emerge sp on taneously from a smooth initial condition. Now, there are imp ortan t geometric ﬂows that exhibit a sp ontaneous emergence of singularities from any smo oth initial state. One of the most meaningful examples that is also the main inspiration for the present w ork is the Camassa-Holm e quation (CH) [CaHo1993] u t + 2 κu x − u xxt + 3 uu x = 2 u x u xx + uu xxx In particular this work fo cuses on the case when κ = 0 and considers the case when b oundary terms do not con tribute to in tegration b y parts (p erio dic boundary conditions or fast deca y at inﬁnit y). It has been shown [HoMa2004] that this equation is a ge o desic motion on the gr oup of diﬀe omorphisms (EPDiﬀ ). In fact one ﬁnds that this equation can be reco vered from the following Euler-P oincar´ e v ariational principle deﬁned on X ( R ) δ Z t 1 t 0 L ( u ) d t = 0 with L ( u ) = 1 2 Z u (1 − ∂ 2 x ) u d x In this wa y the CH equation is the Euler-Poincar ´ e equation for a purely quadratic La- grangian. Thus the CH equation is again a geo desic ﬂow, which is given b y the geo desic equation on the group of diﬀeomorphisms. It is easy to ﬁnd the Lie-Poisson formulation, via the Legendre transform m = δ L δ u = u − u xx ⇒ u = (1 − ∂ 2 x ) − 1 m where m = m ( x ) dx ⊗ dx ∈ X ∗ ( R ), the space of one-form densities. The Hamiltonian b ecomes H ( m ) = 1 2 Z m (1 − ∂ x ) − 1 m d x CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 37 and the Lie-Poisson equation is m t = − £ u m = − um x − 2 mu x . The main result on this equation is its in tegrability which is guaranteed b y its bi-hamiltonian structure. How ever there is another imp ortant statement, whic h is called the ste ep ening lemma [CaHo1993]: Supp ose the initial pr oﬁle of velo city u ( x, t = 0) has an inﬂe ction p oint at x = ¯ x to the right of its maximum, and otherwise it de c ays to zer o in e ach dir e ction suﬃciently r apid ly for the Hamiltonian H ( m ) to b e ﬁnite. Then the ne gative slop e at the inﬂe ction p oint wil l b e c ome vertic al in ﬁnite time. This fact is shown in ﬁg. 1.4 and is particularly relev an t when one fo cuses on the b ehavior of singular solutions in PDE’s. Moreov er these solutions (called p e akons in the v elo city repre- sen tation) presen t soliton b ehavior and this fact mak es their m utual interaction particularly in teresting. Figure 1.1: Peak ons emerging from Gaussian initial conditions for diﬀerent Gaussian widths w . Figure from [HoSt03]. All the ab o ve can be easily generalized to more dimensions, so that the equation becomes m t − u × curl m + ∇ ( u · m ) + m (div u ) = 0 . and one takes the follo wing Hamiltonian on X ∗ ( R 3 ) H ( m ) = 1 2 Z m · (1 − α 2 ∆) − 1 m d x where one has inserted the length-scale α , that determines the smo othing of the v elo city CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 38 u = (1 − α 2 ∆) − 1 m . The singular solutions ( pulsons ) are written in this representation as m ( x , t ) = X i Z P i ( s, t ) δ ( x − Q i ( s, t )) d s (1.7) where s is a v ariable of dimension k < 3. These solutions represent pulse ﬁlamen ts or sheets, when s has dimension 1 or 2 resp ectively . Another generalization is to tak e another k ernel that deﬁnes the norm of m and substitute (1 − α 2 ∆) − 1 m with the general con volution G ∗ m = R G ( x − x 0 ) m ( x 0 ) d x 0 with s ome Green’s function G . The dynamics of ( Q i , P i ) is given by canonical Hamiltonian dynamics with the Hamiltonian H = 1 2 X i,j Z Z P i ( s, t ) · P j ( s 0 , t ) G ( Q i ( s, t ) − Q j ( s 0 , t )) d s d s 0 . An imp ortant result is the theorem stating that the singular solution (1.7) is a momentum map [HoMa2004]: giv en a Poisson manifold (i.e. a manifold P with a Poisson brac k et {· , ·} deﬁned on the functions F ( P )) and a Lie group G acting on it by Poisson maps (so that the Poisson brac k et is preserved), a momen tum map is deﬁned as a map J : P → g ∗ so that { F ( p ) , h J ( p ) , ξ i} = ξ P [ F ( p )] ∀ F ∈ F ( P ) , ∀ ξ ∈ g where F ( P ) denotes the functions on P and ξ P is the vector ﬁeld given b y the inﬁnitesimal generator ξ P ( p ) = d dt     t =0 e tξ · p ∀ p ∈ P No w, ﬁx a k -dimensional manifold S immersed in R n and consider the embedding Q i : S → R n . Suc h em b eddings form a smo oth manifold and th us one can consider its cotangent bundle ( Q i , P i ) ∈ T ∗ Em b( S, R n ). Now consider Diﬀ( R n ) acting on Emb( S, R n ) on the left by com- p osition ( g Q = g ◦ Q ) and lift this action to T ∗ Em b( S, R n ): this giv es the singular solution momen tum map for EPDiﬀ J : T ∗ Em b( S, R n ) → X ∗ ( R n ) with J ( Q , P ) = Z P ( s, t ) δ ( x − Q ( s, t )) ds . This result is extensiv ely presen ted in [HoMa2004], where diﬀeren t proofs are given in v arious cases. A key fact in this regard is that this momentum map is equiv ariant, which means it is also a Poisson map. This explains why the co ordinates ( Q , P ) undergo Hamiltonian dynamics. CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 39 The EPDiﬀ equation has b een applied in several con texts to turbulence mo deling [F oHoTi01] and imaging tec hniques [HoRaT rY o2004, HoT rY o2007] and its CH form (also with disp er- sion) is widely studied in terms of its integrabilit y prop erties. Again the idea of geo desic ﬂow pla ys a central role in the b ehavior of the pulson so- lutions. This suggests that a further inv estigation of geo desic equations on Lie groups is needed with relation with the emergence of singularities and integrabilit y issues. Chapter 2 considers the group of canonical transformations (through momen t dynamics) and chapter 3 form ulates a geo desic ﬂow on it. The results are encouraging for further inv estigation, since this ﬂow includes the EPDiﬀ equation as a special case and pro vides an extension to its multi-component v ersions (some of which are kno wn to b e in tegrable). Ho wev er, singular solutions do not app ear only in Hamiltonian dynamics. There is another class of systems, which undergo a dissipativ e dynamics with a deep geometrical meaning. In fact chapters 4,5 and 6 will sho w that coadjoin t motion does not necessarily need to b e Hamiltonian. This concept is related to the so called double br acket dissip ation , whic h is extensively analyzed in the second part of this work. In order to introduce how singular solutions arise in dissipativ e con tinuum dynamics, the next section reviews the main ideas by follo wing the presentation in [HoPu2006]. 1.5 Darcy’s la w for aggregation dynamics Man y ph ysical processes can b e understoo d in terms of aggregation of individual comp onents in to a “ﬁnal product”. This phenomenon is recognizable at diﬀeren t scales: from galaxy clus- tering [Chandra60, BiT r88] to particles in nano-sensors [MePuXiBr]. Thus self-aggregation is not necessarily dep endent on the particular kind of interaction. A related paradigm arises in biosciences, particularly in c hemotaxis: the study of the inﬂuence of chemical substances in the environmen t on the motion of mobile sp ecies which secrete these substances. One of the most famous among such mo dels is the Keller-Segel system of partial diﬀeren tial equations [KelSeg1970], which w as introduced to explore the eﬀects of nonlinear cross diﬀusion in the formation of aggregates and patterns b y c hemotaxis in the aggregation of the slime mold Dictyostelium disc oidium . The Keller-Segel (KS) model CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 40 consists of tw o strongly coupled reaction-diﬀusion equations ρ t = div  ρ µ ( ρ ) ∇ Φ[ ρ ] + D ∇ ρ   Φ t + ˆ L Φ = γ ρ . expressing the coupled evolution of the concentration of organism density ( ρ ) and the con- cen tration of chemotactic agen t p otential (Φ). The constants , D , γ > 0 are assumed to b e p ositiv e, and the linear operator ˆ L is taken to be p ositiv e and symmetric. F or example, one ma y c ho ose it to b e the Laplacian ˆ L = ∆ or the Helmholtz op erator ˆ L = 1 − α 2 ∆. Historically , it seems that Deby e and H ¨ uck el in 1923 were the ﬁrst to establish this mo del. They derived the KS evolutionary system in their article [DeHu1923] on the theory of electrolytes. In particular, they present the simpliﬁed model with  = 0. Consequently , the simpliﬁed evolutionary KS system with  = 0 may also b e called the Deb ye-H¨ uc kel equations. Later, the same mo del app eared for mo deling aggregation at diﬀerent scales. Chan- drasekhar form ulated the Smolucho wski-Poisson equation for stellar formation and the “Nernst-Planc k” (NP) equations in the same form as KS re-emerged in the biophysics com- m unity , for example, in the study of ion transport in biological channels [BaChEi]. The same system had also surfaced earlier as the drift-diﬀusion equations in the semiconductor device design literature; see Selb erherr (1984) [Se84]. A v ariant of the KS system re-app eared even more recently as a model of the self-assembly of particles at nano-scales [MePuXiBr]. In order to understand the geometric framework for this kind of equations, one can consider the Deby e-H ¨ ukel system (  = 0) in the limit when the diﬀusion is negligible ( D = 0). This system is a conserv ation equation ρ t = − div ( ρ V ) with V = − µ ( ρ ) ∇ δ E δ ρ (1.8) where V is called Dar cy’s velo city , E ( ρ ) is the energy functional, µ ( ρ ) is called “mobilit y” and in general it dep ends on ρ . The ph ysical meaning of these equations is that when the inertia of the particles is negligible, the p article velo city is pr op ortional to the for c e . This happ ens in particular for friction dominate d systems . Under this approximation, one can interpret this mo del as a sort of unifying principle for aggregation and self-assembly of highly dissipative systems at diﬀerent scales. The fact that the energy is dissipated is CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 41 readily seen by calculating dE dt =  ∂ ρ ∂ t , δ E δ ρ  =  div  ρ µ ∇ δ E δ ρ  , δ E δ ρ  = −   µ ∇ δ E δ ρ  ] ,  ρ ∇ δ E δ ρ   = − Z ρ µ ( ρ )     ∇ δ E δ ρ     2 d n x . so that the energy is monotonically decreasing when the mobility is a p ositive deﬁnite quan tity . The equation (1.8) is called Dar cy’s law and in some cases is also known as the p orous media equation. A t this point one starts discussing the existence of singular solutions. In fact, Holm and Putk aradze [HoPu2005, HoPu2006] hav e shown that in 1D this equation allo ws for solutions of the form ρ ( x, t ) = w ρ δ ( x − q ( x, t )) with ˙ w ρ = 0 , ˙ q = − V ( x, t ) | x = q where V is the Darcy’s velocity in tro duced ab ov e. In particular, Holm and Putk aradze hav e sho wn that, for µ = (1 − α 2 ∂ 2 x ) − 1 ρ and E = 1 2 R ρ  1 − β 2 ∂ 2 x  − 1 ρ d x , this equations p ossess sp on taneously emergent singular solutions from any conﬁned initial distribution, just as it happ ens for EPDiﬀ in the Hamiltonian case. So, again for a purely quadratic energy functional, this system p ossesses singular δ -like solutions, whic h emerge sp ontaneously . In particular, a set of singularities emerge from the initial condition and, after a ﬁnite amount of time, these singularities merge in only one ﬁnal singular solution, as shown in ﬁg. 1.5. This is the reason why these solutions hav e b een named clump ons . This b eha vior is particularly meaningful for physical applications, since the merging pro- cess is directly related to the concept of aggregation and self-assem bly . Thus the emergence of singular solutions in Darcy’s law will con trol their p otential application to self-assembly , esp ecially in nano-science. In a more general mathematical setting, this equation can be extended to an y geometric quan tity as follo ws. As a ﬁrst step, one sees that the equation for ρ is an adv ection relation of the type ρ t + £ V ρ = 0, so that the Darcy’s velocity V acts on the densit y ρ as a v ector ﬁeld, as it happ ens for ordinary ﬂuid dynamics. Now take the follo wing pairing with a function φ  £ V ρ, φ  := h div( ρV ) , φ i = − h ρ ∇ φ, V i =: h ρ  φ, V i CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 42 Figure 1.2: Singularities emerging from a Gaussian initial condition. It is shown ho w these singularities merge together after their formation. This ﬁgure is taken from [HoPu2005, HoPu2006]. where one introduces the diamond op eration  : ( ρ, φ ) 7→ ρ ∇ φ ∈ X ∗ ( R ) which is understo o d as the “dual” of the Lie deriv ativ e. If now Darcy’s velocity is written as V = ( µ  δ E /δ ρ ) ] , then Darcy’s law becomes written in the more abstract wa y ρ t + £ ( µ  δE δρ ) ] ρ = 0 This enables one to extend Darcy’s la w to an y geometric order parameter (GOP). Indeed, giv en a tensor κ , one can write the GOP equation κ t + £ ( µ  δE δκ ) ] κ = 0 whic h is the generalization of the ordinary Darcy’s la w for the density ρ . Holm, Putk aradze and the author [HoPuT r2007] hav e sho wn how these equations alwa ys admit singular δ - lik e solutions for an y geometric quantit y , when the mobilit y is taken as a ﬁltered quan tity µ = K ∗ κ , through some ﬁlter K . The reason is that whenever Darcy’s velocity is smo oth, then the adv ection equation admits the single particle solution. The tra jectory of the single particle has an imp ortant geometric meaning, since it reﬂects the geometry underlying the macroscopic contin uum description. Chapter 4 will sho w how this equations are reco vered b y a symmetric dissipativ e brack et and its geometric prop erties will b e connected with Riemannian manifolds. CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 43 A considerable part of this work is devoted to formulate a microscopic kinetic theory that reco vers Darcy’s la w at the macroscopic ﬂuid level. This pro cess again inv olves the theory of kinetic momen ts (introduced in the next section) as a crucial step in deriving ﬂuid equations. Chapter 6 extends this treatmen t to particles with anisotropic in teractions. Again from a suitable kinetic theory , it is p ossible to derive macroscopic equations that extend Darcy’s law to orien ted particles. The next section presents a sligh t introduction to kinetic momen ts and sho ws how ﬂuid dynamics is recov ered from a truncation of the whole momen t hierarc hy . 1.6 The Vlaso v equation in kinetic theory The ev olution of N identical particles in phase space T ∗ M with co ordinates ( q i , p i ) i = 1 , 2 , . . . , N , may b e described b y an ev olution equation for their joint probability distribution function. In tegrating ov er all but one of the particle phase-space co ordinates yields an evolu- tion equation for the single-particle probabilit y distribution function (PDF) [MaMoW e1984]. This is the Vlaso v equation, which ma y b e expressed as an adv ection equation for the phase- space density f along the Hamiltonian vector ﬁeld X h corresp onding to single-particle mo- tion with Hamiltonian h ( q , p ): f t =  f , h  = − div ( q ,p )  f X h  = − £ X h f with X h ( q , p ) =  ∂ h ∂ p , − ∂ h ∂ q  = ∂ h ∂ p ∂ ∂ q − ∂ h ∂ q ∂ ∂ p The solutions of the Vlasov equation reﬂect its heritage in particle dynamics, which ma y b e reclaimed by writing its many-particle PDF as a product of delta functions in phase space f ( q , p, t ) = X j δ ( q − Q j ( t )) δ ( p − P j ( t )) . An y n umber of these delta functions ma y be in tegrated out un til all that remains is the dynamics of a single particle in the collective ﬁeld of the others. In the mean-ﬁeld approximation of plasma dynamics, this collective ﬁeld generates the total electromagnetic properties and the self-consisten t equations ob ey ed by the single parti- cle PDF are the Vlasov-Maxw ell equations. In the electrostatic approximation, these reduce CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 44 to the Vlasov-P oisson (VP) equations, which gov ern the statistical distributions of particle systems ranging from charged-particle b eams [V enturini], to the distribution of stars in a galaxy [Ka1991]. Remark 6 A class of singular solutions of the VP e quations c al le d the “c old plasma” so- lutions have a p articularly b e autiful exp erimental r e alization in the Malmb er g-Penning tr ap. In this exp eriment, the time aver age of the vertic al motion closely p ar al lels the Euler ﬂuid e quations. In fact, the c old plasma singular Vlasov-Poisson solution turns out to ob ey the e quations of p oint-vortex dynamics in an inc ompr essible ide al ﬂow. This c oincidenc e al lows the discr ete arr ays of “vortex crystals” envisione d by J. J. Thomson for ﬂuid vortic es to b e r e alize d exp erimental ly as solutions of the Vlasov-Poisson e quations. F or a survey of these exp erimental c old-plasma r esults se e [DuON1999]. The Vlasov equation is a Lie-P oisson system that may be expressed as ∂ f ∂ t = −  f , δ H δ f  = ∂ f ∂ p ∂ ∂ q δ H δ f − ∂ f ∂ q ∂ ∂ p δ H δ f =: − ad ∗ δ H/δ f f (1.9) Here the canonical P oisson brack et { · , · } is deﬁned for smo oth functions on phase space with coordinates ( q , p ). The v ariational deriv ative δ H/δ f regulates the particle motion and the quantit y ad ∗ δ h/δf f is explained as follows. A functional G ( f ) of the Vlasov distribution f evolv es according to dG dt = Z Z δ G δ f ∂ f ∂ t d q d p = − Z Z δ G δ f  f , δ H δ f  d q d p = Z Z f  δ G δ f , δ H δ f  d q d p =:   f ,  δ G δ f , δ H δ f   =:  G , H  (1.10) where one denotes with {· , ·} b oth the canonical and the non-canonical Poisson brac kets. In this calculation b oundary terms were neglected up on integrating b y parts in the third step and the notation h h · , · i i is introduced for the L 2 pairing in phase space. The quantit y { G , H } deﬁned in terms of this pairing is the Lie-Poisson Vlasov (LPV) brac ket [W eMo]. This Hamiltonian evolution equation ma y also b e expressed as dG dt =  G , H  = −   f , ad δ H/δ f δ H δ f   =: −   ad ∗ δ H/δ f f , δ G δ f   (1.11) whic h deﬁnes the Lie-algebraic op erations ad and ad ∗ in this case in terms of the L 2 pairing on phase space h h · , · i i : s ∗ × s 7→ R . The notation ad ∗ δ H/δ f f expresses c o adjoint action of δ H /δ f ∈ s on f ∈ s ∗ , where s is the Lie algebra of single particle Hamiltonian vec- tor ﬁelds and s ∗ is its dual under L 2 pairing in phase space. This is the sense in which CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 45 the Vlaso v equation represen ts coadjoin t motion on the gr oup of symple ctic diﬀe omor- phisms (symplectomorphisms). In order to give an explicit deriv ation of the LPV structure from the Jacobi-Lie brack et for Hamiltonian vector ﬁelds (here denoted by X can ), one can follo w the same steps as in section 1.3 for the volume preserving v ector ﬁelds X vol and use the following relation  X h , X k  J L = − X { h, k } = − Ω ] d { h, k } where Ω = Ω ij d q i ∧ d p j is the symplectic form and Ω ] = (Ω ] ) ij ∂ q i ∧ ∂ p j is its inv erse. In what follo ws we will consider canonical transformations on the cotangent bundle T ∗ R n , so that Ω ij = J ij = (Ω ] ) ij , where J is the symplectic matrix. Thus, pairing the result with a one-form density Y ∈ X ∗ can and integrating b y parts yields D Y ,  X h , X k  J L E = − D Y , X { h, k } E = − D Y , J ∇{ h, k } E = D ∇{ h, k } , J Y E = −  div  J Y  , { h, k }  =: − D f ,  h, k  E where f := div  J Y  ∈ F ∗ is evidently a density v ariable dual to the space F of functions. Thus, not only do es one iden tify any Hamiltonian function h with its asso ciated vector ﬁeld X h , but also one asso- ciates a density v ariable f with a one-form density Y = Y f , which is deﬁned mo dulo exact one-forms. Finally one chec ks the isomorphism X can ' F , so that h Y f , X h i = h f , h i . In order to av oid confusion, one denotes the Lie algebra of the symplectic group simply by s . In higher dimensions, particularly n = 3, one may tak e the direct sum of the Vlaso v Lie- P oisson brack et, together with with the Poisson brac ket for an electromagnetic ﬁeld (in the Coulom b gauge) where the electric ﬁeld E and magnetic vector potential A are canonically conjugate. F or discussions of the Vlaso v-Maxwell equations from a geometric viewp oin t in the same spirit as the present approac h, see [W eMo, MaW e83, Ma82, MaW eRaScSp, CeHoHoMa1998]. The Vlasov Lie-P oisson structure was also extended to include Y ang- Mills theories in [GiHoKu1982] and [GiHoKu1983]. In statistical theories such as kinetic theory , the introduction of statistic al moments is a usual to ol for extracting useful information from the probability distribution. It is in teresting to see how the dynamics of moments is also a kind of Lie P oisson dynamics. First, consider moments of the form g b m,m ( t ) = Z Z q b m p m f ( q , p, t ) d q d p . (1.12) CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 46 These moments g b m,m are often used in treating the collisionless dynamics of plasmas and particle b eams [Ch83, Ch90, Dragt, DrNeRa92, Ly95, LyP a97]. This is usually done by considering low-order truncations of the p oten tially inﬁnite sum o ver phase space moments, G ( t ) = ∞ X b m,m =0 a b mm g b m,m , H ( t ) = ∞ X b n,n =0 b b nn g b n,n , (1.13) with b m, m, b n, n = 0 , 1 , . . . . If H is the Hamiltonian, the sum ov er moments evolv es under the Vlasov dynamics according to the Lie-P oisson brac ket relation dG dt = { G , H } = ∞ X b m,m, b n,n =0  ∂ G ∂ g b m,m  b m m − b n n  ∂ H ∂ g b n,n  g b m + b n − 1 , m + n − 1 =: ∞ X b m,m, b n,n =0  g b m + b n − 1 , m + n − 1 ,  ∂ G ∂ g b m,m , ∂ H ∂ g b n,n  , (1.14) where the Lie brack et  a b mm , b b nn  := a b mm  b m m − b n n  b b nn has b een deﬁned. Consequently , the P oisson brack et among the moments is giv en by [Ch90, Ch95, LyPa97, ScW e] { g b m,m , g b n,n } =  b m m − b n n  g b m + b n − 1 , m + n − 1 and the moment equations are written as dg b m,m dt = { g b m,m , H } = ∞ X b n,n =0 ad ∗ ∂ H ∂ g b n,n g b m + b n − 1 , m + n − 1 = ∞ X b n,n =0  b m m − b n n  ∂ H ∂ g b n,n g b m + b n − 1 , m + n − 1 where the inﬁnitesimal coadjoint action ad ∗ has b een deﬁned as usual ∞ X b m,m, b n,n =0  g b n,n ,  a b mm , b b n − b m +1 , n − m +1  = ∞ X b m,m, b n,n =0  ad ∗ a c mm g b n,n , b b n − b m +1 , n − m +1  so that ad ∗ a c mm g b n,n =  b m  b n − b m + 1  − m  n − m + 1   a b mm g b n,n =  b m b n − m n  a b mm g b n,n −  b m 2 − m 2  a b mm g b n,n +  b m − m  a b mm g b n,n . The symplectic inv ariants asso ciated with Hamiltonian ﬂows of these moments [LyOv88, DrNeRa92] were disco vered and classiﬁed in [HoLySc1990]. Finite dimensional appro xi- mations of the whole momen t hierarch y were discussed in [ScW e, Ch95]. F or discussions CHAPTER 1. SINGULAR SOLUTIONS IN CONTINUUM DYNAMICS 47 of the Lie-algebraic approach to the control and steering of charged particle b eams, see [Dragt, Ch83, Ch90]. Other than the statistical moments, also the kinetic moments can b e introduced as pro jection integrals of the PDF o ver the momen tum co ordinates only . In particular, in one dimension one deﬁnes the n -th moment as A n ( q , t ) = Z + ∞ - ∞ p n f ( q , p, t ) d p . Kinetic moments arise as imp ortant v ariables not only in kinetic theory , but also in the theory of integrable shallo w w ater equations [Be1973, Gi1981]. The zero-th kinetic moment is the spatial mass density of particles as a function of space and time. The ﬁrst kinetic momen t is the mean ﬂuid momentum densit y . The next c hapter shows ho w kinetic moment equations are also Lie-Poisson equations and in vestigates the geometric meaning of these quan tities. Connections are established with w ell kno wn in tegrable systems in the con text of shallo w w ater theory and plasma dynamics. Later, a ge o desic motion on the momen ts is constructed which generalizes the Camassa-Holm equation and its multi-component v ersions, recov ering singular solutions. Chapter 2 Dynamics of kinetic momen ts 2.1 In tro duction This chapter reviews the moment Lie-Poisson dynamics in the Kup ershmidt-Manin form [KuMa1978, Ku1987, Ku2005] and pro vides a new geometric interpretation of the moments, whic h shows how the Lie-Poisson brack et is determined by the Sc houten symmetric brac ket on contra v ariant symmetric tensors [GiHoT r2008]. New v ariational formulations of momen t dynamics are pro vided and the Euler-Poincar ´ e momen t equations are formulated as a result. This chapter also considers the action of cotangen t lifts of diﬀeomorphisms on the mo- men ts. The resulting geometric dynamics of the Vlasov kinetic moments possesses singular solutions. These equations turn out to be related to the so called b -hier ar chy [HoSt03] ex- hibiting the sp ontaneous emergence of singularities. Moreov er, when the kinetic momen t equations are closed at the lev el of the ﬁrst-order momen t, their singular solutions are found to recov er the p eaked soliton of the in tegrable Camassa-Holm (CH) equation for shallow w ater w av es [CaHo1993]. These singular Vlasov moment solutions ma y also corresp ond to individual particle motion. The same treatmen t is extended to include the dynamics of the zero-th moment, reco v ering the geometric structures of ﬂuid dynamics [MaRa99]. 48 CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 49 2.2 Momen t Lie-P oisson dynamics 2.2.1 Review of the one dimensional case One of the most remark able features of moment dynamics is that the Lie-P oisson dynamics is inherited from the Vlasov equation [Gi1981]. That is, the evolution of the moments of the Vlasov PDF is also a form of Lie-Poisson dynamics. This fact has b een used also in Y ang-Mills theories b y Gibb ons, Holm and Kup ershimdt [GiHoKu1982, GiHoKu1983]. In order to show wh y this happ ens one considers functionals deﬁned by , G = ∞ X m =0 Z Z α m ( q ) p m f d q d p = ∞ X m =0 Z α m ( q ) A m ( q ) d q =: ∞ X m =0 D A m , α m E , H = ∞ X n =0 Z Z β n ( q ) p n f d q d p = ∞ X n =0 Z β n ( q ) A n ( q ) d q =: ∞ X n =0 D A n , β n E , where h · , · i is the L 2 pairing on p osition space. The functions α m and β n with m, n = 0 , 1 , . . . are assumed to b e suitably smooth and in tegrable against the Vlasov momen ts. T o ensure these prop erties, one may relate the momen ts to the previous sums of Vlasov statistical moments b y choosing α m ( q ) = ∞ X b m =0 a b mm q b m and β n ( q ) = ∞ X b n =0 b b nn q b n . (2.1) F or these choices of α m ( q ) and β n ( q ), the sums of kinetic moments will recov er the full set of Vlasov statistical moments. Thus, as long as the statistical moments of the distribution f ( q , p ) contin ue to exist under the Vlasov ev olution, one ma y assume that the dual v ariables α m ( q ) and β n ( q ) are smo oth functions whose T aylor series expands the kinetic momen ts in the statistical momen ts. These functions are dual to the kinetic momen ts A m ( q ) with m = 0 , 1 , . . . under the L 2 pairing h· , ·i in the spatial v ariable q . In what follows one again assumes b oundary conditions giving zero contribution under integration b y parts. This means, for example, that one can ignore b oundary terms arising from integrations by parts. In what follows the term “momen t” means kinetic moment, unless otherwise speciﬁed. The Poisson brack et among the functionals G = h A m , α m i and H = h A n , β n i (sum- mation ov er m, n ) is obtained from the Lie-Poisson brac k et for the Vlasov equation via the CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 50 follo wing explicit calculation, { G , H } = ∞ X m,n =0 Z Z f h α m ( q ) p m , β n ( q ) p n i d q d p = ∞ X m,n =0 Z Z h nβ n α 0 m − mα m β 0 n i f p m + n − 1 d q d p = ∞ X m,n =0 Z A m + n − 1 ( q ) h nβ n α 0 m − mα m β 0 n i d q =: ∞ X m,n =0 D A m + n − 1 , ad α m β n E = − ∞ X m,n =0 Z h nβ n A 0 m + n − 1 + ( m + n ) A m + n − 1 β 0 n i α m d q =: − ∞ X m,n =0 D ad ∗ β n A m + n − 1 , α m E where one in tegrates b y parts assuming homogeneous b oundary conditions and introduces the notation ad and ad ∗ for adjoint and c oadjoin t action, resp ectively . Up on recalling the dual relations α m = δ G δ A m and β n = δ H δ A n the LPV brack et in terms of the momen ts may b e expressed as { G , H } ( { A } ) = ∞ X m,n =0 Z A m + n − 1  n δ H δ A n ∂ ∂ q δ G δ A m − m δ G δ A m ∂ ∂ q δ H δ A n  d q =: ∞ X m,n =0  A m + n − 1 ,  δ G δ A m , δ H δ A n  (2.2) where one introduces the compact notation { A } := { A n } with n a non-negativ e in teger. This is the Kupershmidt-Manin Lie-P oisson (KMLP) brac ket [KuMa1978], whic h is deﬁned for functions on the dual of the Lie algebra with brack et [ α m , β n ] = nβ n ∂ q α m − mα m ∂ q β n . (2.3) This Lie algebra brack et inherits the Jacobi identit y from its deﬁnition in terms of the canonical Hamiltonian vector ﬁelds. Also, for n = m = 1 this Lie brack et reduces to minus the Jacobi-Lie brack et for the v ector ﬁelds α 1 and β 1 . Thus, one has recov ered the follo wing Theorem (Gibbons [Gi1981]) The op er ation of taking kinetic moments of Vlasov solutions is a Poisson map. It takes CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 51 the LPV br acket describing the evolution of f ( q, p ) into the KMLP br acket, describing the evolution of the kinetic moments A n ( x ) . A result related to this, for the Benney hierarc hy [Be1973], w as also presented by Leb edev and Manin [Le1979, LeMa]. Although the moment brack et is a Lie-Poisson brack et, strictly sp eaking the solutions for the moments cannot y et be claimed to undergo coadjoint motion, as in the case of the Vlasov PDF solutions, b ecause the group action underlying the Lie- P oisson structure of the moments is not y et understo o d and thus the Ad ∗ group op eration is not deﬁned. F or example, it is not p ossible to express the Ad ∗ op eration on the moments b y simply starting from the coadjoint motion on the PDF, as sho wn b y the follo wing calculation: h A n ( t ) , β n i = h h f ( t ) , p n β n i i =   Ad ∗ g − 1 f (0) , p n β n   =  Z p n Ad ∗ g − 1 f (0) d p, β n  so that A n ( q , t ) = Z p n Ad ∗ g − 1 f (0) d p = Z p n  f (0) ◦ g − 1 ( q , p )  d p and the righ t hand side cannot b e expressed as an ev olution map for the sequence of momen ts { A n } . The evolution of a particular momen t A m ( q , t ) is obtained from the KMLP brac ket b y ∂ A m ∂ t = − ad ∗ δ H δ A n A m + n − 1 = { A m , H } = − ∞ X n =0  n ∂ ∂ q A m + n − 1 + mA m + n − 1 ∂ ∂ q  δ H δ A n (2.4) The KMLP brack et among the moments is giv en by { A m , A n } = − n ∂ ∂ q A m + n − 1 − mA m + n − 1 ∂ ∂ q expressed as a diﬀeren tial op erator acting to the right. This op eration is skew-symmetric under the L 2 pairing and the general KMLP brack et can then b e written as [Gi1981] { G , H } ( { A } ) = ∞ X m,n =0 Z δ G δ A m { A m , A n } δ H δ A n d q so that ∂ A m ∂ t = ∞ X n =0 { A m , A n } δ H δ A n . CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 52 Remark 7 The moments have an imp ortant ge ometric interpr etation, which has never ap- p e ar e d in the liter atur e so far. Inde e d one c an write the moments as A n = Z p ⊗ n ( p dq ) f ( q, p ) dq ∧ dp = A n ( q ) ⊗ n d q ⊗ d V ol (2.5) wher e ⊗ n d q := d q ⊗ · · · ⊗ d q n times and d V ol is the volume element in physic al sp ac e. Thus, moments A n b elong to the ve ctor sp ac e dual to the c ontr avariant tensors of the typ e β n = β n ( q ) ⊗ n ∂ q . These tensors ar e given a Lie algebr a structur e by the Lie br acket [ α m , β n ] =  n β n ( q ) α 0 m ( q ) − m α m ( q ) β 0 n ( q )  ⊗ n + m − 1 ∂ q =: ad α m β n (2.6) so that the ad ∗ op er ator is deﬁne d by h ad ∗ β n A k , α k − n +1 i := h A k , ad β n α k − n +1 i and is given explicitly as ad ∗ β n A k =  n β n ∂ A k ∂ q + ( k + 1) A k ∂ β n ∂ q  ⊗ k − n +1 d q ⊗ d V ol . (2.7) The equations for the ideal compressible ﬂuid are recov ered by the momen t hierarch y b y simply truncating at the ﬁrst order moment. In fact the moment equations b ecome in this case ∂ A 0 ∂ t = − ad ∗ β 1 A 0 = ∂ ∂ q  A 0 δ H δ A 1  ∂ A 1 ∂ t = − ad ∗ β 1 A 1 − ad ∗ β 0 A 0 = − δ H δ A 1 ∂ A 1 ∂ q − 2 A 1 ∂ ∂ q δ H δ A 1 − A 0 ∂ ∂ q δ H δ A 0 . whic h are the equations for ideal compressible ﬂuids when the Hamiltonian is written as H = 1 2 R A 2 1 / A 0 d x , so that the ﬂuid velocity is u = δ H /δ A 1 = A 1 / A 0 . Giv en the b eauty and utilit y of the solution b ehavior for ﬂuid equations for the ﬁrst momen ts, one is intrigued to know more ab out the dynamics of the other momen ts of Vlaso v’s equation. Of course, the dynamics of the moments of the Vlasov-P oisson equation is one of the mainstream sub jects of plasma physics and space ph ysics, which are the main inspiration and motiv ation for the present w ork. 2.2.2 Multidimensional treatment I: bac kground One can sho w that the KMLP brack et and the equations of motion ma y b e written in three dimensions in multi-index notation. By writing p 2 n +1 = p 2 n p , and chec king that: p 2 n = X i + j + k = n n ! i ! j ! k ! p 2 i 1 p 2 j 2 p 2 k 3 CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 53 it is easy to see that the multidimensional treatment can b e p erformed in terms of the quan tities p σ =: p σ 1 1 p σ 2 2 p σ 3 3 where σ = ( σ 1 , σ 2 , σ 3 ) ∈ N 3 . Let A σ b e deﬁned as [Ku1987, Ku2005] A σ ( q , t ) =: Z p σ f ( q , p , t ) d 3 p and consider functionals of the form G = X σ Z Z α σ ( q ) p σ f ( q , p , t ) d 3 q d 3 p =: X σ ∈ N 3 h A σ , α σ i H = X ρ Z Z β ρ ( q ) p ρ f ( q , p , t ) d 3 q d 3 p =: X ρ ∈ N 3 h A ρ , β ρ i The ordinary LPV brack et leads to: { G, H } = X σ,ρ Z Z f [ α σ ( q ) p σ , β ρ ( q ) p ρ ] d 3 q d 3 p = = − X σ,ρ X j Z Z f  α σ p ρ ∂ p σ ∂ p j ∂ β ρ ∂ q j − β ρ p σ ∂ p ρ ∂ p j ∂ α σ ∂ q j  d 3 q d 3 p = = − X σ,ρ X j Z Z f  σ j α σ p ρ p σ − 1 j ∂ β ρ ∂ q j − ρ j β ρ p σ p ρ − 1 j ∂ α σ ∂ q j  d 3 q d 3 p = = − X σ,ρ X j Z A σ + ρ − 1 j  σ j α σ ∂ β ρ ∂ q j − ρ j β ρ ∂ α σ ∂ q j  d 3 q = =: − X σ,ρ X j  A σ + ρ − 1 j ,  ad β ρ  j α σ  = = − X σ,ρ X j Z  ρ j β ρ ∂ ∂ q j A σ + ρ − 1 j + ( σ j + ρ j ) A σ + ρ − 1 j ∂ β ρ ∂ q j  α σ d 3 q = =: − X σ,ρ X j   ad ∗ β ρ  j A σ + ρ − 1 j , α σ  where the sum is extended to all σ, ρ ∈ N 3 and one introduces the notation, 1 j =: (0 ,... , 1 , ↑ z }| { j th elemen t ..., 0) so that (1 j ) i = δ j i . CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 54 The LPV brack et in terms of the momen ts may then b e written as ∂ A σ ∂ t = − X ρ ∈ N 3 X j  ad ∗ δh δA ρ  j A σ + ρ +1 j where the Lie brack et is now expressed as  δ G δ A σ , δ H δ A ρ  j = ρ j δ H δ A ρ ∂ ∂ q j δ G δ A σ − σ j δ G δ A σ ∂ ∂ q j δ H δ A ρ . Prop erties of the multidimensional moment brack et. The ev olution of a particular momen t A σ is obtained by ∂ A σ ∂ t = { A σ , h } = = − X ρ X j  ρ j δ h δ A ρ ∂ ∂ q j A σ + ρ − 1 j + ( σ j + ρ j ) A σ + ρ − 1 j ∂ ∂ q j δ h δ A ρ  and the KMLP br acket among moments is given b y { A σ , A ρ } = − X j  σ j ∂ ∂ q j A σ + ρ − 1 j + ρ j A σ + ρ − 1 j ∂ ∂ q j  . Inserting the previous op erator in this multi-dimensional KMLP brack et leads to { g , h } ( { A } ) = X σ,ρ Z δ g δ A σ { A σ , A ρ } δ h δ A ρ d 3 q and the corresp onding evolution equation becomes ∂ A σ ∂ t = X ρ { A σ , A ρ } δ h δ A ρ . Th us, in multi-index notation, the form of the Hamiltonian ev olution under the KMLP brac ket is essen tially unchange d in going to higher dimensions. 2.2.3 Multidimensional treatment I I: a new result Besides the multi-index notation, it is also possible to extend the discussion in remark 7 so to emphasize the tensor in terpretation of the moments. Indeed, one can deﬁne the momen ts as A n ( q , t ) = Z T ∗ q Q ⊗ n ( p · d q ) f ( q , p , t ) d 3 q ∧ d 3 p (2.8) CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 55 whic h can b e written in tensor notation as [GiHoT r2008] A n ( q , t ) = Z T ∗ q Q  p i d q i  n f ( q , p , t ) d 3 q ∧ d 3 p = Z T ∗ q Q p i 1 . . . p i n d q i 1 . . . d q i n f ( q , p , t ) d 3 q ∧ d 3 p = ( A n ( q , t )) i 1 ...i n d q i 1 . . . d q i n d 3 q This in terpretation is consisten t with the momen t Lie-P oisson brac ket. In fact one can follo w the same steps { G , H } = Z Z f h α m ( q ) ⊗ m p , β n ( q ) ⊗ n p i d 3 q ∧ d 3 p = Z Z f  p i 1 . . . p i m ∂ ( α m ) i 1 ,...,i m ∂ q k ∂ p j 1 . . . p j n ∂ p k ( β n ) j 1 ,...,j n − p j 1 . . . p j n ∂ ( β n ) j 1 ,...,j n ∂ q h ∂ p i 1 . . . p i m ∂ p h ( α m ) i 1 ,...,i m  d 3 q ∧ d 3 p = Z Z f  n p i 1 . . . p i m p j 1 . . . p j n − 1 ( β n ) j 1 ,..., j n − 1 , k ∂ ( α m ) i 1 ,...,i m ∂ q k − m p j 1 . . . p j n p i 1 . . . p i m − 1 ( α m ) i 1 ,..., i m − 1 , h ∂ ( β n ) j 1 ,...,j n ∂ q h  d 3 q ∧ d 3 p = Z Z f p j 1 . . . p j m + n − 1  n ( β n ) j m +1 ,..., j m + n − 1 , k ∂ ( α m ) j 1 ,...,j m ∂ q k − m ( α m ) j n +1 ,..., j m + n − 1 , h ∂ ( β n ) j 1 ,...,j n ∂ q h  d 3 q ∧ d 3 p = ∞ X m,n =0  A m + n − 1 , h n ( β n ∇ ) α m − m ( α m ∇ ) β n i  := ∞ X m,n =0 D A m + n − 1 , ad α m β n E where the notation β n ∇ stands for contraction of indexes β n ∇ = ( β n ) i 1 ,...,i n ∂ i n and the square brac ket in the p enultimate step identi ﬁes a Lie brack et ad α m β n , also known as symmetric Schouten br acket [BlAs79, Ki82, DuMi95] (see remark b elow). The last expression is the Lie-Poisson brac ket on the momen ts in terms of symmetric tensors. CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 56 One also chec ks that { G, H } = ∞ X n,m =0 D A m + n − 1 ,  α m , β n  E = − Z Z p j 1 . . . p j m + n − 1  n ( α m ) j 1 ,...,j m ∂ ∂ q k  f ( β n ) j m +1 ,..., j m + n − 1 , k  + m f ( α m ) j n +1 ,..., j m + n − 1 , h ∂ ( β n ) j 1 ,...,j n ∂ q h  d 3 q ∧ d 3 p = − Z Z p j 1 . . . p j m + n − 1  n ( α m ) j 1 ,...,j m ( β n ) j m +1 ,..., j m + n − 1 , k ∂ f ∂ q k + n f ( α m ) j 1 ,...,j m ∂ ( β n ) j m +1 ,..., j m + n − 1 , k ∂ q k + m f ( α m ) j n +1 ,..., j m + n − 1 , h ∂ ( β n ) j 1 ,...,j n ∂ q h  d 3 q ∧ d 3 p = − ∞ X m,n =0  n D ( β n ∇ ) A m + n − 1 , α m E + n D ( ∇ β n ) A m + n − 1 , α m E + m D A m + n − 1 ∇ β n , α m E  =: − ∞ X m,n =0 D ad ∗ β n A m + n − 1 , α m E Consequen tly , w e ha ve pro ven the follo wing Prop osition 8 ([GiHoT r2008]) The tensor interpr etation of the moments (2.8) le ads to a Lie-Poisson structur e, which involves a Lie br acket that gener alizes the Jac obi- Lie br acket to symmetric contra v arian t tensors . This Lie br acket is c al le d symmetric Sc houten brack et and the c orrsp onding Lie-Poisson structur e is given by { F , G } = ∞ X n,m =0  A m + n − 1 ,  n  δ E δ A n ∇  δ F δ A m − m  δ F δ A m ∇  δ E δ A n  In particular, all the considerations made for the one-dimensional case are v alid also in the tensor interpretation of the higher dimensional treatment. The tensor equation for the n -th moment will then in volv e 3 n comp onen ts, which are symmetric so that the num b er of equations for each momen t ma y b e appropriately reduced to 1 / 2 · ( n + 2)! /n ! = ( n + 2)( n + 1) / 2. An interesting example is given by A 2 , which is the pressure tensor such that T r( A 2 )/2 is the density of kinetic energy . Ho wev er, giv en CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 57 the level of diﬃcult y of this problem, the follo wing discussion will mainly restrict to the one-dimensional case. Remark 9 (The symmetric Sc houten brack et) The tensor interpr etation of the mo- ments pr ovides a dir e ct identiﬁc ation b etwe en the moment Lie br acket and the so c al le d symmetric Schouten br acket (or c onc omitant). This br acket was known to Schouten as an invariant diﬀer ential op er ator and its r elation with the p olynomial algebr a of the phase-sp ac e functions is very wel l known. The symmetric Schouten br acket has b e en obje ct of some studies in diﬀer ential ge ometry [BlAs79, DuMi95] and its c onne ction with the Lie-Poisson dynamics for the Vlasov moments has never b e en establishe d so far. However it is imp ortant to notic e that the Lie-Poisson br acket functional was known to Kiril lov [Ki82], although not in r elation with Vlasov dynamics, but r ather he studie d such structur es in c onne ction with invariant diﬀer ential op er ators. In p articular, Kiril lov was the only author who notic e d how this br acket functional c an gener ate what has b e en her e c al le d c o adjoint op er ator ( ad ∗ β h A k ), “which, app ar ently, has so far not b e en c onsider e d”, he claime d in 1982 (the Kup ershmidt- Manin op er ator was known sinc e 1977). What he claime d had b e en c onsider e d wer e the c ase h = 1 , which is the Lie derivative, and the c ase h = k , which is often c al le d “L agr angian Schouten c onc omitant”. It is e asy to c alculate fr om e q. (2.7) that this op er ation with h = k is the diamond op eration ad ∗ β h A k = β k  A k , which wil l b e deﬁne d in chapter 4 as the dual of the Lie derivative. 2.3 New v ariational principles for moment dynamics This section shows how the momen t dynamics can be deriv ed from Hamilton’s principle b oth in the Hamilton-P oincar ´ e and Euler-Poincar ´ e forms. These v ariational principles are deﬁned, resp ectiv ely , on the dual Lie algebra g ∗ con taining the moments, and on the Lie algebra g itself. F or further details ab out these dual v ariational form ulations, see [CeMaPeRa] and [HoMaRa]. Summation ov er rep eated indices is intended in this section 2.3.1 Hamilton-P oincar ´ e hierarc hy One b egins with the Hamilton-Poincar ´ e principle for the p − moments written as δ Z t j t i d t  h A n , β n i − H ( { A } )  = 0 CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 58 (where β n ∈ g ). It is possible to pro ve that this leads to the same dynamics as found in the con text of the KMLP brac ket. T o this purpose, one must deﬁne the n − th moment in terms of the Vlasov distribution function. One chec ks that 0 = δ Z t j t i d t ( h A n , β n i − H ( { A } )) = = Z t j t i d t  δ h h f , p n β n i i −   δ f , δ H δ f   = = Z t j t i d t   δ f ,  p n β n − δ H δ f   + h h f , δ ( p n β n ) i i  No w recall that an y g = δ G/δ f b elonging to the Lie algebra s of the symplectomorphisms (whose dual s ∗ con tains the distribution function itself ) ma y b e expressed as g = δ G δ f = p m δ G δ A m = p m ξ m b y the chain rule. Consequently , one ﬁnds the pairing relationship,   δ f ,  p n β n − δ H δ f   =  δ A n ,  β n − δ H δ A n  Next, recall from the general theory that v ariations on a Lie group induce v ariations on its Lie algebra of the form δ w = ˙ u + [ g , u ] where u, w ∈ s and u v anishes at the endp oints. W riting u = p m η m then leads to Z t j t i d t h h f , δ ( p n β n ) i i = Z t j t i d t h h f , ( ˙ u + [ p n β n , u ]) i i = = − Z t j t i d t D ˙ A m , η m E − h A n + m − 1 , [ [ β n , η m ] ] i  = = − Z t j t i d t D ˙ A m + ad ∗ β n A m + n − 1  , η m E Consequen tly , the Hamilton-P oincar´ e principle may be written entirely in terms of the mo- men ts as δ S = Z t j t i d t  δ A n ,  β n − δ H δ A n  − D ˙ A m + ad ∗ β n A m + n − 1  , η m E  = 0 This expression pro duces the inv erse Legendre transform β n = δ H δ A n (holding for hyperregular Hamiltonians). It also yields the equations of motion ∂ A m ∂ t = − ad ∗ β n A m + n − 1 CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 59 whic h are v alid for arbitrary v ariations δ A m and v ariations δ β m of the form δ β m = ˙ η m + ad β n η m − n +1 where the v ariations η m satisfy v anishing endp oin t conditions, η m | t = t i = η m | t = t j = 0 Th us, the Hamilton-Poincar ´ e v ariational principle reco vers the hierarch y of the evolution equations derived in the previous section using the KMLP brack et. 2.3.2 Euler-P oincar ´ e hierarc hy The corresp onding Lagrangian formulation of the Hamilton’s principle now yields δ Z t j t i L ( { β } ) d t = Z t j t i  δ L δ β m , δ β m  d t = = Z t j t i  δ L δ β m , ( ˙ η m + ad β n η m − n +1 )  d t = = − Z t j t i  ∂ ∂ t δ L δ β m , η m  +  ad ∗ β n δ L δ β m , η m − n +1  d t = = − Z t j t i  ∂ ∂ t δ L δ β m , η m  +  ad ∗ β n δ L δ β m + n − 1 , η m  d t = = − Z t j t i  ∂ ∂ t δ L δ β m + ad ∗ β n δ L δ β m + n − 1  , η m  d t up on using the expression previously found for the v ariations δ β m and relab eling indices appropriately . The Euler-P oincar´ e equations may then b e written as ∂ ∂ t δ L δ β m + ad ∗ β n δ L δ β m + n − 1 = 0 with the same constraints on the v ariations as in the previous paragraph. Applying the Legendre transformation A m = δ L δ α m yields the Euler-Poincar ´ e equations (for h yp erregular Lagrangians). This again leads to the same hierarch y of equations derived earlier using the KMLP brack et. T o summarize, the calculations in this section hav e prov en the following result. CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 60 Theorem 10 With the ab ove notation and hyp otheses of hyp err e gularity the fol lowing statements ar e e quivalent: 1. The Euler–Poinc ar´ e V ariational Principle. The curves β n ( t ) ar e critic al p oints of the action δ Z t j t i L ( { β } ) dt = 0 for variations of the form δ β m = ˙ η m + ad β n η m − n +1 in which η m vanishes at the endp oints η m | t = t i = η m | t = t j = 0 and the variations δ A n ar e arbitr ary. 2. The Lie–Poisson V ariational Principle. The curves ( β n , A n ) ( t ) ar e critic al p oints of the action δ Z t j t i ( h A n , β n i − H ( { A } )) d t = 0 for variations of the form δ β m = ˙ η m + ad β n η m − n +1 wher e η m satisﬁes endp oint c onditions η m | t = t i = η m | t = t j = 0 and wher e the variations δ A n ar e arbitr ary. 3. The Euler–Poinc ar´ e e quations hold: ∂ ∂ t δ L δ β m + ad ∗ β n δ L δ β m + n − 1 = 0 . 4. The Lie–Poisson e quations hold: ˙ A m = − ad ∗ δH δA n A m + n − 1 CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 61 An analogous result is also v alid in the multidimensional case with sligh t mo diﬁcations. Remark 11 (Hamilton-Poincar ´ e theorems and reduction) The or em 10 b elongs to a class of the or ems, c al le d Hamilton-Poincar ´ e theorems [CeMaPeR a]. These the or ems involve a hyp err e gular Hamiltonian (or L agr angian), which is invariant with r esp e ct to the action of some Lie gr oup. The Hamilton-Poinc ar ´ e r e duction pr o c ess [CeMaPeR a] then al lows to write the Hamiltonian (or L agr angian) on the Lie algebr a of that Lie gr oup, by fol lowing the same lines as in the ﬁrst se ction of chapter 1. This r e duction pr o c ess is not cle ar in the c ase of moment dynamics, sinc e it would r e quir e the explanation of moment Lie-Poisson dynamics as c o adjoint motion on a Lie gr oup. The latter has not b e en identiﬁe d yet and thus it is not p ossible to write the unreduced Hamiltonian [MaR a99] on the moment Lie gr oup. Remark 12 (Legendre transformation) In the c ase of moments, the hyp othesis of hy- p err e gularity is a str ong assumption. In physic al applic ations, for example, this hyp othesis often fails, as it happ ens for the Poisson-Vlasov system, whose Hamiltonian is given by H = 1 2 R  A 2 + A 0 ∆ − 1 A 0  d q . This failur e is intr o duc e d by the single p article kinetic en- er gy, which pr o duc es the term R A 2 d q . This term c annot b e L e gendr e-tr ansforme d, sinc e the quantity δ H /δ A 2 is not deﬁne d as a dynamic al variable (the moment algebr a do es not in- clude c onstants). Nevertheless, se ction 3.3 wil l show that for the c ase of geodesic motion on the moments, this hyp othesis is always satisﬁe d when the metric is diagonal [GiHoT r2007]. This pr o duc es the Euler-Poinc ar ´ e e quations on the moment algebr a, which extend the CH e quation to its multic omp onent versions (cf. chapter 3). Remark 13 (Euler-Poincar ´ e equations for statistical moments) By fol lowing the same ar guments as in the pr o of of the the or em ab ove, one se es, that similar r esults hold also for the statistical momen ts pr esente d in chapter 1. This yields the Euler-Poinc ar´ e e quations arising fr om a moment L agr angian L ( { a b mm } ) . Such moment e quations on the L agr angian fr amework have never b e en c onsider e d so far and it would b e inter esting to study their dy- namics, for example by using simple Lie sub-algebr a closur es. Such an appr o ach is fol lowe d in the next se ction for kinetic moments. 2.4 Some results on momen ts and cotangen t lifts As explained in the in tro duction, a ﬁrst order closure of the moment hierarc hy leads to the equations of ideal ﬂuid dynamics. Such equations represent coadjoint motion with resp ect CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 62 to the Lie group of smo oth inv ertible maps (diﬀeomorphisms). This coadjoint ev olution ma y be in terpreted in terms of Lagrangian v ariables, whic h are in v ariant under the action of diﬀeomorphisms. This section inv estigates how the entire moment hierarch y may b e expressed in terms of the ﬂuid quantities ev olving under the diﬀeomorphisms and expresses the conserv ation la ws in this case. 2.4.1 Bac kground on Lagrangian v ariables In order to lo ok for Lagrangian v ariables, one considers the geometric interpretation of the momen ts, regarded as ﬁb er inte gr als on the cotangent bundle T ∗ Q of some conﬁguration manifold Q . A moment is deﬁned as a ﬁb er integral; that is, an integral on the single ﬁb er T ∗ q Q with base p oint q ∈ Q kept ﬁxed A n ( q ) = Z T ∗ q Q p n f ( q , p ) d p (2.9) A similar approach is follo wed for gyrokinetics in [QiT a2004]. Now, the problem is that in general the in tegrand do es not sta y on a single ﬁb er under the action of canonical trans- formations, i.e. s ymplectomorphisms are not ﬁb er-pr eserving in the general case. Ho wev er, one ma y a void this problem by restricting to a subgroup of these canonical transformations whose action is ﬁb er preserving. The transformations in this subgroup (indicated with T ∗ Diﬀ( Q )) are called p oint tr ans- formations or c otangent lifts of diﬀeomorphisms and they arise from diﬀeomorphisms on p oin ts in conﬁguration space [MaRa99], such that q t = q t ( q 0 ) (2.10) The ﬁb er preserving nature of cotangen t lifts is expressed b y the preserv ation of the canonical one-form: p t dq t = p t ( q 0 , p 0 ) dq t ( q 0 ) = p 0 dq 0 (2.11) This fact also reﬂects in the particular form assumed b y the generating functions of cotangent lifts, which are linear in the momen tum co ordinate [MaRa99], i.e. h ( q , p ) = β ( q ) ∂ ∂ q p dq = p β ( q ) . (2.12) where the symbol denotes con traction betw een the vector ﬁeld β and the momentum one- form p . Restricting to cotangent lifts represents a limitation in comparison with considering the whole symplectic group. How ever, this is a natural wa y of recov ering the Lagrangian approac h, starting from the full moment dynamics. CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 63 2.4.2 Characteristic equations and related results Once one restricts to cotangent lifts, Lagrangian moment v ariables ma y b e deﬁned and conserv ation la ws ma y b e found, as in the context of ﬂuid dynamics. The key idea is to use the preserv ation of the canonical one-form for constructing inv ariant quantities. Indeed one ma y tak e n times the tensor pro duct of the canonical one-form with itself and write: p n t ( dq t ) n = p n 0 ( dq 0 ) n (2.13) One then considers the preserv ation of the Vlaso v density f t ( q t , p t ) dq t ∧ dp t = f 0 ( q 0 , p 0 ) dq 0 ∧ dp 0 (2.14) and writes p n t f t ( q t , p t ) ( dq t ) n ⊗ dq t ∧ dp t = p n 0 f 0 ( q 0 , p 0 ) ( dq 0 ) n ⊗ dq 0 ∧ dp 0 (2.15) In tegration ov er the canonical particle momen ta yields the follo wing c haracteristic equations d dt h A ( t ) n ( q t ) ( dq t ) n ⊗ dq t i = 0 along ˙ q t = ∂ h ∂ p = β ( q ) (2.16) whic h recov er the well known conserv ations for ﬂuid densit y and momen tum ( n = 0 , 1) and can be equiv alently written in terms of the Lie-Poisson equations arising from the KMLP brac ket, as sho wn in the next section. Indeed, if the v ector ﬁeld β is iden tiﬁed with the Lie algebra v ariable β = β 1 = δ H /δ A 1 and h ( A 1 ) is the momen t Hamiltonian, the KMLP form (2.4) of the moment equations is ∂ A n ∂ t + ad ∗ β 1 A n = 0 . (2.17) In this case, the KM ad ∗ β 1 op eration coincides with the Lie deriv ative £ β 1 ; so, one may write it equiv alen tly as ∂ A n ∂ t + £ β 1 A n = 0 . (2.18) F or n = 0 , 1, one reco vers the advection relations for the densit y A 0 and the momentum A 1 in ﬂuid dynamics. How ever, unlik e ﬂuid dynamics, all the momen ts are conserved quan tities. This equation is reminiscent of the so called b-e quation in tro duced in [HoSt03], for which the vector ﬁeld β is nonlocal and may be tak en as β ( q ) = G ∗ A n (for any n ), where G is the Green’s function of the Helmholtz op erator. When the vector ﬁeld β is suﬃcien tly smo oth, this equation is known to possess singular solutions of the form A n ( q , t ) = K X i =1 P n, i ( t ) δ ( q − Q i ( t )) (2.19) CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 64 where the i − th p osition Q i and weigh t P n, i of the singular solution for the n − th momen t satisfy the following equations ˙ Q i = β ( q ) | q = Q i , ˙ P n, i = − n P n, i ∂ β ( q ) ∂ q     q = Q i (2.20) One can easily see how these solutions are obtained by pairing the equation (2.18) with the con trav ariant tensor ϕ n . One calculates h ˙ A n , ϕ n i = X i Z dq ϕ n ( q , t ) d dt [ P n, i ( t ) δ ( q − Q i ( t ))] = X i Z dq ϕ n  ˙ P n, i δ ( q − Q i ) − P n, i ( t ) ˙ Q i δ 0 ( q − Q )  = = X i Z dq  b ϕ n ˙ P n, i + P n, i ˙ Q i b ϕ 0 n  where the hat denotes ev aluation at the p oin t q = Q i ( t ). Analogously one calculates h £ β 1 A n , ϕ n i =  ad ∗ β 1 A n , ϕ n  = h A n , ad β 1 ϕ n i = P n, i  n b ϕ n b β 0 1 − b β 1 b ϕ 0 n  and equating the corresp onding terms in b ϕ n and b ϕ 0 n yields the equations for Q i and P n, i . In terestingly , for n = 1 (with β ( q ) = G ∗ A 1 ), these equations reco ver the peakon solutions of the Camassa-Holm equation [CaHo1993], which play an imp ortan t role in the follo wing discussion. Moreov er the particular case n = 1 represen ts the single particle solution of the Vlasov equation. How ev er, when n 6 = 1 the interpretation of these solutions as single- particle motion requires the particular c hoice P n, i = ( P i ) n . F or this c hoice, the n − th weigh t is identiﬁed with the n − th pow er of the particle momentum. Higher dimensions and the b -equation. The generalization to higher dimensions when considering the tensorial nature of the momen ts from section 2.2.2, lead to rather complicated tensor equations. In the one dimensional case, one has β 1 = G ∗ A n , so the con volution maps the tensor quantit y A n to the vector ﬁeld β 1 . In higher dimensions, one has to b e careful in order to let dimensions matc h in the expression β 1 = G ∗ A n . One can think of the kernel G ( q − q 0 ) as a con trav ariant k -tensor itself G i 1 ,...,i k , so that the con volution op erator b ecomes written as R G ( q − q 0 ) A n ( q ) d N q and a vector ﬁeld ma y b e constructed b y letting k = n + 1, so that β i n +1 1 = G i 1 ,...,i n +1 ∗ ( A n ) i n ,...,i n (the density d N q do es not app ear in β 1 b ecause it has b een in tegrated out in the con volution). F or example, the EPDiﬀ equation in higher dimensions is reco vered in the case n = 1 by writing β i 1 = G ij ∗ ( A 1 ) j = G δ ij ∗ ( A 1 ) j = G ∗ ( A ] 1 ) i . This tensorial interpretation of the kernel will be adopted later in this Chapter, when dealing with quadratic moment Hamiltonians. CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 65 A t this p oin t, the equation ∂ A n ∂ t + £ G ∗ A n A n = 0 (2.21) is v alid in an y num b er of dimensions and it has the same singular solutions (2.20) as ab o ve, pro vided the v ariable P n is now a symmetric cov ariant tensor on the conﬁgu- ration manifold P n = ( P n ) i 1 ,...,i n d q i 1 . . . d q i n . How ever, in more generality the higher dimensional equations allow for solutions of the form A n ( q , t ) = X i Z P n, i ( s, t ) δ ( q − Q i ( s, t )) d s for which the tensor ﬁeld A n ( q , t ) is supp orted on a submanifold of R 3 (a ﬁlament if s is a one-dimensional co ordinate, a sheet if s is tw o-dimensional). One also reco vers the single particle tra jectory when P n = ⊗ n P . The previous discussion has shown how the one-dimensional version of this equation coincides with the b -equation in [HoSt03] for b = n + 1. Ho wev er in higher dimensions this equation substan tially diﬀers from the three-dimensional v ersion prop osed in [HoSt03], whic h is a c haracteristic equation for the tensorial quantit y m · d q ⊗ b d 3 q = m i d q i ⊗ b d 3 q along the nonlo cal v ector ﬁeld G ∗ m . This c haracteristic equation for m has b een shown to p ossess emergen t singular solutions and it would b e in teresting to chec k whether this prop ert y is shared with the tensorial b -equation (2.21) prop osed here. KMLP brack et and cotangen t lifts. The previous arguments ha ve shown that re- stricting to cotangen t lifts leads to a Lagrangian ﬂuid-lik e formulation of the dynamics of the resulting p -momen ts. In this case, the momen t equations are giv en b y the KMLP brac ket when the Hamiltonian dep ends only on the ﬁrst moment ( β 1 = δ H /δ A 1 ) { G, H } = X n  A n ,  δ G δ A n , δ H δ A 1  (2.22) If one now restricts the brack et to functionals of only the ﬁrst moment, one may chec k that the KMLP brack et yields the well kno wn Lie-Poisson brac ket on the group of diﬀeomor- phisms { G, H } [ A 1 ] = −  A 1 ,  δ G δ A 1 ∂ ∂ q δ H δ A 1 − δ H δ A 1 ∂ ∂ q δ G δ A 1   (2.23) This is a very natural step since diﬀeomorphisms and their cotangent lifts are isomorphic. In fact, this is the brack et used for ideal incompressible ﬂuids as well as for the construction CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 66 of the EPDiﬀ equation, whic h will b e discussed later as an application of moment dynamics. 2.4.3 Momen ts and semidirect pro ducts Another interesting example of ho w the Kup ershmidt-Manin brack et reduces to interesting structures is given by considering Hamiltonian functionals depending only on the ﬁrst tw o momen ts A 0 and A 1 , instead of only A 1 . In this case one mo diﬁes equation (2.18) as ∂ A n ∂ t + £ δH δA 1 A n + n A n − 1 ∂ ∂ q δ H δ A 0 = 0 . (2.24) The last term corresp onds to ad ∗ δ H/δ A 0 A n − 1 and is not completely understoo d as an inﬁnites- imal action, unless one considers the case n = 1 for which h ad ∗ δ H/δ A 0 A 0 , ξ i = h A 0  δ H /δ A 0 , ξ i , where ξ ∈ X ( R ) is a v ector ﬁeld on the real line and £ ξ A 0 = ∂ q ( A 0 ξ ). It is w orth noticing that this hierarch y of equations also allows for singular solutions of the form A n ( q , t ) = K X i =1 P n, i ( t ) δ ( q − Q i ( t )) . Ho wev er the dynamics of Q i and P n, i sligh tly diﬀers from that previously found and is written as ˙ Q i = β 1 ( q )    q = Q i , ˙ P n, i = − n  P n, i ∂ β 1 ( q ) ∂ q + P n − 1 , i ∂ β 0 ( q ) ∂ q  q = Q i (2.25) where the notation β n = δ H /δ A n has b een used. Again, if P n, i = P n i , then one reco vers the single-particle dynamics undergoing Hamiltonian motion with a Hamiltonian function giv en b y H N = P i P i β 1 ( Q i ) + P i β 0 ( Q i ). Analogous formulas also hold in more dimensions. Also the momen t brack et for functionals of only A 0 and A 1 p ossesses an interesting structure, which is a peculiar feature of ﬂuid systems. Indeed one calculates { G, H } =  A 0 ,  δ G δ A 1 , δ H δ A 0  +  A 0 ,  δ G δ A 0 , δ H δ A 1  +  A 1 ,  δ G δ A 1 , δ H δ A 1  = −  A 1 ,  δ G δ A 1 ∂ ∂ q δ H δ A 1 − δ H δ A 1 ∂ ∂ q δ G δ A 1   −  A 0 ,  δ G δ A 1 ∂ ∂ q δ H δ A 0 − δ H δ A 1 ∂ ∂ q δ G δ A 0   whic h is the w ell known Lie-P oisson semidirect pro duct structure [HoMaRa] on Diﬀ ( R ) s Den( R ) where Den( R ) := F ∗ ( R ) indicates the vector space of densities on the real line. The moment brack et and con tinuum mo dels. At this point it is easy to see how the Kup ershmidt-Manin brack et is a usual to ol for deriving contin uum ﬂuid mo dels from kinetic equations. This machinery is extended in the next sections to include extra degrees of freedom such as orien tation and magnetic momen t for each particle. CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 67 2.5 Discussion After a review of the moment Kupershmidt-Manin brack et, this c hapter has sho wn how this Lie-P oisson structure can b e extended to take into accoun t of the tensorial nature of the momen ts. The result is a Lie-Poisson brack et determined by the symmetric Schouten br acket [BlAs79, Ki82, DuMi95]. The moments hav e th us a purely geometric meaning in terms of symmetric c ovariant tensors [GiHoT r2008]. The Lie-P oisson structures for the momen ts hav e b een deriv ed from variational prin- ciples , in terms of Hamilton-Poinc ar´ e dynamics [CeMaPeRa]. As a result, the Euler- Poinc ar ´ e e quations [MaRa99, HoMaRa] hav e b een derived from a momen t Lagrangian. In the second part, this chapter has fo cused on the moment dynamics generated by diﬀe omorphisms and their cotangen t lifts on the phase space. In one dimension, the resulting moment equations ha ve the same form as the b -equation [HoSt03], which th us ma yb e interpreted as a char acteristic e quation for a single Vlasov kinetic moment. This concept extends to higher dimensions , thereby generating a higher dimensional version of the b -equation in terms of characteristic equations for symmetric tensors. These higher dimensional tensor equations are diﬀer ent from those prop osed in [HoSt03]. The same treatmen t has b een extended to consider semidirect pro ducts of diﬀeomorphisms and the corresp onding equations hav e b een presen ted. Singular solutions and future w ork. The moment equations obtained in this chapter ha ve b een shown to p ossess singular solutions , which reduce to the single particle tra- jectory in one dimension. The dynamics of these singularities has been studied and future w ork will b e fo cusing on the b ehavior of singular solutions in some simple cases of the ten- sorial b -equation in higher dimensions. F or example, one can consider the characteristic equation for A 2 : in two dimensions , this equation p ossesses thr e e indep endent c omp onents . This problem would represent an in teresting opportunity for analyzing the b eha viour of the singular solutions. In particular, one is in trigued to know whether these solutions emerge sp on taneously in a ﬁnite time, as it happ ens in some cases of b -equation in one dimension. P ersp ectives on momentum maps. The search for Lagrangian v ariables in moment dynamics turns out to be a c hallenging task and is strictly connected to the geometric nature of the momen ts. It is w ell kno wn they are P oisson maps, but one may wonder whether they are actually momentum maps [Ma82, W eMo, MaW eRaScSp] arising from a group action on CHAPTER 2. D YNAMICS OF KINETIC MOMENTS 68 the Vlasov Hamiltonian H [ f ]. This question has nev er been answ ered. There are reasons to b eliev e that the geometric iden tiﬁcation of momen ts with symmetric tensors is a k ey step in the construction of momentum maps. In particular, this construction would require the complete description of the momen t Lie-P oisson dynamics in terms of c o adjoint motion after the identiﬁcation of the symmetry group acting on the moments. F or example, the tensorial description w ould in v olve the symmetric group of permutations, as it happ ens also in the theory of statistical moments [HoLySc1990] BBGKY moments [MaMoW e1984]. Chapter 3 Geo desic ﬂo w on the momen ts: a new problem 3.1 In tro duction This chapter reviews some direct applications of momen t dynamics to physical problems and, as a new result, sho ws ho w the one-dimensional system of Benney long w av e equa- tions [Be1973] describ es the dynamics of coasting b eams in particle accelerators [V en turini]. The Benney momen t hierarch y is integrable and this explains the nature of the coherent structures observed in the experiments [KoHaLi2001, CoDaHoMa04]. This c hapter also form ulates the moment dynamics generated b y quadr atic Hamiltonians. This dynamics is a certain t yp e of geodesic motion on the symplectic diﬀeomorphisms, whic h are smo oth inv ertible symplectic maps acting on the phase space and p ossessing smo oth in verses. In some cases, the theory of momen t dynamics for the Vlaso v equation turns out to b e related to the theory of shallo w water equations. Indeed, the geo desic equations for the ﬁrst tw o momen ts reco ver b oth the integrable CH equation [CaHo1993] and its tw o- comp onen t v ersion [F alqui06, ChLiZh2005, Ku2007], which is again an integrable system of PDE’s. The study of suc h geo desic momen t equations is a new problem, which is here approac hed for the ﬁrst time. Singular solutions are presen ted as well as an extension of momen t geo desic motion to anisotropic interactions. 69 CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 70 3.2 Applications of the momen ts and quadratic terms 3.2.1 The Benney equations and particle b eams: a new result The KMLP brack et (2.2) was ﬁrst derived in the context of Benney long wa ves, whose Hamiltonian is H = 1 2 Z ( A 2 ( q ) + g A 2 0 ( q )) d q . (3.1) The Hamiltonian form ∂ t A n = { A n , H } with the KMLP brack et 2.4 leads to the moment equations ∂ A n ∂ t + ∂ A n +1 ∂ q + g nA n − 1 ∂ A 0 ∂ q = 0 (3.2) deriv ed b y Benney [Be1973] as a description of long wa ves on a shallow p erfect ﬂuid, with a free surface at y = h ( q , t ). In this interpretation, the A n w ere vertical momen ts of the horizon tal com ponent of the velocity p ( q, y , t ): A n = Z h 0 p n ( q , y , t ) d y . The corresp onding system of ev olution equations for p ( q , y , t ) and h ( q , t ) is related b y ho do- graph transformation, y = R p −∞ f ( q , p 0 , t ) d p 0 , to the Vlasov equation ∂ f ∂ t + p ∂ f ∂ q − g ∂ A 0 ∂ q ∂ f ∂ p = 0 . (3.3) The most imp ortan t fact ab out the Benney hierarch y is that it is completely in tegrable [KuMa1978]. Applications to coasting accelerator b eams. Interestingly , the equation that reg- ulates coasting proton b eams in particle accelerators takes exactly the same form as the Vlasov-Benney equation (3.3). (See for example [V en turini] where a linear bunching term is also included.) The in tegrability of the Vlasov-Benney equation implies coheren t structures. These structures are indeed found exp erimentally at CERN [KoHaLi2001], BNL [BlBr&Al.], LANL [CoDaHoMa04] and F ermiLab [MoBa&Al.]. (In the last case coheren t structures are shown to app ear ev en when a bunching force is present.) These structures hav e attracted the attention of the accelerator communit y and considerable analytical w ork has b een carried out ov er the last decade (see for example [ScF e2000]). The existence of coherent structures in coasting proton b eams has nev er b een related CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 71 to the integrabilit y of the gov erning Vlasov equation via its connection to the Benney hierarc hy . How ev er, this connection would explain v ery naturally why robust coherent structures are seen in these exp eriments as fully nonlinear excitations. 3.2.2 The wak e-ﬁeld mo del and some sp ecializations Besides in tegrability of the Vlasov-Benney equation, there are other imp ortant applications of the Vlasov equation that ha ve in common the presence of a quadratic term in A 0 within the Hamiltonian: H = 1 2 Z A 2 ( q ) d q + 1 2 Z Z A 0 ( q ) G ( q , q 0 ) A 0 ( q 0 ) d q d q 0 . (3.4) F or example, when G =  ∂ 2 q  − 1 , this Hamiltonian leads to the Vlasov-P oisson system, whic h is of fundamental importance in many areas of plasma physics. Remark ably , the ﬂuid closure of this system has b een shown to b e in tegrable in [Pa05]. More generally , this Hamiltonian is widely used for b eam dynamics in particle accelerators: in this case G is related to the electromagnetic interaction of a b eam with the v acuum cham b er. The wake ﬁeld is originated by the image charges induced on the w alls by the passage of a moving particle: while the b eam passes, the charges in the walls are attracted tow ards the inner surfaces and generate a ﬁeld that acts back on the b eam. This aﬀects the dynamics of the b eam, thereby causing several problems such as b eam energy spread and instabilities. In the literature, the wake function W is introduced so that [V en turini] G ( q , q 0 ) = Z q −∞ W ( x, q 0 ) dx (3.5) W ake functions usually depend only on the prop erties of the accelerator c hamber. An interesting wak e-ﬁeld mo del has b een presented in [ScF e2000] where G is c hosen to b e the Green’s function of the Helmholtz op erator  1 − α 2 ∂ 2 q  : this generates a Vlasov- Helmholtz (VH) equation [CaMaPu2002] that is particularly in teresting for future work. Connections of this equation with the well known integrable KdV equation hav e b een pro- p osed. How ev er this is not a natural step since in tegrability appears already with no further appro ximations in the Vlaso v-Benney (VB) system that go verns the collectiv e motion of the b eam. In particular one w ould lik e to understand the VH equation as a special deformation of the integrable VB case that allows the existence of singular solutions . Indeed, the CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 72 presence of the Green’s function G ab ov e is a key ingredien t for the existence of the single- particle solution, which is not allow ed in the VB case. In particular, the single-particle solutions for the Vlaso v-Helmholtz equation may b e of great interest, since these singular solutions arise from a deformation of an integrable system. In the limit as the deformation parameter α in the Helmholtz Green’s function passes to zero ( α → 0), one reco vers the in te- grable Vlaso v-Benney case. Also, in the limit α → ∞ the ﬂuid closure of this system reduces again to an integrable system [Pa05]. Thus the study of the VH equation and its single par- ticle solution can provide useful understanding of t wo integrable limits, the Vlasov-Benney equation ( α → 0) and the ﬂuid closure of the Vlasov-P oisson system ( α → ∞ ). 3.2.3 The Maxwell-Vlaso v system In higher dimensions, particularly N = 3, one takes the direct sum of the KMLP brack et, together with with the P oisson brack et for an electromagnetic ﬁeld (in the Coulomb gauge) where the electric ﬁeld E and magnetic vector p oten tial A are canonically conjugate; then the Hamiltonian (in multi-index notation) H [ { A } , A , E ; ϕ ] = 1 2 Z X j  A 2 j ( q ) − 2 A j ( q ) A 1 j ( q )  d 3 q + 1 2 Z  | A ( q ) | 2 + 2 ϕ ( q )  A 0 ( q ) d 3 q + 1 2 Z  | E ( x ) | 2 + 2 E ( x ) · ∇ ϕ ( x ) + |∇ × A ( x ) | 2  d 3 x yields the Maxw ell-Vlasov (MV) equations for systems of interacting charged particles. In the Hamiltonian, ϕ plays the role of a Lagrange multiplier that constraints the v aria- tional principle in order to include Gauss’ law. F or a discussion of the MV equations from a geometric viewp oin t in the same spirit as the present approac h, see [Ma82, MaW e81, CeHoHoMa1998]. 3.2.4 The EPDiﬀ equation and singular solutions Another in teresting momen t equation is giv en by the Euler-Poincar ´ e equation on the group of diﬀeomorphisms (EPDiﬀ ) [CaHo1993]. In this case, the Hamiltonian is purely quadratic in the ﬁrst moments: H = 1 2 Z Z A 1 ( q ) G ( q , q 0 ) A 1 ( q 0 ) d q d q 0 (3.6) CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 73 and the EPDiﬀ equation [HoMa2004] ∂ A 1 ∂ t + ∂ A 1 ∂ q Z G ( q , q 0 ) A 1 ( q 0 , t )d q 0 + 2 A 1 ∂ ∂ q Z G ( q , q 0 ) A 1 ( q 0 , t )d q 0 = 0 (3.7) comes from the closure of the KMLP brack et given b y cotangent lifts. (Without this restric- tion one w ould obtain again the equations (2.16) with β = G ∗ A 1 .) Thus this EPDiﬀ equation is a ge o desic equation on the group of diﬀeomorphisms. The Camassa-Holm equation is a particular case in whic h G is the Green’s function of the Helmholtz op erator 1 − α 2 ∂ 2 q . Both the CH and the EPDiﬀ equations are completely integrable and hav e a large n umber of ap- plications in ﬂuid dynamics (shallow water theory , a veraged ﬂuid mo dels, etc.) and imaging tec hniques [HoRaT rY o2004] (medical imaging, contour dynamics, etc.). Besides the complete in tegrability of the CH equation, the connection b et ween the CH (EPDiﬀ ) equation and momen t dynamics lies in the fact that singular solutions app ear in b oth con texts. The existence of this kind of solution for EPDiﬀ leads to in vestigate its origin in the context of Vlasov momen ts. More particularly it is a reasonable question whether there is a natural extension of the EPDiﬀ equation to all the moments. This would again b e a geo desic (hierarc hy of ) equation, whic h would perhaps explain how the singular solutions for EPDiﬀ arise in this larger context. Remark 14 It should b e p ointe d out that the KMLP and VLP formulations ar e not whol ly e quivalent; in p articular the map fr om the distribution function f ( q , p ) to the moments { A n } is explicit, but it is not a trivial pr oblem to r e c onstruct the distribution fr om its moments. Simple ﬂuid-like closur es of the system arise very natur al ly in the KMLP fr amework, as with the example in Se ction 3. 3.3 A new geo desic ﬂo w and its singular solutions 3.3.1 F orm ulation of the problem: quadratic Hamiltonians The previous examples sho w how quadratic terms in the Hamiltonian pro duce interesting b eha vior in v arious contexts. This suggests that a deeper analysis of the role of quadratic terms ma y b e worth while particularly in connections b etw een Vlasov momen t dynamics and the EPDiﬀ equation, with its singular solutions. Purely quadratic Hamiltonians are consid- ered in [GiHoT r05], leading to the problem of geo desic motion on the space of moments. In this problem the Hamiltonian is the norm on the momen ts given by the follo wing CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 74 metric and inner pro duct, H = 1 2 k A k 2 = 1 2 ∞ X n,s =0 Z Z A n ( q ) G ns ( q , q 0 ) A s ( q 0 ) d q d q 0 (3.8) The metric G ns ( q , q 0 ) in (3.8) is c hosen to b e p ositive deﬁnite, so it deﬁnes a norm for { A } ∈ g ∗ . The corresponding geodesic equation with respect to this norm is found as in the previous section to b e, ∂ A m ∂ t = { A m , H } = − ∞ X n =0  nβ n ∂ ∂ q A m + n − 1 + ( m + n ) A m + n − 1 ∂ ∂ q β n  (3.9) with Lie algebra v ariables β n ∈ g deﬁned by β n = δ H δ A n = ∞ X s =0 Z G ns ( q , q 0 ) A s ( q 0 ) d q 0 = ∞ X s =0 G ns ∗ A s . (3.10) Th us, evolution under (3.9) may be rewritten as formal coadjoint motion on the dual Lie algebra g ∗ ∂ A m ∂ t = { A m , H } =: − ∞ X n =0 ad ∗ β n A m + n − 1 (3.11) This system comprises an inﬁnite system of nonlinear, nonlocal, coupled evolutionary equa- tions for the moments. In this system, evolution of the m th momen t is go verned b y the p oten tially inﬁnite sum of contributions of the velocities β n asso ciated with n th momen t sw eeping the ( m + n − 1) th momen t b y a t yp e of coadjoint action. Moreov er, b y equation (3.10), each of the β n p oten tially dep ends nonlo cally on all of the moments. Equations (3.8) and (3.10) may b e written in three dimensions in multi-index notation, as follows: the Hamiltonian is given b y H = 1 2 || A || 2 = 1 2 X µ,ν Z Z A µ ( q , t ) G µν ( q , q 0 ) A ν ( q 0 , t ) d 3 q d 3 q 0 so the dual v ariable is written as β ρ = δ H δ A ρ = X ν Z Z G ρν ( q , q 0 ) A ν ( q 0 , t ) d 3 q d 3 q 0 = X ν G ρν ∗ A ν . Ho wev er the equations (3.8) and (3.10) are already v alid in higher dimensions if one considers the tensor in terpretation of the momen ts. This is another case in whic h the tensor in terpretation is helpful. In this case, the metric is written as G nm = G i 0 ,...,i n , j 0 ,...,j m nm ( q , q 0 ) whic h tak es into accoun t for the tensor nature of the moment equations. CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 75 Remark 15 (Euler-Poincar ´ e form ulation) When the metric G nm is diagonal ( G nm = K nm δ m n =: G n ), the Hamiltonian b e c omes hyp err e gular and one c an ﬁnd the inverse L e g- endr e tr ansform. In or der to se e this explicitly one c an write the Lie algebr a variable β n in one dimension as β n = δ H δ A n = Z G n ( q , q 0 ) A n ( q 0 ) d q 0 = G n ∗ A n . so that, if the n -th kernel G n is the Gr e en ’s function c orr esp onding to the inverse of some op er ator b Q n (so that G n = b Q − 1 n ), then one c alculates the inverse the L e gendr e-tr ansform as A n = b Q n β n and the pr oblem admits a L agr angian formulation in terms of the Euler-Poinc ar ´ e variational principle δ Z t 2 t 1 β n b Q n β n d t = 0 (3.12) and the c orr esp onding Euler-Poinc ar ´ e hier ar chy fol lows. The construction of this geo desic motion on the moments is motiv ated by the examples pro vided b y Euler and CH equation and is justiﬁed b y its Lie-Poisson structure. Ho wev er the search for singular solutions requires more insight into the geometric meaning of this inﬁnite hierarch y of equations. In particular, since the Lie-P oisson dynamics has not been fully in terpreted in terms of coadjoint motion and the underlying Lie group has not b een iden tiﬁed, this geo desic ﬂow needs further in vestigation. 3.3.2 A ﬁrst result: the geo desic Vlaso v equation (EPSymp) Imp ortan tly , geo desic motion for the momen ts is equiv alent to geo desic motion for the Euler- P oincar´ e equations on the symplectomorphisms (EPSymp). This is generated by the follo wing quadratic Hamiltonian H [ f ] = 1 2 Z Z f ( q , p ) G ( q , p, q 0 , p 0 ) f ( q 0 , p 0 ) d q d p d q 0 d p 0 (3.13) The equiv alence with EPSymp emerges when the function G is written as G ( q , q 0 , p, p 0 ) = X n,m p n G nm ( q , q 0 ) p 0 m . (3.14) CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 76 and the corresp onding Vlasov equation reads as ∂ f ∂ t + n f , G ∗ f o = 0 (3.15) where {· , ·} denotes the canonical Poisson brac ket. Th us, whenever the metric G for EPSymp has a T a ylor series, its solutions ma y b e expressed in terms of the geo desic motion for the moments. More particularly the geo desic Vlasov equation presented here is nonlo cal in b oth p osition and momen tum. How ev er this equation extends to more dimensions [GiHoT r05] and to any kind of geo desic motion, no matters ho w the metric is expressed explicitly . Such an equation reduces to the 2D Euler’s equation for G = ∆ − 1 , as sho wn in c hapter 1, and is surprisingly similar in construction to another imp ortan t integrable geodesic equation on the linear Hamiltonian vector ﬁelds (Hamiltonian matrices), which has been recently proposed [BlIs, BlIsMaRa05]. F or a more extensive analysis, one can relate the geo desic Vlaso v equation EPSymp with its corresp ondent equation on Hamiltonian v ector ﬁelds. T o this purp ose one restricts the EPDiﬀ Lagrangian to the symplectic algebra on T ∗ Q L [ X h ] = 1 2  ˆ Q X h , X h  where ˆ Q : X can → X ∗ can is an in vertible symmetric diﬀerential op erator. Up on in tegration b y parts, this Lagrangian is written on the Hamiltonian functions as L = 1 2 D ˆ Q X h , X h E = 1 2 D ˆ Q J ∇ h , J ∇ h E = 1 2 D div  J ˆ Q J ∇ h  , h E = L [ h ] . The Legendre transform f = δ L δ h = div  J ˆ Q J ∇ h  ⇒ h =  div J ˆ Q J ∇  − 1 f yields the EPSymp Hamiltonian in the Vlasov form H [ f ] = 1 2  f , ˆ O − 1 f  with ˆ O := div J ˆ Q J ∇ . This makes clear the connection betw een the geodesic Vlasov equation and the geo desic motion on the Hamiltonian v ector ﬁelds. An interesting case o ccurs when ˆ Q is the ﬂat CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 77 op eration ˆ Q X h = ( X h ) [ , so that div J ( J ∇ h ) [ = − ∆ h . Then ˆ O reduces to minus the Laplacian ˆ O = − ∆ and in t wo dimensions one obtains the Euler Hamiltonian H [ ω ] = 1 / 2  ω , ( − ∆) − 1 ω  with ω = f . This analysis explains ho w the geo desic motion on the symplectic group is related to the geo desic motion on the volume-preserving diﬀeomorphisms in the vorticit y representation in tro duced in chapter 1. In the more general case when ˆ Q is a purely diﬀerential op erator, one has that ˆ Q and J commute and thus ˆ O = − div ˆ Q ∇ . Also if ˆ Q commutes with the div ergence, then, one has ˆ O = − ˆ Q ∆. How ever in the most general case, ˆ Q is a matrix diﬀeren tial op erator that do es not commute with J . 3.3.3 The nature of singular geo desic solutions The geometric meaning of the momen t equations is no w explained in terms of coadjoint geo desic motion on the symplectic group and one can therefore c haracterize singular solu- tions, since the geo desic Vlasov equation (EPSymp) essentially describ es advection in phase space. Indeed, the geo desic Vlaso v equation p ossesses the single particle solution f ( q , p, t ) = X j δ ( q − Q j ( t )) δ ( p − P j ( t )) (3.16) whic h is a well known singular solution that is admitted whenever the phase-space density is advected along a smooth Hamiltonian vector ﬁeld. This happ ens, for example, in the Vlaso v-Poisson system and in the general wak e-ﬁeld mo del. On the other hand, these singular solutions do not app ear in the Vlasov-Benney equation. In any num b er of spatial dimensions, the geodesic equation (3.9) p ossesses exact solutions whic h are singular ; that is, they are supp orted on delta functions in q − space: equation (3.9) admits singular solutions of the form A n ( q , t ) = N X j =1 Z ⊗ n P j ( a, t ) δ  q − Q j ( a, t )  d a (3.17) CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 78 in which the integral o ver co ordinate a is p erformed ov er an embedded subspace of the q − space and the parameters ( Q j , P j ) satisfy canonical Hamiltonian equations in which the Hamiltonian is the norm H in (3.8) ev aluated on the singular solution Ansatz (3.17). In one dimension, the co ordinates a j are absent and the equation (3.9) admits singular solutions of the form A n ( q , t ) = N X j =1 P n j ( t ) δ  q − Q j ( t )  (3.18) In order to show this is a solution in one dimension, one c hecks that these singular solu- tions satisfy a system of partial diﬀerential equations in Hamiltonian form, whose Hamilto- nian couples all the moments H N = 1 2 ∞ X n,s =0 N X j,k =1 P s j ( t ) P n k ( t ) G ns ( Q j ( t ) , Q k ( t )) (3.19) Explicitly , one takes the pairing of the coadjoint equation ˙ A m = − X n,s ad ∗ G ns ∗ A s A m + n − 1 with a sequence of smo oth functions { ϕ m ( q ) } , so that: h ˙ A m , ϕ m i = X n,s h A m + n − 1 , ad G ns ∗ A s ϕ m i One expands each term and denotes e ϕ m := ϕ m ( q , t ) | q = Q j : h ˙ A m , ϕ m i = X j  dP m j dt ϕ m + P m j ˙ Q j ϕ 0 m  Similarly expanding D A m + n − 1 , ad G ns ∗ A s ϕ m E = X j,k P s k P m + n − 1 j  n e ϕ 0 m G ns ( Q j , Q k ) − m e ϕ m ∂ G ns ( Q j , Q k ) ∂ Q j  leads to dQ j dt = X n,s X k n P s k P n − 1 j G ns ( Q j , Q k ) dP j dt = − X n,s X k P s k P n j ∂ G ns ( Q j , Q k ) ∂ Q j so that one ﬁnally obtains equations for Q j and P j in canonical form, dQ j dt = ∂ H N ∂ P j , dP j dt = − ∂ H N ∂ Q j . CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 79 Remark 16 These singular solutions of EPSymp ar e also solutions of the Euler-Poinc ar ´ e e quations on the diﬀe omorphisms (EPDiﬀ ). In the latter c ase, the single-p article solutions r e duc e to the pulson solutions for EPDiﬀ [CaHo1993]. Thus, the singular pulson solutions of the EPDiﬀ e quation arise natur al ly fr om the single-p article dynamics on phase-sp ac e. 3.3.4 Some results on the dynamics of singular solutions This section presents the problem of the interaction b et ween t wo singular solutions. It is easy to show how this system preserves the total momentum P = P 1 + P 2 . Indeed, one observ es that ˙ P 1 = − X n,m p n 1 P m 1 ∂ ∂ Q     Q = Q 1 G nm ( Q − Q 1 ) + P m 2 ∂ ∂ Q     Q = Q 1 G nm ( Q − Q 2 ) ! = − X n,m P n 1 P m 2 ∂ Q 1 G nm ( Q 1 − Q 2 ) under the assumption that ( ∂ G nm ( Q ) /∂ Q ) Q =0 = 0. and ∂ Q 1 G nm ( Q 1 − Q 2 ) = − ∂ Q 2 G nm ( Q 1 − Q 2 ). Thus ˙ P 1 + ˙ P 2 = 0 since G nm ( Q 1 − Q 2 ) = G mn ( Q 1 − Q 2 ) . One can also see this by writing the Hamiltonian H N = 1 2 X n,m p n 1 p m 1 + 1 2 X n,m p n 2 p m 2 + 1 2 X n,m p n 1 K nm ( q 1 − q 2 ) p m 2 + 1 2 X n,m p n 2 K nm ( q 1 − q 2 ) p m 1 = 1 2 X n,m   P n + m 1 + P n + m 2  G nm (0) + 2 G nm ( Q 1 − Q 2  P n 1 P m 2  and by c hec king that ˙ P 1 = − X n,m P n 1 P m 2 ∂ Q 1 G nm ( Q 1 − Q 2 ) so that ˙ P 1 + ˙ P 2 = 0. Con vergence of the Hamiltonian. The problem with the Hamiltonian H N is that it eviden tly diverges in the case when G nm is the Helmholtz k ernel G nm ( x ) = e | x | /α nm , whic h is the case of the Camassa-Holm equation. Ho wev er, one can solve this problem by deﬁning the kernels G nm through the in tro duction of a sequence of co eﬃcients c nm suc h that c nm → ∞ with n, m → ∞ . F or example, one can deﬁne G nm ( x ) = 1 c nm  1 − α nm ∂ 2  − 1 = 1 c nm e | x | α nm (3.20) CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 80 In this case the Hamiltonian H N b ecomes H N = 1 2 X n,m  1 c nm  P n + m 1 + P n + m 2  + 2 G nm ( Q 1 − Q 2 ) P n 1 P m 2  (3.21) and if c nm → ∞ suﬃcien tly rapidly , then the Hamiltonian con verges. A particular choice inspired by T a ylor series could b e c nm = ( n + m )!. F or example one ev aluates the sum 1 2 X n,m P n + m ( n + m )! = 1 2  1 + ( P 1+0 + P 0+1 ) + 1 2 ( P 1+1 + P 2+0 + P 0+2 ) + 1 3! ( P 3+0 + P 0+3 + P 1+2 + P 2+1 ) + 1 4! ( P 4+0 + P 0+4 + P 1+3 + P 3+1 + P 2+2 ) + · · · + n + 1 n ! P n  whic h evidently div erges. Consequen tly , the right c hoice for c nm b ecomes c nm = ( n + m + 1)! so that 1 2 X n,m P n + m ( n + m )! = 1 2 X n n + 1 ( n + 1)! P n = 1 2 X n 1 n ! P n = 1 2 e P Th us, up on redeﬁning c nm = ( n + m + 1)! / 2 for conv enience, the Hamiltonian b ecomes H N = e P 1 + e P 2 + X n,m G nm ( Q 1 − Q 2 ) P n 1 P m 2 (3.22) whic h yields a particle velocity of the form ˙ Q 1 = e P 1 + 2 X n,m n G nm ( Q 1 − Q 2 ) P n − 1 1 P m 2 Th us one has the following Prop osition 17 With the choic e of metric (3.20) and for c nm = ( n + m + 1)! / 2 , the two p article Hamiltonian (3.21) c onver ges to the expr ession (3.22). No w one can sp ecializes to the case when the metric G nm is diagonal ( G nm = G n δ nm ), so that the Hamiltonian b ecomes H N = 1 2 X n X i,j P n i G n ( Q i − Q j ) P n j (3.23) that is, for i, j = 1 , 2 H N = 1 2 X n  1 c n  P 2 n 1 + P 2 n 2  + 2 G n ( Q 1 − Q 2 ) P n 1 P n 2  and if one c ho oses c n = (2 n )! / 2 (the factor 2 is just a conv enient c hoice), then one can write H N = cosh( P 1 ) + cosh( P 2 ) + X n G n ( Q 1 − Q 2 ) P n 1 P n 2 (3.24) This result can b e summarized as CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 81 Prop osition 18 The two p article Hamiltonian (3.23) with the metric G n ( x ) = 2 (2 n )! e | x | α n c onver ges to the expr ession in (3.24). The quadrature formulas for these systems are left for further study as well as the expressions for phase shifts in the collisions. How ev er it is interesting to notice the particular forms assumed by the Hamiltonian H N whic h are very diﬀerent from the usual expression used in physics H = T + V = 1 / 2 g kh ( Q ) P k P h + V ( Q ) . Remark 19 (Remark ab out higher dimensions) The singular solutions (3.17) with the inte gr al over c o or dinate a exist in higher dimensions. The higher dimensional singular solu- tions satisfy a system of c anonic al Hamiltonian inte gr al-p artial diﬀer ential e quations, inste ad of or dinary diﬀer ential e quations. Remark 20 (Connections with EPDiﬀ ) The singular solutions of EPSymp ar e also so- lutions of the Euler-Poinc ar´ e e quations on the diﬀe omorphisms (EPDiﬀ ), pr ovide d one c on- siders only the ﬁrst or der moment [HoMa2004]. In this c ase, the singular solutions r e duc e in one dimension to the pulson solutions for EPDiﬀ [CaHo1993]. Thus the pulson solution for EPDiﬀ has b e en shown to arise very natur al ly as the closur e of single-p article dynamics given by c otangent lifte d diﬀe omorphisms on phase-sp ac e. 3.3.5 Connections with the cold plasma solution A more general kind of singular solution for the momen ts may b e obtained by considering the c old-plasma solution of the Vlasov equation f ( q , p, t ) = X j ρ j ( q , t ) δ ( p − P j ( q , t )) (3.25) F or example, the single particle solution is recov ered b y putting ρ j ( q , t ) = δ ( q − Q j ( t )). Moreo ver exc hanging the v ariables q ↔ p in the single particle PDF leads to the following expression f ( q , p, t ) = X j ψ j ( p, t ) δ ( q − λ j ( p, t )) (3.26) whic h is alw a ys a solution of the Vlasov equation b ecause of the symmetry in q and p . This leads to the following singular solutions for the moments: A n ( q , t ) = X j Z d p p n ψ j ( p, t ) δ ( q − λ j ( p, t )) (3.27) CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 82 A t this p oint, if one considers a Hamiltonian dep ending only on A 1 (i.e. one considers the action of cotangent lifts of diﬀeomorphism), then it is p ossible to drop the p -dep endence in the λ ’s and thereby recov er to the singular solutions previously found for eq. (2.16). In order to understand this point, one can proceed as follows. Let λ j b e independent of p and deﬁne λ j =: Q j ( t ); thus one writes the momen ts as A n ( q , t ) = X j Z p n ψ j ( p, t ) d p δ ( q − Q j ( t )) =: X j P n, j ( t ) δ ( q − Q j ( t )) where one deﬁnes P n, j ( t ) := R p n ψ j ( p, t ) d p . In order to calculate the dynamics of P n and Q , it suﬃces to substitute the expression ab o ve in the momen t equations (3.11) and to calculate the pairing with contra v arian t n -tensors ϕ n . This pro cedure leads to ˙ P n = − n X m P m + n − 1 b β 0 m P n ˙ Q = X m m P m + n − 1 b β m whic h hold for all non-negative in tegers n . In particular, ﬁxing n = 0 yields ˙ P 0 = 0 = R ˙ ψ ( p, t ) d p , consistently with the h yp othesis R f d q d p = 1 = P 0 . More imp ortan tly , ﬁxing n = 0 yields the dynamics for the co ordinate Q ˙ Q = X m m P m − 1 b β m . Multiplying b y P n again leads to the pow ers P n = P n . In fact, since Q is indep endent on n one obtains P n X m m P m − 1 b β m = X m m P n + m − 1 b β m whic h means that P k P h = P h + k ⇒ P n = P n ∀ n ≥ 0 and th us obtaining the singular solutions (3.18), corresp onding to single particle dynamics, that is ψ ( p, t ) = δ ( p − P ( t )). It is worth noticing that the single particle solution arises only when considering moment dynamics, while it is alw ays p ossible to allow for a Vlasov solution of the form f = ψ ( p, t ) δ ( q − Q ( t )). This happ ens b ecause the p o wer P n in the singular solutions restricts the moments to b e necessarily symmetric, when they are considered as co v ariant tensors. This result diﬀers from that obtained for eq. (2.16), whic h alwa ys allows the solution A n ( q , t ) = P n ( t ) δ ( q − Q ( t )). The reason is that eq. (2.16) is generated by the action of diﬀeomorphisms, whic h alw a ys preserv es the symmetric nature of the tensor P n in CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 83 the dynamics. This is not true for all the canonical transformations, whose general action do es not keep P n symmetric during its evolution; rather the tensor P n b ecomes a tensor p o wer P n = P n , which is symmetric b y deﬁnition. In this spirit, the solution A n ( q , t ) = X j Z p n ψ j ( p, t ) δ ( q − λ j ( p, t )) d p represen ts a more general singular solution than the solutions (2.20), since it em b o dies the action of more general canonical transformations, which are not cotangent lifts of diﬀeomor- phisms on the conﬁguration manifold. 3.3.6 A result on truncations: the CH-2 equation The problem presented by the coadjoint motion equation (3.11) for geo desic ev olution of momen ts under EPDiﬀ may be simpliﬁed, by truncating the Poisson brack et to a ﬁnite set. Such truncations are not in general consistent with the full dynamics; in the rarer cases where they are consisten t, they will b e referred to as “reductions” [GiTs1996]. These moment dynamics may b e truncated to a Hamiltonian system, at an y stage by simply modifying the Lie algebra in the KMLP brack et to v anish for w eights m + n − 1 greater than a chosen cut-oﬀ v alue. F or example, if one truncates the sums to m, n = 0 , 1 , 2 only , then equation (3.11) pro duces the coupled system of partial diﬀerential equations, ∂ A 0 ∂ t = − ∂ q ( A 0 β 1 ) − 2 A 1 ∂ q β 2 − 2 β 2 ∂ q A 1 ∂ A 1 ∂ t = − A 0 ∂ q β 0 − 2 A 1 ∂ q β 1 − β 1 ∂ q A 1 − 3 A 2 ∂ q β 2 − 2 β 2 ∂ q A 2 (3.28) ∂ A 2 ∂ t = − 2 A 1 ∂ q β 0 − 3 A 2 ∂ q β 1 − β 1 ∂ q A 2 The ﬂuid closure of system (3.28), which ma y be called EPSymp ﬂuid , neglects A 2 and ma y b e written as ∂ A 0 ∂ t = − ∂ q ( A 0 β 1 ) ∂ A 1 ∂ t = − A 0 ∂ q β 0 − 2 A 1 ∂ q β 1 − β 1 ∂ q A 1 (3.29) When A 1 = (1 − α 2 ∂ 2 q ) β 1 and β 0 = A 0 , this system becomes the tw o-comp onen t Camassa-Holm system (CH-2) studied in [ChLiZh2005, F alqui06, Ku2007]. CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 84 F or this case, the ﬂuid closure system (3.29) is equiv alent to the compatibilit y for dλ/dt = 0 of a system of tw o linear equations, ∂ 2 x ψ +  − 1 4 + A 1 λ + A 2 0 λ 2  ψ = 0 (3.30) ∂ t ψ = −  1 2 λ + β 0  ∂ x ψ + 1 2 ψ ∂ x β 1 (3.31) The ﬁrst of these (3.30) is an eigen v alue problem kno wn as the Schr¨ odinger e quation with ener gy dep endent p otential . Because the eigenv alue λ is time indep endent, the ev olution of the nonlinear ﬂuid closure system (3.29) is said to b e isosp e ctr al . The second equation (3.31) is the evolution equation for the eigenfunction ψ . The ﬂuid closure system for geo desic ﬂow of the ﬁrst tw o Vlaso v moments also has a semidirect product structure on Diﬀ( R 3 ) s Den( R 3 ) [HoMaRa] whic h allows for singular so- lutions for b oth A 0 and A 1 in the case that β s = G ∗ A s , s = 0 , 1. The b ehavior of these singular solutions will b e inv estigated in future work. In particular one w ould like to un- derstand whether these singularities may emerge spontaneously as for the EPDiﬀ equation. Remark 21 (CH-2 equation for imaging) R emarkably, a similar system of e quations also arises in the study of imaging using a pr o c ess of template matching with active templates, known as metamorphosis [HoT rY o2007]. In this c ontext these e quations ar e c al le d EP G s H, which emphasizes the semidir e ct pr o duct structur e. Remark 22 (Euler-Poincar ´ e equations for the EPSymp ﬂuid) As mentione d in se c- tion 3.3.1, the moment e quations for EPSymp have an Euler-Poinc ar´ e formulation, which is given by the hier ar chy of e quations (3.12). This hier ar chy c an b e trunc ate d to obtain the Euler-Poic ar ´ e e quations for the ﬂuid closur e (3.29). In or der to ke ep close to the formulation of the Camassa-Holm e quation, one c an cho ose ˆ Q n = 1 − α 2 n ∂ 2 q in the e quations (3.12). If α 1 = 1 , then one obtains λ t − α 2 0 λ q qt = −  uλ − α 2 0 uλ q q  q u t − u q qt = − 3 uu q + 2 u q u q q + uu q qq − λ q  λ − α 2 0 λ q q  with A 1 =  1 − ∂ 2 q  β 1 and one intr o duc es the notation ( β 0 , β 1 ) = ( λ, u ) . This yields an extension of the two c omp onent Camassa-Holm e quation, which is nonlocal in b oth densit y and momentum . A gain, for α 0 → 0 , one r e c overs the r esults in [ChLiZh2005, F alqui06, Ku2007]. Of c ourse, inte gr ability issues for this system r emain to b e pursue d elsewher e. CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 85 Remark 23 (Singular solutions) The inter action of two singular solutions of the EP- Symp ﬂuid may b e e asily analyze d by trunc ating the Hamiltonian (3.19) to c onsider only n = 0 , 1 . This yields H 2 = 1 2  P 2 1 + P 2 2 + 2 G 1 ( Q 1 − Q 2 ) P 1 P 2 + 2 G 0 ( Q 1 − Q 2 )  By pr o c e e ding in the same way as in [HoSt03], one deﬁnes P = P 1 + P 2 , Q = Q 1 + Q 2 , p = P 1 − P 2 , q = Q 1 − Q 2 so that, the Hamiltonian c an b e written as H = 1 2 P 2 − 1 4 ( P 2 − p 2 ) (1 − G 1 ( q )) + G 0 ( q ) A t this p oint one writes the e quations dP dt = − 2 ∂ H ∂ Q = 0 , dQ dt = 2 ∂ H ∂ P = P (1 + G 1 ( q )) dp dt = − 2 ∂ H ∂ q = − 1 2  P 2 − p 2  G 0 1 ( q ) − 2 G 0 0 ( q ) , dq dt = 2 ∂ H ∂ p = − p (1 − G 1 ( q )) that yield  dq dt  2 = P 2 (1 − G 1 ( q )) 2 − 4 ( H − G 0 ( q )) (1 − G 1 ( q )) and ﬁnal ly le ad to the quadr atur e dt = dG 1 G 0 1 q P 2 (1 − G 1 ( q )) 2 − 4 ( H − G 0 ( q )) (1 − G 1 ( q )) . Setting A 0 and A 2 b oth initially to zero in (3.28) reduces these three equations to the single equation ∂ A 1 ∂ t = − β 1 ∂ q A 1 − 2 A 1 ∂ q β 1 . (3.32) Finally , if one assumes that G in the conv olution β 1 = G ∗ A 1 is the Green’s function for the op erator relation A 1 = (1 − α 2 ∂ 2 q ) β 1 (3.33) for a constan t lengthscale α , then the evolution equation for A 1 reduces to the in tegrable Camassa-Holm (CH) equation [CaHo1993] in the absence of linear dispersion. This is the one-dimensional EPDiﬀ equation, which has singular (p eak on) solutions. Th us, ev en very drastic restrictions of the momen t system still lead to in teresting sp ecial cases, some of which are in tegrable and p ossess emergent coherent structures among their CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 86 solutions. That such strong restrictions of the moment system leads to suc h interesting sp ecial cases bo des w ell for future inv estigations of the EPSymp moment equations. Before closing, it is useful to men tion other op en questions ab out the solution b ehavior of the momen ts of EPSymp. 3.3.7 Extending EPSymp to anisotropic in teractions An example of how the geo desic motion on the moments can b e extended to include extra degrees of freedom is provided by the work of Gibb ons, Holm and Kup ershmidt [GiHoKu1982, GiHoKu1983], where the authors consider a Vlasov distribution dep ending also on a dual Lie algebra v ariable g ∈ g ∗ undergoing Lie-P oisson dynamics in ﬁnite dimen- sions. F ollowing the treatmen t in [GiHoKu1983], tak e the purely quadratic Hamiltonian on s ∗ ( T ∗ R ⊕ g ∗ ) (with s := T e Symp) deﬁned by H [ f ] = Z Z Z f ( q , p, g ) ( G ∗ f )( q, p, g ) d q d p d 3 g with notation g = g a e a , pairing h e a , e b i = δ a b and Lie brack et [ g b , g c ] =  a bc g a ( G ∗ f )( q , p, g ) = Z Z Z G ( q , q 0 , p, p 0 , g , g 0 ) f ( q 0 , p 0 , g 0 ) d q 0 d p 0 d 3 g 0 The geo desic Vlasov equation is giv en in [GiHoKu1982] as ∂ f ∂ t = − n f , G ∗ f o 1 , where { · , ·} 1 is the sum of the canonical brack et on T ∗ R and the Lie-Poisson brack et on g ∗ , n f , k o 1 = n f , k o +  g ,  ∂ f ∂ g , ∂ k ∂ g   , in vector notation for elements of so (3) ∗ . Now, assume that the kernel G can b e expanded as G ( q , q 0 , p, p 0 , g , g 0 ) = K 0 ( q , q 0 ) + p K 1 ( q , q 0 ) p 0 + g a ¯ K ab ( q , q 0 ) g 0 b so that the quadratic Hamiltonian b ecomes H = Z ρ ( q ) ( K 0 ∗ ρ )( q ) d q + Z M ( q ) ( K 1 ∗ M )( q ) d q + Z  G ( q ) , ( ¯ K • G )( q )  d q where one deﬁnes ¯ K • G ( q ) := Z ¯ K ab ( q , q 0 ) G b ( q 0 ) d q 0 e a ∈ so (3) . The moment equations for mass density ρ ( q , t ) = R f d p d 3 g , momentum density M ( q , t ) = R pf d p d 3 g and orientation densit y G ( q , t ) = R g f d p d 3 g are presented in [GiHoKu1982]. CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 87 F or the quadratic Hamiltonian ab ov e these b ecome ∂ ρ ∂ t = − ∂ ∂ q ( ρ u ) (3.34) ∂ G ∂ t = − ∂ ∂ q ( G u ) + ad ∗ ¯ K • G G (3.35) ∂ M ∂ t = − £ u M − ρ ∂ ∂ q ( K 0 ∗ ρ ) −  G, ∂ ∂ q  ¯ K • G   (3.36) where u = K 1 ∗ M . When G ∈ F ∗ ( R ) ⊗ so (3) ∗ , then ad ∗ ¯ K • G G = − ( ¯ K • G ) × G and one recognizes the Hamiltonian part of the Landau-Lifschitz equation on the righ t hand side in the second equation ∂ G ∂ t = − ∂ ∂ q ( G u ) + G × δ H δ G . F or K 1 = (1 − ∂ 2 q ) − 1 and K 0 = δ ( q − q 0 ), this extends the Camassa-Holm system to sev eral comp onen ts. Suc h an approac h will b e also follow ed in Chapter 6 for aggregation and self-assem bly of oriented nano-particles in the con text of double brack et dissipation. Singular solutions. This section presen ts the interaction of tw o singular solutions of the equations presen ted in this section in the particular case of g = so (3) in the simple case when the densit y v ariable ρ is neglected. The result generalizes the pulson solutions to the p ossibilit y of oriente d pulsons, which ma y b e called “orientons”. One starts with the Hamiltonian H = 1 2 h M , K ∗ M i + 1 2 h G , H ∗ G i and by inserting the singular solution ansatz M ( q , t ) = X i P i ( t ) δ ( q − Q i ( t )) , G ( q , t ) = X i µ i ( t ) δ ( q − Q i ( t )) the Hamiltonian b ecomes H = 1 2 X i,j p i p j K ij + 1 2 X ij  µ i , H ij µ j  with K ij = K ( Q i − Q j ) and H ij = H ( Q i − Q j ) CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 88 equations of motions ˙ Q i = ∂ H ∂ p i = X j K ( Q i − Q j ) p j ˙ p i = − ∂ H ∂ Q i = − p i X j K 0 ( Q i − Q j ) p j − X j  µ i , H 0 ( Q i − Q j ) µ j  ˙ µ i = ad ∗ ∂ H ∂ µ i µ i = X j ad ∗ H ( Q i − Q j ) µ j µ i F or simplicity , we restrict to the case µ ∈ so (3). This do es not aﬀect the v alidit y of the follo wing res ult, whic h is true for any ﬁnite-dimensional Lie-algebra. It is straightforw ard to v erify that the orienton–orien ton system has the following eight constan ts of motion H , P = p 1 + p 2 , µ = µ 1 + µ 2 , θ ij = µ i · µ j ∀ i, j = 1 , 2 In order to prov e the conserv ation of P , take the equation for p 1 : ˙ p 1 = − p 1 p 1 ∂ ∂ q     q = Q 1 K ( Q 1 − q ) + p 2 ∂ ∂ q     q = Q 1 K ( Q 2 − q ) ! − * µ 1 , ∂ ∂ q     q = Q 1 H ( Q 1 − q ) µ 1 + ∂ ∂ q     q = Q 1 H ( Q 2 − q ) µ 2 !+ = − p 1 p 2 ∂ Q 1 K ( Q 2 − Q 1 ) − h µ 1 , ∂ Q 1 H ( Q 2 − Q 1 ) µ 2 i so that ˙ p 1 + ˙ p 2 = 0, since ∂ Q 1 K ( Q 2 − Q 1 ) = − ∂ Q 2 K ( Q 2 − Q 1 ) (analogously for H ). Also one prov es ˙ µ 1 + ˙ µ 2 = ad ∗ H 12 µ 2 µ 1 + ad ∗ H 21 µ 1 µ 2 = ad ∗ H 12 µ 2 µ 1 − ad ∗ H 12 µ 2 µ 1 = 0 . The conserv ation of θ is another simple result. This conclusion is not aﬀected by the insertion of the density v ariable ρ = R f d p d µ in the dynamics. 3.4 Op en questions for future w ork Singular solutions for EPSymp. Sev eral open questions remain for future work. The ﬁrst of these is whether the singular solutions found here will emerge sp ontaneously in EPSymp dynamics from a smooth initial Vlasov PDF. This sp ontaneous emergence of the singular solutions do es o ccur for EPDiﬀ. Namely , one sees the singular solutions of EPDiﬀ CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 89 emerging from any conﬁned initial distribution of the dual v ariable. What happ ens w ith the singular solutions for EPSymp? Will they emerge from a conﬁned smooth initial distribution, or will they only exist as an inv ariant manifold for sp ecial initial conditions? Of course, the in teractions of these singular solutions in higher dimensions and for v arious metrics and the prop erties of their collective dynamics is a question for future w ork. The same questions apply to the case of anisotropic interactions. F or example, the interaction of tw o ﬁlaments carrying an extra degree of freedom in tw o or three dimensions would be a very in teresting problem, which could also shed ligh t on the questions arising in chapter 6. Similarities with the Blo ch-Iserles equation. A ﬁnite dimensional integrable equa- tion has b een recen tly prop osed by Bloch and Iserles, which may be written in the even- dimensional case as the geo desic equation on the group of the linear canonical transforma- tions Sp( R , 2 n ) [BlIsMaRa05]. Giv en an antisymmetric matrix N , this equation is usually written on the space of symmetric matrices as ˙ X = [ X 2 , N ] where the brack et is the usual matrix commutator. On the other hand, it is w ell known that b X = N X ∈ sp ( R , 2 n ) is a Hamiltonian matrix asso ciated to the symplectic form N − 1 (if N is not inv ertible, this system is still integrable). At this p oint a Lie algebra isomorphism can b e constructed b e- t ween symmetric and Hamiltonian matrices [BlIsMaRa05], through the Lie brack et relation N [ X , Y ] N = [ b X , b Y ] with [ X, Y ] N := X N Y − Y N X The Blo ch-Iserles equation arises no w as the Euler-P oincar´ e equation on the Lie algebra [ · , · ] N of symmetric matrices, where the Lagrangian l ( X ) is given b y l ( X ) = 1 2 T r( X 2 ) By the isomorphism ab ov e, this equation is then equiv alen t to the Euler-Poincar ´ e equation on the Hamiltonian matrices sp ( R , 2 n ): th us one wonders what connections there ma y b e b et ween this equation and the geodesic Vlaso v equation (EPSymp) whic h has been proposed in this paper, given the surprisingly similar nature of these t wo equations. In particular one w onders whether in tegrability prop erties ma y arise also for EPSymp, with a certain c hoice of metric. In ﬁnite dimensions, a certain class of geo desic ﬂo ws on Lie groups is well known to CHAPTER 3. GEODESIC FLOW ON THE MOMENTS: A NEW PROBLEM 90 b e integrable from the work of Mi ˘ s˘ cenk o and F omenk o [MiF o1978]. Nev ertheless, the Bloch- Iserles system do es not b elong to the Mi ˘ s ˘ cenko-F omenk o class [BlIsMaRa05]. In inﬁnite dimensions, some important examples of geodesic ﬂo ws on Diﬀ vol (Euler’s equation) and Diﬀ (CH equation) are also integrable. Thus it is a reasonable question whether the geo desic momen t hierarc hy corresp onding to EPSymp may exhibit integrable dynamics. A p ositiv e answ er is already av ailable for the ﬂuid closure, recov ering the CH and CH-2 equations. An in vestigation of the relations b etw een the EPSymp equation and the Blo c h-Iserles system w ould b e fundamen tal to answer suc h questions. Chapter 4 GOP theory and geometric dissipation 4.1 In tro duction The approac h to a critical p oin t in free energy of a con tinuum material ma y pro duce pattern formation and self-organization. Diverse examples of such pro cesses include the formation of stars and galaxies at large scales, gro wth of colonies of organisms at mesoscales and self- assem bly of proteins or micro/nanodevices at micro- and nanoscales [Whitesides2002]. Some of these pro cesses, such as nano-scale self-assembly of molecules, are of great tec hnological in terest. Due to the large num b er of particles inv olved in nano-scale self-assembly , the dev elopment of con tin uum descriptions for aggregation or self-assembly is a natural approach to ward its theoretical understanding and modeling. This c hapter sho ws ho w suc h con tinuum descriptions may be formulated in order to allow the existence of singular solutions. A useful concept for deriving a con tinuum description of macroscopic pattern formation (e.g., aggregation) due to microscopic pro cesses is the notion of or der p ar ameter . Order parameters are con tinuum v ariables that describ e macroscopic eﬀects due to microscopic v ariations of the in ternal structure [Ho2002]. They take v alues in a vector space called the order parameter space that respects the underlying geometric structure of the microscopic v ariables. The canonical example is the description of the local directional asymmetries of nematic liquid crystal molecules b y a spatially and temp orally v arying macroscopic contin- uum ﬁeld of unsigned unit vectors called “directors”, see, e.g. Chandrasekhar [Ch1992] and 91 CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 92 de Gennes and Prost [deGePr1993]. The classic examples of contin uous equations for aggregation are those of Deby e-H ¨ uck el [DeHu1923] and Keller-Segel (KS) [KelSeg1970] for whic h the order parameter is the densit y of particles. The ph ysics of these models consists of Darcy’s la w, in tro duced in chapter 1: ∂ t ρ = div ( ρ u ), coupled with an evolution equation for velocity u whic h depends on the densit y ρ through a free energy E as u ' µ ∇ δ E /δ ρ (velocity prop ortional to force), in whic h ‘mobilit y’ µ may also depend on the densit y . The idea of a v elo city prop ortional to the force has its ro ots in the w ork of George Gabriel Stok es, who formulated his famous drag law for the resistance of spherical particles moving in a viscous ﬂuid at low Reynolds n umbers (dominance of viscous forces). It is commonly assumed that all pro cesses in ﬂuids at micro- and nano-scales are dominated b y viscous forces and the Stok es appro ximation applies. The Stok es result states that a round particle mo ving through am bient ﬂuid will exp erience a resistance force that is prop ortional to the v elo city of the particle. Conv ersely , in the absence of inertia, the v elo city of a particle will be prop ortional to the force applied to it since resistance force and applied force m ust balance. This law, that “force is proportional to velocity” is also known as Dar cy’s law . At this p oin t it is clear how dissipation and friction are key concepts in the dev elopment of this theory . As a result, friction dominate d systems describ ed by Darcy’s law exhibit aggregation and self-assembly phenomena that can b e recognized mathematically through the formation of singularities clumping together in a ﬁnite time [HoPu2005, HoPu2006]. Previous inv estigation b y Holm and Putk aradze [HoPu2007] extended Darcy’s Law to incorp orate nonlo cal, nonlinear and anisotropic eﬀects in self-organization of aggregating particles of ﬁnite size . This theory pro duces a whole family of geometric ﬂows that describe certain dissipative dynamics. In particular, the Holm-Putk aradze theory form ulates a form of ge ometric dissip ation for con tinuu m systems describing the evolution of order parameters (geometric order parameter (GOP) equations). The main goals of this chapter are • to present the author’s contribution to the Holm-Putk aradze theory; • to present a particular application to vorticit y dynamics. The ﬁrst allo ws for the existence of singular solutions in order to capture coheren t struc- tures. This is done b y introducing a spatial a veraging that follows these coheren t structures CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 93 in a Lagrangian sense. In principle, the av eraging process can b e inserted in t wo diﬀerent w ays, although only one of them allows the formation of singularities. The second scope is to present an application that prepares for the developmen ts in the next chapters. It is shown how the Euler vorticit y equation (for an exact tw o-form) can b e extended to include a Darcy-lik e dissipation term. This formalism generalizes earlier mo diﬁed ﬂuid equations of this type in Blo ch et al. [BlKrMaRa1996, BlBrCr1997] and V allis et al. [V aCaY o1989] so that the theory now allo ws for p oint v ortex solutions or vortex ﬁlamen ts (and sheets) in three dimensions. Applications of this general frame w ork also include the dynamics of geometric quantities suc h as scalars, densities (Darcy’s law), one- and tw o-forms. Each ﬂow reco vers the singular solutions. Dep ending on the geometric type of the order parameter, the space of singular solutions ma y either form an in v arian t manifold, or these solutions ma y emerge from smooth conﬁned initial conditions. In the latter case, the singular solutions dominate the long-term aggregation dynamics. F rom the physical p oint of view, suc h localized, or quenc hed solutions w ould form the core of the pro cesses of self-assembly and are therefore of great practical in terest. The formation of these localized solutions is driv en b y a combination of nonlinearit y and nonlo cality . Their ev olution admits a reduced description, expressed completely in terms of co ordinates on their singular embedded subspaces. 4.2 Theory of geometric order parameter equations 4.2.1 Bac kground: geometric structure of Darcy’s la w Darcy’s law for the geometric order parameter ρ (densit y) [HoPu2005, HoPu2006] is written in terms of an energy functional E = E [ ρ ] and a mobility µ which tak es in to account of the t ypical size of the particles in the system (in general it dep ends on ρ ). In form ulas, one has the equation ∂ ρ ∂ t = div  ρ µ [ ρ ] ∇ δ E δ ρ  . (4.1) This may be stated in terms of Lie deriv atives in t wo possible wa ys as ∂ ρ ∂ t = £ “ ρ ∇ δE δρ ” ] µ [ ρ ] or ∂ ρ ∂ t = £ “ µ [ ρ ] ∇ δE δρ ” ] ρ (4.2) where sharp ( · ) ] denotes raising the v ector index from cov ariant to con trav ariant, so its div ergence may b e taken (the sign in the right hand side is taken in agreement with the CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 94 dissipativ e nature of the dynamics, as it is sho wn in Sec. 4.2.5). The evident diﬀerence b et ween these tw o forms is that, unlik e the ﬁrst form, the second equation can b e written as the characteristic equation dρ dt ( x ( t ) , t ) = 0 along d x dt = u [ ρ ] =  µ [ ρ ] ∇ δ E δ ρ  ] (4.3) so that velocity u dep ends on density ρ through the gradient of the v ariation of the free energy E (v elo city prop ortional to thermo dynamic force with mobility µ [ ρ ]) [HoPu2005, HoPu2006, HoPu2007]. The Holm-Putk aradze (HP) theory [HoPu2007] generalizes this type of geometric ﬂow metho d underlying Darcy’s Law approac h to apply to other order parameters (denoted b y κ ) with diﬀeren t geometrical meaning (not just densities). The k ey question for understanding the ph ysical mo deling that w ould b e needed in making such a generalization is, “What is the corresp onding Darcy’s law for an order parameter κ ?” Namely , how do es one determine the corresp onding geometric ﬂo w for an arbitrary geometrical quantit y κ ? The ﬁrst problem is that there is no reason to consider only one of the tw o geometric form ulations in (4.2). Although a characteristic form w ould be preferable b ecause of its reac her geometric meaning, no choice can be p erformed a priori. As a further step in the in vestigation of the geometric structure in Darcy’s la w (4.1), one seeks a v ariational formulation of the equations (4.2), that could shed more light on how these formulations arise. Thus one takes the L 2 pairing of (4.1) with a test function φ and sets it equal to the v ariation δ E of the free energy [HoPu2006, HoPu2007]  ∂ ρ ∂ t , φ  =  δ ρ , δ E δ ρ  where the v ariation δ ρ satisﬁes δ ρ = − div  ρ µ ∇ φ  in order to recov er equation (4.1) (the calculation pro ceeds by in tegration by parts, [HoPu2005, HoPu2006]). In order to analyze the geometric structure, one needs to express the v ariational principle in terms of geometric cov ariant quantities and this leads to the same am biguity as in (4.2). Tw o p ossibilities are av ailable: δ ρ = − £ ( µ ∇ φ ) ] ρ or δ ρ = − £ ( ρ ∇ φ ) ] µ (4.4) whic h are determined by the relativ e p osition of ρ and µ in the formulas. CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 95 No w, in order to express (4.1) and (4.2) in a completely geometric co v ariant form, one writes out the integrations by parts explicitly and makes use of the diamond op er ation in tro duced in c hapter 1 (see b elo w). Up on performing the second choice in (4.4), one obtains [HoPu2007, HoPuT r2007] D ∂ ρ ∂ t , φ E =  δ ρ , δ E δ ρ  = −  £ v ( φ ) µ , δ E δ ρ  =: −  v ( φ ) , µ  δ E δ ρ  =  div µ ( ρ ∇ φ ) ] , δ E δ ρ  = −  ( ρ ∇ φ ) ] , µ ∇ δ E δ ρ  =: −  ( ρ  φ ) ] , µ  δ E δ ρ  , (4.5) while p erforming the ﬁrst choice in (4.4) switches ρ ↔ µ in the last tw o lines, so that (4.1) ma y b e written in the following geometric forms ∂ ρ ∂ t = − £ “ ρ  δE δρ ” ] µ [ ρ ] or ∂ ρ ∂ t = − £ “ µ [ ρ ]  δE δρ ” ] ρ (4.6) corresp onding to the tw o diﬀerent cases in (4.2). As in chapter 1, the third equalit y on the ﬁrst line deﬁnes the diamond (  ) op eration as the dual of the Lie deriv ative under in tegration b y parts for any pair ( κ, b ) of dual v ariables and an y v ector ﬁeld v [HoMaRa]. That is h κ  b, v i = h κ, − £ v b i . (4.7) It is readily seen how the geometric prop erties of the result in (4.5) are unchanged b y switching ρ ↔ µ and the only diﬀerence is that the second c hoice in (4.4) yields a c haracteristic equation for ρ . How ever, at this stage there is no particular reason to choose b et ween the t wo possibilities. 4.2.2 GOP equations: a result on singular solutions The arguments in the previous section show that Darcy’s law can b e applied to any tensor quan tity κ , since equations (4.6) do not dep end on the particular nature of ρ as a densit y v ariable. The Lie deriv ativ e is deﬁned for an y tensor along a generic v ector ﬁeld and thus the substitution ρ → κ is completely justiﬁed in geometric terms. Th us one obtains ∂ κ ∂ t = − £ “ κ  δE δκ ” ] µ [ κ ] or ∂ κ ∂ t = − £ “ µ [ κ ]  δE δκ ” ] κ (4.8) It is in teresting to notice that the t wo p ossibilities are iden tical when µ ∝ κ , sa y for simplicit y µ = κ . In this case one obtains a type of ge ometric or der p ar ameter e quation (GOP) CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 96 whic h is written as ∂ κ ∂ t = − £ “ κ  δE δκ ” ] κ (4.9) This equation indeed iden tiﬁes the t ype of ﬂo w for the order parameter κ arising from the geometric extension of Darcy’s law. How ever, equation (4.1) with generic mobilit y µ = µ [ ρ ] has one more feature, besides its purely geometric character. This feature is the emergence of singular solutions . F or example, in one dimension the equation (4.1) admits particle-lik e solutions of the form [HoPu2005, HoPu2006] ρ ( x, t ) = N X n =1 w n ( t ) δ ( x − Q n ( t )) corresp onding to the tra jectories of N particles in the system (one has ˙ w n = 0). The sp on- taneous emergence of this kind of solution [HoPu2005, HoPu2006] is a remark able result on its own, within the context of blow-up phenomena in nonlinear PDE’s. How ever, the b eha viour of these solutions exhibits one more interesting feature: these particle-like struc- tures merge together in ﬁnite time [HoPu2005, HoPu2006], thereb y recov ering aggregation and self-assembly phenomena. This p oint leads to the question: is it p ossible to generalize the existence of singular solutions to GOP theory? F or example, in the one dimensional case, one would expect solutions of the GOP equation for κ of the form κ ( x, t ) = N X n =1 p n ( t ) δ ( x − Q n ( t )) . It is a direct veriﬁcation that this type of solution nev er exists for an y equation of the form (4.9). Thus one is motiv ated to lo ok at one of the forms in (4.8). Upon pairing the ﬁrst equation in (4.8) with a dual element φ , direct substitution of the singular solution ansatz yields  ∂ κ ∂ t , φ  = X n ∂ p n ∂ t · φ ( Q n ( t )) + X n ∂ Q n ∂ t · φ 0 ( Q n ( t )) = −  £ “ κ  δE δκ ” ] µ , φ  = −  µ  φ,  κ  δ E δ κ  ]  = −  κ, £ ( µ  φ ) ] δ E δ κ  = − N X n =1 p n ( t ) £ ( µ  φ ) ] δ E δ κ    x = Q ( t ) where the dot symbol denotes con traction of indexes. In order for the singular solutions to exist, one w ould match terms in φ and φ 0 and obtain the ev olution equations for p n and Q n , as it happens for the densit y v ariable ρ in Darcy’s law [HoPu2005, HoPu2006]. Ho wev er, CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 97 in general the term in the last line may inv olve higher deriv ativ es, not just ﬁrst order (for example, if κ is a one-form densit y , then diamond is again a Lie deriv ative, whic h generates second order deriv ativ es in φ ). Therefore, the ﬁrst choice in (4.8) is not suitable to reco ver the singular solutions in the general case of an order parameter κ . Instead, b y follo wing the same pro cedure for the second equation in (4.8), one obtains  ∂ κ ∂ t , φ  = X n ∂ p n ∂ t · φ ( Q n ( t )) + X n ∂ Q n ∂ t · φ 0 ( Q n ( t )) = −  £ “ µ  δE δκ ” ] κ , φ  = − N X n =1 p n ( t ) £ “ µ  δE δκ ” ] φ      x = Q ( t ) (4.10) No w, from the general theory of Lie diﬀeren tiation [AbMaRa] one recognizes that the last term on the second line con tains only terms that are line ar in φ and its ﬁrst order deriv atives and do es not in volv e any higher order deriv atives of φ . Th us, in higher dimensions one ﬁnds the following conclusion [HoPuT r2007] Theorem 24 The se c ond e quation of (4.8) always al lows for singular solutions of the form κ ( x , t ) = N X n =1 Z p n ( s, t ) δ ( x − q n ( s, t )) d s (4.11) for any tensor ﬁeld κ , pr ovide d µ and δ E /δ ρ ar e suﬃciently smo oth. As in earlier chapters, the v ariable s is a co ordinate on a s ubmanifold of R 3 : if s is a one- dimensional co ordinate, then κ is s upported on a curve (ﬁlament), if s is tw o dimensional, then κ is supp orted on a surface (sheet) immersed in ph ysical space. The proof pro ceeds by direct substitution. A t this p oin t, one deﬁnes GOP equations as characteristic equations of the type dκ dt ( x ( t ) , t ) = 0 along d x dt =  µ [ κ ] ∇ δ E δ κ  ] (4.12) or, in Eulerian co ordinates, ∂ κ ∂ t = − £ “ µ [ κ ]  δE δκ ” ] κ (4.13) CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 98 The fact that Darcy’s la w (4.1) is symmetric in ρ and µ is the reason why singular solutions are recov ered for b oth p ossibilities in (4.6). This prop erty is p eculiar of Darcy’s law and do es not hold in general. The distinction b et ween the tw o cases identiﬁes the geometric structure of the GOP family of equations. Moreo ver, it is important to notice that the geometry underlying this dynamics is uniquely determined by the group of diﬀeomorphisms, whose inﬁnitesimal generator co- incides with the Lie deriv ativ e, as explained in chapter 1. Ho wev er, these equations can b e further generalized to consider diﬀer ent Lie group actions, such as the rotations S O (3) [HoPu2007]. Indeed, if κ b elongs to a generic g -mo dule V (i.e. a vector space acted on by the Lie algebra g ), then the GOP equation b ecomes ∂ κ ∂ t = −  µ [ κ ]  δ E δ κ  ] κ where ξ κ ∈ V denotes the action of the Lie algebra elemen t ξ ∈ g on the order parameter κ ∈ V and the diamond is now deﬁned as h κ  b, ξ i := h κ, ξ κ i . In order to distinguish b et ween the v arious Lie groups, the next c hapters will use diﬀerent sym b ols for the diamond op eration. The next question in the form ulation of GOP theory is the particular meaning assumed b y the gener alize d mobility µ [ κ ]. This quantit y has b een related to the typical particle size in Darcy dynamics, but the physical meaning of this quan tity is not y et clear in the case of a generic GOP equation for the order parameter κ . The next section presen ts the mobility as a smoothed quantit y that k eeps in to account the dynamics of jammed states in the system, b y introducing a typical length-scale [HoPu2005, HoPu2006, HoPu2007]. 4.2.3 More background: m ulti-scale v ariations One seeks a v ariational principle for a contin uum description of coherent structures. This includes the evolution of particles of ﬁnite size that may clump together under crowded condi- tions. In crowded conditions, ﬁnite-sized particles t ypically reach jammed states, sometimes called rafts, that ma y b e lo cally lo c ked together o ver a coherence length of several particle- size scales. Thus, a v ariational principle for the ev olution of coheren t structures suc h as jammed states in particle aggregation m ust accommo date more than one length scale. A m ulti-scale v ariational principle may b e derived by considering the v ariations as b eing ap- plied to rafts, or patc hes, of jammed states of a certain size (the coherence length). How ev er, the approac h of Holm and Putk aradze [HoPu2007] is based on applying a Lagrangian co- CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 99 ordinate a verage that mov es with the clumps of particles. In this approach, the v ariation ( δ κ ) of the order parameter ( κ ) at a given ﬁxed point in space is determined b y a family of smo oth maps ϕ ( s ) dep ending contin uously on a parameter s and acting on the av erage v alue ¯ κ deﬁned by ¯ κ = H ∗ κ = Z H ( y − y 0 ) κ ( y 0 ) dy 0 (4.14) in the frame of motion of the jammed state. Thus, y is a Lagrangian coordinate, deﬁned in that frame. This motion itself is to b e determined b y the v ariational principle. The kernel H represents the t ypical size and shap e of the coheren t structures, which in this example w ould be rafts of close-pack ed ﬁnite-size particles. The mobility µ of the rafts dep ends on ¯ κ , and so the corresp onding v ariation of the lo cal quantit y κ at a ﬁxed p oint in space may b e mo deled as δ κ = d ds    s =0  µ ( ¯ κ ) ϕ − 1 ( s )  . (4.15) Here ϕ ( s ) y = x ( s ) is a p oint in space, which ϕ − 1 ( s ) returns to its Lagrangian lab el y and ϕ (0) is the identit y op eration. The av erage ¯ κ = H ∗ κ is applied in a L agr angian sense, follo wing a locally lo c ked raft of particles along a curv e parameterized b y time t in the family of smo oth maps. The latter represen ts the motion of the raft as ϕ ( t ) y = x ( t ), whose velocity tangen t vector is still to b e determined. When comp osed from the right the deriv ative at the iden tity of the action of ϕ ( s ) results in a v ariation δ κ at a ﬁxed p oin t in space given by δ κ = − £ v ( ϕ ) µ [ κ ] with v ( ϕ ) = ϕ 0 ϕ − 1   s =0 , (4.16) thereb y recov ering the prop er expression for the v ariation, i.e. the second equation of (4.4). In the GOP theory of Holm and Putk aradze [HoPu2007] µ [ κ ] is a general functional of κ , not just a function of ¯ κ . 4.2.4 Properties of the diamond op eration The diamond op er ation  is deﬁned in (4.7) for Lie deriv ativ e £ η acting on dual v ariables a ∈ V and b ∈ V ∗ ( V b eing a vector space) b y h b  a , η i ≡ − h b , £ η a i =: − h b , a η i , (4.17) where Lie deriv ativ e with resp ect to right action of the diﬀeomorphisms on elements of V is also denoted b y concatenation on the right. The diamond op erator takes tw o dual CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 100 quan tities a and b and pro duces a quantit y dual to a vector ﬁeld, i.e. , a a  b is a one-form densit y . In abstract notation  : V × V ∗ → X ∗ . The  op eration is also known as the “dual represen tation” of this right action of the Lie algebra of vector ﬁelds on the represen tation space V [HoMaRa]. When paired with a vector ﬁeld η , the diamond op eration has the follo wing three useful prop erties [HoMaRa, HoPu2007]: 1. It is antisymmetric h b  a + a  b , η i = 0 . 2. It satisﬁes the pro duct rule for Lie deriv ative h £ ξ ( b  a ) , η i = h ( £ ξ b )  a + b  ( £ ξ a ) , η i . 3. It is antisymmetric under in tegration by parts h db  a + b  da , η i = 0 . These three properties of  are useful in computing the explicit forms of the v arious geometric ﬂo ws for order parameters (4.5). Of course, when the order parameter is a density undergoing a gradient ﬂo w, then one recov ers Darcy’s law (4.1) from (4.13). 4.2.5 Energy dissipation in GOP theory As mentioned in the ﬁrst section of this chapter, the ph ysical nature of Darcy’s law resides in energy dissipation and friction [HoPu2005, HoPu2006, HoPu2007]. Th us, a faithful gen- eralization in the context of GOP theory needs to accommo date energy dissipation. This allo ws the in tro duction of the dissip ation br acket [HoPu2007], so that the equations can b e written in an alternativ e brack et form. The corresponding energy equation follows from (4.13) as dE dt = D ∂ κ ∂ t , δ E δ κ E =  − £ ( µ [ κ ]  δE δκ ) ] κ, δ E δ κ  = −   µ [ κ ]  δ E δ κ  ,  κ  δ E δ κ  ]  . (4.18) Holm and Putk aradze [HoPu2007] observed that equation (4.18) deﬁnes the follo wing brack et notation for the time deriv ative of a functional F [ κ ], dF [ κ ] dt = D ∂ κ ∂ t , δ F δ κ E =  − £ ( µ [ κ ]  δE δκ ) ] κ , δ F δ κ  = −   µ [ κ ]  δ E δ κ  ,  κ  δ F δ κ  ]  =: { { E , F } } [ κ ] (4.19) CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 101 The prop erties of the GOP brack ets { { E , F } } deﬁned in equation (4.19) are determined b y the diamond op eration and the choice of the mobility µ [ κ ]. F or physical applications, one should choose a mobility that satisﬁes strict dissipation of energy , i.e. { { E , E } } ≤ 0 . A particular example of mobility that satisﬁes the energy dissipation requirement is µ [ κ ] = κM [ κ ], where M [ κ ] ≥ 0 is a non-negativ e scalar functional of κ [HoPu2007]. (That is, M [ κ ] is a num b er.) Requiring the mobility to pro duce energy dissipation do es not limit the mathematical prop erties of the GOP brac ket. F or example, Holm and Putk aradze show ed that the dissipative brac ket p ossesses the Leibnitz prop erty with any c hoice of mobility [HoPu2007]. That is, it satisﬁes the Leibnitz rule for the deriv ativ e of a pro duct of functionals Prop osition 25 (Leibnitz prop erty [HoPu2007]) The GOP br acket (4.19) satisﬁes the L eibnitz pr op erty. That is, it satisﬁes {{ E F , G }} [ κ ] = F {{ E , G }} [ κ ] + E {{ F , G }} [ κ ] for any functionals E , F and G of κ . Pro of. F or arbitrary scalar functionals E and F of κ and any smooth vector ﬁeld η , the Leibnitz prop ert y for the functional deriv ative and for the Lie deriv ativ e together imply  µ   δ ( E F ) δ κ  , η  =  µ   E δ F δ κ + F δ E δ κ  , η  =  µ , − £ η  E δ F δ κ + F δ E δ κ   = E  µ , − £ η δ F δ κ  + F  µ , − £ η δ E δ κ  = E  µ  δ F δ κ , η  + F  µ  δ E δ κ , η  Cho osing η =  κ  δ G δ κ  ] then prov es the prop osition that the brack et (4.19) is Leibnitz. In addition, the dissipativ e brac ket form ulation (4.19) allo ws one to reformulate the GOP equation (4.13) in terms of ﬂow on a Riemannian manifold with a metric deﬁned through the dissipation brack et. The follo wing section reviews the results in [HoPu2007]. Connection to Riemannian geometry . F ollowing [Ot2001], Holm and Putk aradze [HoPu2007] used their GOP brac ket to in tro duce a metric tensor on the manifold connecting a “v ector” CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 102 ∂ t κ and “co-vector” δ E /δ κ . That is, one expresses the ev olution equation (4.13) in the w eak form as  ∂ κ ∂ t , ψ  = − g κ  δ E δ κ , ψ  (4.20) for an arbitrary elemen t ψ of the space dual to the κ space, and where g κ ( · , · ) is a symmetric p ositiv e deﬁnite function – metric tensor – deﬁned on vectors from the dual space. First one notice that for any choice of mobilit y pro ducing a symmetric brack et (in par- ticular, µ [ κ ] = κM [ κ ]) { { E , F } } = { { F , E } } , so that that symmetric brack et deﬁnes an inner pro duct b et ween the functional deriv ativ es [HoPu2007], { { E , F } } =: − g κ  δ E δ κ , δ F δ κ  = −  µ  δ E δ κ ,  κ  δ F δ κ  ]  (4.21) Alternativ ely , (4.21) can be understo o d as a symmetric p ositive deﬁnite function of tw o elemen ts of dual space φ, ψ : g κ  φ , ψ  =  µ  φ , ( κ  ψ ) ]  . (4.22) Notice that g ( φ, φ ) ≥ 0, since { E , E } ≤ 0 and these arguments maybe summarized in the follo wing Prop osition 26 (Metric prop erty [HoPu2007]) F or the choic e of metric tensor (4.22), the GOP e quation (4.13) may b e expr esse d as the metric r elation (4.20). This approach harnesses the p ow erful machinery of Riemannian geometry to the math- ematical analysis of the GOP equation (4.13). This op ens a wealth of p ossibilities, but it also limits the analysis to mobilities µ for which the GOP brack et (4.19) is symmetric and p ositiv e deﬁnite, as in the mo deling choice µ [ κ ] = κM [ κ ]. Previous dissipativ e brack ets. Historically , the use of symmetric brac kets for in tro duc- ing dissipation into Hamiltonian systems seems to ha ve originated with works of Grmela [Gr1984], Kaufman [Ka1984] and Morrison [Mo1984]. See [Ot05] for references and further engineering developmen ts. This approach in tro duces a sum of tw o brack ets, one describing the Hamiltonian part of the motion and the other obtained by represen ting the dissipation with a symmetric brac ket op eration inv olving an en tropy deﬁned for that purp ose. Being CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 103 expressed in terms of the diamond op eration (  ) for an arbitrary geometric order param- eter κ , the dissipative brack et in equation (4.19) diﬀers from symmetric brack ets prop osed in earlier work. The geometric advection law (4.13) for the order parameter will b e shown b elo w to arise from thermo dynamic principles that naturally yield the dissipative brac ket (4.19). Moreov er, b eing written as a Lie deriv ative, the equation of motion (4.13) respects the geometry of the transported quan tity . The dissipativ e brac k ets from the earlier literature do not app ear to b e expressible as a geometric transp ort equation in Lie deriv ative form. 4.2.6 A general principle for geometric dissipation Equations (4.13) ma y b e justiﬁed b y more general principles. Consider using an arbitrary functional F in (4.19) as a basis for the deriv ation of an equation for κ . Supp ose κ is an observ able quan tity for a physical system, and that system evolv es due to the inherent free energy E [ κ ] in the absence of external forces. This is the ph ysical picture one envisions, for example, when thinking ab out pro cesses of self-assem bly in nanotechnology . Supp ose one would like to measure the time ev olution of a functional F [ κ ], which may for example represen t as total energy or total momentum. F or an arbitr ary functional F [ κ ] and for a given free energy E [ κ ], the GOP brack et yields dF dt =  ∂ t κ , δ F δ κ  =  δ κ , δ E δ κ  = δ E . (4.23) The main postulate here is that, in principle, one can determine the ev olution of the system indirectly by probing many diﬀeren t glob al quantities F [ κ ] (for example, the moments of a probabilit y distribution). It is only natural to assume that the law for the evolution of κ should b e indep enden t of the choice of whic h quantities F are used to determine it. Surprisingly , this rather general sounding assumption sets severe restrictions on the na- ture of the v ariation δ κ . In particular, 1. The v ariation δ κ must b e linear in δ F /δ κ , since the left hand side of (4.23) is also linear in δ F /δ κ . 2. The v ariation δ κ must transform the same wa y as κ , as it must b e dual to δ F /δ κ . This introduces the mobility µ that m ust b e of the same type as κ . 3. The v ariation δ κ m ust sp ecify a quan tity at the tangent space to the space of all p ossible κ . The prop er geometric wa y to sp ecify this quan tity is through the Lie CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 104 deriv ativ e £ v with resp ect to some vector ﬁeld v . There are only tw o wa ys to sp ecify δ κ so that it ob eys these three thermo dynamic and geometric constraints, when one insists that only a single new physical quantit y µ [ κ ] is in tro duced. Namely , either δ κ = − £ v κ with v = −  µ [ κ ]  δ F δ κ  ] , (4.24) or δ κ = − £ v µ [ κ ] with v = −  κ  δ F δ κ  ] . (4.25) Both of (4.24) and (4.25) are consistent with all three geometric and thermo dynamics re- quiremen ts. Ho wev er, the ﬁrst p ossibility (4.24) prev ents formation of measure-v alued solu- tions in κ , when κ is c hosen to be a 1-form, a 2-form or a v ector ﬁeld. In contrast, the second p ossibilit y (4.25) yields the c onservation law (4.13), which is a characteristic equation admit- ting measure-v alued solutions for an arbitrary geometric quantit y κ . The remainder of this c hapter deals with (4.25) and in vestigates the corresponding ev olution equation (4.12). The alternativ e c hoice (4.24) would ha v e rev ersed the roles of κ and µ [ κ ] in the Lie deriv ative. 4.3 Review of scalar GOP equations The fundamen tal example is an active scalar, for which κ = f is a function. F or this particular example, the exposition follows the w ork by Holm and Putk aradze [HoPu2007]. The evolution of a scalar b y equation (4.13) ob eys ∂ t f = − £ ( µ [ f ]  δE δf ) ] f = −  δ E δ f ∇ µ [ f ]  ] · ∇ f . (4.26) Equation (4.26) can b e rewritten in characteristic form as d f /dt = 0 along d x /dt =  δ E δ f ∇ µ [ f ]  ] . (4.27) The characteristic sp eeds of this equation are nonlo c al when δ E /δ f and µ are c hosen to dep end on the aver age value , ¯ f . It is interesting that suc h problems arise commonly in the theory of quasi-geostrophic con vection and may lead to the developmen t of singularities in ﬁnite time [Co2003, Chae2005, Cordoba2005]. Explicit equations for the evolution of strengths p n and coordinates q n for a sum of δ -functions in (4.11) may b e deriv ed using (4.10) when µ [ f ] = H ∗ f = ¯ f . The singular CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 105 solution parameters satisfy [HoPu2007] ∂ p n ( t, s ) ∂ t = p n ( t, s ) div  δ E δ f ∇ µ [ f ]  ]     x = q n ( t,s ) (4.28) p n ( t, s ) ∂ q n ( t, s ) ∂ t = p n ( t, s )  δ E δ f ∇ µ [ f ]  ]     x = q n ( t,s ) (4.29) for n = 1 , 2 , . . . , N . F or the c hoice µ [ f ] = ¯ f , a solution containing a single δ -function satisﬁes ˙ p = − Ap 3 , so an initial condition p (0) = p 0 , evolv es according to 1 /p ( t ) 2 = 1 /p 2 0 + 4 α 2 t [HoPu2007]. 4.4 New GOP equations for one-forms and tw o-forms 4.4.1 Results on singular solutions As particular examples, this section develops nonlo cal characteristic equations for the evo- lution of one- and tw o-forms. So one sp ecializes equation (4.13) to consider the diﬀerential 1-form κ = A · d x and the 2-form κ = B · d S in three-dimensional space. F or this, one b egins by computing the the Lie deriv ativ e and the diamond operation for these cases. In Euclidean co ordinates, the Lie deriv atives for these t wo c hoices of κ are: − £ v ( A · d x ) = −  ( v · ∇ ) A + A j ∇ v j  · d x = ( v × curl A − ∇ ( v · A )) · d x , − £ v ( B · d S ) = − d  v ( B · d S )  − v d ( B · d S ) = − d  ( v × B ) · d x  − v (div B d 3 x ) = (curl ( v × B ) − v div B ) · d S Both of these expressions are familiar from ﬂuid dynamics, particularly magnetoh ydro dy- namics (MHD). F rom these formulas for Lie deriv ative in v ector form and the deﬁnition of diamond in equation (4.7), one computes explicit expressions for the diamond op eration with 1-forms and 2-forms,  µ [ A ]  δ E δ A , u  =  δ E δ A × curl µ [ A ] − µ [ A ] div δ E δ A , u   µ [ B ]  δ E δ B , u  =  µ [ B ] × curl δ E δ B − δ E δ B div µ [ B ] , u  for any v ector ﬁeld u . CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 106 The explicit forms of the GOP equations (4.13) are ∂ A ∂ t = − ∇ ( v 1 · A ) + v 1 curl A , v 1 :=  δ E δ A × curl µ [ A ] − µ [ A ] div δ E δ A  ] (4.30) ∂ B ∂ t = curl ( v 2 × B ) − v 2 div B , v 2 :=  µ [ B ] × curl δ E δ B − δ E δ B div µ [ B ]  ] (4.31) and in vector notation one has the following result Prop osition 27 The ge ometric or der p ar ameter e quations (4.30) and (4.31) for any one- forms A and any two-form B have singular solutions of the form (4.11), wher e ˙ q a ( t, s ) = v 1 ( x ) | x = q a ˙ p a ( t, s ) = p a ( t, s ) ( ∇ · v 1 ( x )) | x = q a − ∇ v 1 ( x ) | x = q a · p a ( t, s ) (4.32) for close d one-forms A and ˙ q a ( t, s ) = v 2 ( x ) | x = q a ˙ p a ( t, s ) = p a T ( t, s ) · ∇ v 2 ( x ) | x = q a for close d two-forms B . Pro of. Consider equation (4.30) for A . Pairing this equation with a smo oth vector ﬁeld φ , substituting the singular solution ansatz (4.11), integrating b y parts where necessary and matc hing all terms in φ on the tw o sides yields equations (4.32) for the q ’s and p ’s. The result for closed 2-forms is prov en b y noticing that curl ( v 2 × B ) = B T · ∇ v 2 − v 2 T · ∇ B − B div v 2 + v 2 div B then following the same steps as for the case of exact 1-forms. 4.4.2 Exact diﬀerential forms When considering the GOP equations (4.13) for exact diﬀerential forms (curl A = 0 = div B ), the singular solutions also exist satisfy analogous relations. One may see this, by follo wing the same procedure. An imp ortan t simpliﬁcation in this case is to tak e curl µ [ A ] = 0 = div µ [ B ]. In this case the GOP equations for A and B take the simpler form ∂ A ∂ t = ∇ A ·  µ [ A ] div δ E δ A  ] ! (4.33) ∂ B ∂ t = − curl B ×  µ [ B ] × curl δ E δ B  ] ! (4.34) CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 107 where the expressions for v 1 and v 2 ha ve be en inserted explicitly . Moreo ver, for exact one- and tw o-forms, the v ector equations abov e can b e reduced to nonlo cal nonlinear scalar c haracteristic equations (4.26) for the potentials [HoPu2007]. Note that in R 3 (whic h is of interest to us here) every closed form is exact since curl A = 0 giv es A = ∇ ψ for some scalar ψ and div B = 0 necessitates B = curl C for some vector C . The c haracteristic equations for the p otentials are deriv ed in the following Prop osition 28 (GOP equations for scalar potentials [HoPu2007]) The ve ctor e qua- tions (4.33) and (4.34) for exact 1-forms A = ∇ ψ and exact 2-forms B = curl (Ψ ˆ z ) ar e e quivalent to sc alar GOP e quations of the typ e (4.26), in terms of the p otentials ψ and Ψ . Sp e ciﬁc al ly, one ﬁnds ∂ ψ ∂ t =  δ E δ ψ ∇ ϑ [ ψ ]  ] · ∇ ψ , (4.35) and ∂ Ψ ∂ t =  δ E δ Ψ ∇ Φ[Ψ]  ] · ∇ Ψ , (4.36) wher e one deﬁnes µ [ A ] := ∇ ϑ [ ψ ] and µ [ B ] := curl (Φ[Ψ] ˆ z ) . Pro of. Inserting the expression A = ∇ ψ in eq. (4.33) yields ∂ ψ ∂ t =  µ [ A ] div δ E δ A  ] · ∇ ψ =  ∇ ϑ [ ψ ] δ E δ ψ  ] · ∇ ψ with nonlo cal δ E /δ ψ and µ [ ψ ]. Similarly , the evolution of 2-form ﬂuxes B · d S = B x d y ∧ d z + B y d z ∧ d x + B z d x ∧ d y also simpliﬁes, when B = ∇ Ψ × ˆ z where Ψ only dep ends on t wo spatial co ordinates ( x, y ). Then, curl δ E δ B = ˆ z δ E δ Ψ . and µ [ B ] × curl δ E δ B = ( ∇ Φ × ˆ z ) × ˆ z δ E δ Ψ = − δ E δ Ψ ∇ Φ . Equation (4.26) ma y b e written for the stream function Ψ (remo ving the curl from b oth sides of (4.34)) ˆ z ∂ Ψ ∂ t = −  δ E δ Ψ ∇ Φ  ] × B (4.37) Then, simpliﬁcation of tw o cross pro ducts leads to ∂ Ψ ∂ t =  δ E δ Ψ ∇ Φ  ] · ∇ Ψ . (4.38) CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 108 Hence, c ho osing δ E /δ Ψ and Φ to dep end on the av erage v alue ¯ Ψ again yields a nonlo cal c haracteristic equation. 4.4.3 Singular solutions for exact forms and their p oten tials Equations (4.35) and (4.36) do allo w singular δ -like solutions of the form (4.11) for ψ and Ψ. These solutions, ho wev er, lead to δ 0 -lik e singularities in the forms A and B . One ma y understand this p oin t by deriving the expressions for ψ and Ψ corresp onding to the clump on solutions of the form (4.11) for A and B . F or example, taking the div ergence of an exact one-form A = ∇ ψ yields ∇ · A = ∆ ψ . Up on using the Green’s function of the Laplace op erator G ( x , y ) = − | x − y | − 1 , an expression for ψ emerges in terms of A : ψ ( x , t ) = − Z ∇ x 0 G ( x , x 0 ) · A ( x 0 , t ) d x 0 . Inserting the singular solution (4.11) for A then yields ψ ( x , t ) = − X i Z ds P i ( s, t ) · ∇ Q i G ( x , Q i ( s, t )) . Ho wev er, this singular solution for the p otential is not in the same form as (4.11), since the singularities for ψ do not manifest themselves as δ -functions. A similar pro cedure applies to the case of exact tw o-forms B ( x, y ) = curl(Ψ( x, y ) ˆ z ), so that curl B = ∆ (Ψ ˆ z ). One has Ψ( x ) = ˆ z · X i Z ds P i ( s, t ) × ∇ Q i G ( x , Q i ( s, t )) , where Q is in the plane ( x, y ). Th us, the equations (4.35) and (4.36) for ψ and Ψ allo w for tw o sp ecies of singular solutions. One of them tak es the form (4.11), while the other corresp onds to a δ -like solution of the same form (4.11) for A and B . A deep er explanation of this fact can b e given in a general context as follo ws. Consider the advection equation for an exact form κ = dλ , with p otential λ ( ∂ t + £ u ) dλ = 0 . A t this point, one remembers that the exterior diﬀerential comm utes with the Lie deriv ativ e so that the equation for the potential λ is again an advection equation with the same c haracteristic v elo city ( ∂ t + £ u ) λ = 0 CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 109 A t this p oint, one obtains singular δ -like solutions of the form (4.11) for b oth κ and λ (pro vided the characteristic v elo cit y u is suﬃciently smooth). 4.5 Applications to v ortex dynamics 4.5.1 A new GOP equation for ﬂuid v orticit y The developmen ts ab ov e pro duce an interesting opportunity for the addition of dissipation to ideal ﬂuid equations. This opp ortunit y arises from noticing that the dissipative diamond ﬂo ws that were just deriv ed could just as well be used with any t yp e of evolution operator, not just the Eulerian partial time deriv ative. F or example, if one chooses the geometric order parameter κ to b e the exact tw o-form ω = ω · d S app earing as the v orticity in Euler’s equations for incompressible motion with ﬂuid velocity u , then the GOP equation (4.13) with Lagrangian time deriv ativ e may b e introduced as a mo diﬁcation of Euler’s v orticity equation as follows, ∂ t ω + £ u ω | {z } Euler = £ ( µ [ ω ]  δE δω ) ] ω | {z } Dissipation . (4.39) Euler’s v orticity equation is reco vered when the left hand side of this equation is set equal to zero. This mo diﬁed geometric form of v orticity dynamics supports p oint v ortex solutions, requires no additional boundary conditions, and dissipates kinetic energy for the appropriate c hoices of µ and E . Equation (4.39) will b e derived after making a few remarks ab out the geometry of the vorticit y gov erned by Euler’s equation. The Lie-Poisson structure of the vorticit y equation as been presen ted in chapter 1 and it is written as ∂ t ω = − ad ∗ δ H/δ ω ω = curl  ω × curl ∆ − 1 ω  = curl  ω × curl ψ  In order to write the GOP evolution equation (4.13) for ω one must compute the diamond op eration  for the ad ∗ action, which is deﬁned in terms of Lie deriv ativ e b y ad ∗ ψ ω = £ curl ψ ω . (4.40) The computation of the  operation follows from its deﬁnition in equation (4.7). F or an y t wo v elo city v ector p otentials φ and ψ , and an exact tw o form ω one ﬁnds h φ  ω , ψ i = − h φ , £ curl ψ ω i =  φ , curl  ω × curl ψ  (4.41) = h curl φ × ω , curl ψ i =  curl  curl φ × ω  , ψ  . (4.42) CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 110 Consequen tly , up to addition of a gradient, the diamond op eration is giv en in vector form as φ  ω = curl  curl φ × ω  . (4.43) The insertion of this expression in the brack et (4.19) gives the GOP equation for ω , ∂ t ω = curl  ω × curl curl  µ [ ω ] × curl δ E δ ω  . (4.44) Consequen tly , equation (4.39) emerges in the equiv alen t forms, ∂ t ω = − ad ∗ ψ ω + ad ∗ (ad ∗ ψ µ [ ω ]) ] ω = curl  ω × curl  − δ H δ ω + curl  µ [ ω ] × curl δ E δ ω   . (4.45) The full dynamics for the v orticity in equation (4.39) is sp eciﬁed up to the c hoices of the mobilit y µ [ ω ] and the energy in the dissipativ e brack et E [ ω ]. By deﬁnition, the mobility b elongs to the dual space of volume-preserving v ector ﬁelds whic h is here iden tiﬁed with exact t wo-forms, thus one can write the mobilit y in terms of its v ector p otential as µ = curl λ and rewrite the GOP equation (4.44) as ∂ t ω = curl  ω × curl curl  λ , δ E δ ω  (4.46) This equation raises questions concerning the dynamics of vortex ﬁlaments with nonlo cal dissipation, follo wing the ideas in [Ho03], where connections w ere established betw een the Marsden-W einstein brack et [MaW e83] and the Rasetti-Regge brack et for vortex dynamics [RaRe]. Ideas for dissipative brac ket descriptions in ﬂuids hav e b een in tro duced previously , see Blo ch et al. [BlKrMaRa1996, BlBrCr1997] and references therein. In particular, equation (4.45) recov ers equations (2.2-2.3) of V allis et al. [V aCaY o1989] when E = H and µ = α ω for a constant α . Remark 29 (Coadjoint dissipative dynamics) F r om the ﬁrst line in e quation (4.45), one se es that the vorticity dynamics is a form of c o adjoint motion (cf. chapter 1). Inde e d, the e quation c an also b e written as ∂ ω ∂ t = − ad ∗  ψ − “ ad ∗ ψ µ [ ω ] ” ]  ω which shows how the vorticity evolves on c o adjoint orbits of the gr oup Diﬀ vol , gener ate d by the Lie algebr a element ψ −  ad ∗ ψ µ [ ω ]  ] . In p articular, the Casimir functionals c orr esp onding CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 111 to Hamiltonian dynamics ar e also pr eserve d by the ge ometric dissip ation. This observation is a key step in the the ory of ge ometric dissip ation and it le ads to the pr eservation of entr opy when this the ory is applie d to kinetic e quations (cf. chapter 5). The GOP equation (4.39) may b e expressed as the Lie-deriv ativ e relation for conserv ation of vorticit y ﬂux, ∂ t ( ω · d S ) = − £ u − v ( ω · d S ) , (4.47) in which the velocities u and v may b e written in terms of the com m utator [ · , · ] of div er- genceless vector ﬁelds as, u = curl δ H δ ω , v = curl curl ( µ [ ω ] × ˜ u ) = curl  µ [ ω ] , ˜ u  where ˜ u = curl δ E δ ω , (4.48) The compact form (4.47) clearly underlines the dissipativ e nature of the dynamics, for whic h the transp ort velocity u is appropriately decreased by the nonlocal dissipative v elo city v . Since b oth u and v are div ergenceless, the v orticity equation (4.47) may also be expressed as a commutator of div ergenceless vector ﬁelds, denoted as [ · , · ], ∂ t ω + ( u − v ) · ∇ ω − ω · ∇ ( u − v ) = ∂ t ω + [ u − v , ω ] = 0 . (4.49) Th us, the v orticity is advected b y the total v elo city ( u − v ) and is stretched by the total v elo city gradient. In this form one recognizes that the singular v ortex ﬁlament solutions of (4.49) will mov e with the total velocity ( u − v ), instead of the Biot-Sav art v elo city ( u = curl − 1 ω ) alone. 4.5.2 Results in tw o dimensions: p oint v ortices and steady ﬂo ws The GOP equation (4.39) for vorticit y including b oth inertia and dissipation takes the same form as the Euler vorticit y equation in two dimensions , but with a mo diﬁed stream function. Indeed, by a standard calculation with stream functions in tw o dimensions, equations (4.48) and (4.47) imply the following dynamics, expressed in terms of ω := ˆ z · ω and µ := ˆ z · µ ∂ t ω +  ω , ψ − [ µ [ ω ] , ˜ ψ ]  = 0 , (4.50) where ψ = δ H/δ ω , ˜ ψ = δ E /δ ω and [ f , g ] is the symplectic brac ket, giv en for motion in the ( x, y ) plane by the t wo-dimensional Jacobian determinan t,  f , g  dx ∧ dy = d f ∧ dg . (4.51) Equation (4.50) takes the same form as Euler’s equation for v orticity , but with a modiﬁed stream function, now giv en by the sum ψ − [ µ, ˜ ψ ]. CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 112 Remark 30 The GOP e quation for vorticity in two dimensions (4.50) r e c overs e quation (4.3) of V al lis et al. [V aCaY o1989] when one cho oses µ = αω for a c onstant α and E = 1 2 R ω ψ dxdy . However, for this choic e of mobility, µ , p oint vortex solutions ar e exclude d. Prop osition 31 (Poin t v ortices) The GOP e quation for vorticity in two dimensions (4.50) p ossesses p oint vortex solutions, with any choic es of µ [ ω ] and ˜ ψ for which K = ψ − [ µ [ ω ] , ˜ ψ ] is suﬃciently smo oth. Pro of. Pairing equation (4.50) with a stream function η yields h η , ∂ t ω i =   η , K [ ω ]  , ω  where K [ ω ] = ψ − [ µ [ ω ] , ˜ ψ ] (4.52) Inserting the expression ω ( x, y , t ) = Γ( t ) δ ( x − X ( t )) δ ( y − Y ( t )) in to the previous equation and integrating against a smo oth test function yields ˙ Γ η + Γ ˙ X ∂ η ∂ X + Γ ˙ Y ∂ η ∂ Y = Γ ∂ η ∂ X ∂ K ∂ Y − Γ ∂ K ∂ X ∂ η ∂ Y , where η and K are ev aluated at the p oint ( x, y ) = ( X ( t ) , Y ( t )). Thus, the p oint vortex solutions for equation (4.50) on the ( X , Y ) plane satisfy ˙ Γ = 0 , ˙ X = ∂ K ∂ Y , ˙ Y = − ∂ K ∂ X , (4.53) whose solutions exist provided the function K is suﬃciently smooth. Remark 32 Solutions of the symple ctic Hamiltonian system (4.53) extend for the c ase of evolution of arbitr ary many p oint vortic es for the GOP vorticity e quation (4.50) in two dimensions. These solutions r epr esent a set of N vortic es at p ositions ( X k ( t ) , Y k ( t )) ( k = 1 , . . . , N ) moving in the plane. Pr op erties of the c orr esp onding p oint vortex solutions of Euler’s e quations in the plane ar e discusse d for example in [Sa1992]. Steady states of the dissipativ e v orticit y equation in 2D. The tw o dimensional v er- sion of the vorticit y equation pro vides a simple opportunity for inv estigating the stationary solutions. F or example, it is obvious that the equation  ω , ψ − [ µ [ ω ] , ˜ ψ ]  = 0 , CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 113 is alwa ys satisﬁed when ψ − [ µ [ ω ] , ˜ ψ ] = Φ( ω ), where Φ is a function of the vorticit y ω . In fact, the chain rule yields  ω , Φ( ω )  = ω x Φ 0 ( ω ) ω y − ω y Φ 0 ( ω ) ω x = 0 . This is an evident consequence of the fact that geometric dissipation preserves the coad- join t nature of the Hamiltonian ﬂow thereby recov ering the Casimir functionals C [ ω ] = R Φ( ω ) d x d y . How ever, more can b e said ab out the relation o ccurring b etw een the steady ﬂo ws of the Hamiltonian dynamics and those corresponding to geometric dissipation. Indeed the observ ation ab o ve means that Prop osition 33 If [ µ [ ω ] , ˜ ψ ] dep ends only on ω (say e Φ( ω ) = [ µ [ ω ] , ˜ ψ ] ), and if ¯ ω is a sta- tionary state of the Hamiltonian vorticity e quation (so that [ ¯ ω , ψ ] = 0 ), then the equilibria of the dissipative ﬂo w will coincide with those of the Hamiltonian ﬂow and the level sets of ω and ψ wil l evolve until they c oincide. The same holds if [ µ [ ω ] , ˜ ψ ] = M [ ω ] [ ω , ψ ] , wher e M [ ω ] is a pur e functional of ω and if [ µ [ ω ] , ˜ ψ ] = α [ ω , ψ ] . Remark 34 (Extending to kinetic equations) The validity of this statement c an b e ex- tende d to other systems under going ge ometric dissip ation. The ne c essary c ondition is that the Hamiltonian ﬂow c orr esp onding to the dissip ative system under c onsider ation under go es c o adjoint dynamics, so that Casimir functionals ar e known. An example is pr ovide d in chap- ter 5, wher e the ge ometric dissip ation is applie d to kinetic e quations. The e quilibria of the r esulting dissip ative Vlasov e quation ar e explained exactly by the prop osition ab ov e , which applies in this c ase up on substituting the vorticity ω with the distribution function f on phase sp ac e. 4.5.3 More results in three dimensions As a consequence of the mo diﬁed vorticit y equation (4.49) in comm utator form, one easily c hecks the follo wing prop erties. • Ertel’s the or em is satisﬁed b y the vector ﬁeld ω · ∇ asso ciated to v orticity . By using the commutator notation and the material deriv ativ e D /D t , one can write D α D t := ∂ α ∂ t + ( u − v ) · ∇ α = ω · ∇ α , so that  D D t , ω · ∇  α = 0 , (4.54) for any scalar function α ( x , t ). CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 114 • An analogue of the Kelvin’s circulation theorem holds for equation (4.49). Up on expressing the v orticity as ω = curl u , one writes the follo wing dissipative form of the Euler equation for the velocity u ∂ t u + ( u − v ) · ∇ u − u j ∇ v j = −∇ p , ∇ · u = 0 , (4.55) where v is given in (4.48). This equation may also b e expressed as ∂ t u + u · ∇ u + ∇  p + u · v  = − v × curl u | {z } V ortex force , ∇ · u = 0 , (4.56) b y using a vector iden tity . An equiv alen t alternativ e is the Lie deriv ative form, ( ∂ t + £ u − v ) ( u · d x ) = − d p . (4.57) Hence, one ﬁnds that a mo diﬁe d cir culation the or em is satisﬁed, d dt I C ( u − v ) u · d x = 0 (4.58) for a loop C ( u − v ) moving with the “total” velocity u − v . That is, tw o v elo cities app ear in the mo diﬁed circulation theorem. One is the “transp ort velocity” u − v and the other is the “transp orted velocity” u . • F rom equations (4.58) and (4.47) one c hecks that ( ∂ t + £ u − v ) ( ω · d S ∧ u · d x ) = − ω · d S ∧ d p = − div ( p ω ) d 3 x (4.59) so that the helicity of the vorticity ω is c onserve d d dt Z Z Z V ol ω · u d 3 x = 0 (4.60) One may summarize these remarks as follo ws: All of these classical geometric results for ideal incompressible ﬂuid mec hanics follo w for the modiﬁed Euler equation. These results all persist (including preserv ation of helicit y) when transp ort velocity is replaced as ( u − v ) → − v . This completes the presen t inv estigation of GOP v ortex dynamics. An obvious extension w ould b e to consider GOP vortex patc hes in tw o dimensions. Instead of pursuing such GOP v orticity considerations further, the next section applies GOP theory to diﬀeren t well kno wn cases in contin uum Hamiltonian mechanics. CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 115 4.6 Tw o more examples The developmen ts discussed ab ov e pro duce an interesting opp ortunity for the addition of dissipation to v arious other contin uum equations. F ollowing the introduction of the dissi- pativ e Euler equation ab ov e, one could extend the dissipativ e diamond ﬂows with any t yp e of ev olution op erator. This section sk etches how one might develop this idea further, b y illustrating its application in three more physically relev ant examples. 4.6.1 Dissipative EPDiﬀ equation Consider adding geometric dissipation to the Euler-Poincar ´ e equation on the diﬀeomor- phisms (EPDiﬀ ) [HoMa2004] for the evolution of a one-form density m deﬁned b y m = m · d x ⊗ d 3 x . (4.61) This addition gives the dissip ative EP e quation , ∂ t m + ad ∗ δ H/δ m m = − £  µ [ m ]  δE /δ m  ] m = − ad ∗  µ [ m ]  δE /δ m  ] m . (4.62) When H [ m ] is the Hamiltonian for the Lie-P oisson theory corresp onding to EPDiﬀ, then the v ector ﬁeld δ H /δ m = u is the characteristic v elo city for the Euler-P oincar´ e equation. F or a one-form density m , the diamond op eration is given b y ad ∗ , which is equiv alent to Lie deriv ativ e. That is, µ [ m ]  δ E /δ m = ad ∗ δ E /δm µ [ m ] = £ δ E /δm µ [ m ] . (4.63) The further choice µ [ m ] = αm for a p ositiv e constant α reco vers equation (6.10) of Blo ch et al. [BlKrMaRa1996]. When in addition, µ [ m ] = K ∗ m for a smoothing k ernel K , then equation (4.62) supp orts singular solutions of the type discussed in Holm and Marsden [HoMa2004]. P eakon dynamics for the dissipative Camassa-Holm equation. In one dimension, the GOP version of the EPDiﬀ equation (4.62) reduces to, ∂ t m + ( u − v ) m x + 2 m ( u − v ) x = 0 , (4.64) where u = δ H /δ m for a speciﬁed Hamiltonian H [ m ]. The other v elo city v is given in one dimension by v =  ad ∗ δ E /δm µ [ m ]  ] = δ E δ m ∂ x µ [ m ] + 2 µ [ m ] ∂ x δ E δ m , (4.65) CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 116 for arbitrary (smo oth) c hoices of µ [ m ] and E [ m ]. Now consider the singular solution form for m given b y a sum of N delta functions, m ( x, t ) = N X i =1 p i ( t ) δ ( x − q i ( t )) , (4.66) and take quadratic functionals H [ m ] = 1 / 2 h m, G ∗ m i and E [ m ] = 1 / 2 h m, W ∗ m i so that u ( x ) = G ∗ m = N X j =1 p j G ( x − q j ) and, since µ [ m ] = K ∗ m , v ( x ) = W ∗ m ∂ x K ∗ m + 2 K ∗ m ∂ x W ∗ m = N X j,k =1 p j p k  K ( x − q j ) ∂ x W ( x − q k ) + 2 W ( x − q k ) ∂ x K ( x − q j )  = N X j,k =1 p j p k R ( x − q j , x − q k ) where one deﬁnes R for compactness of notation. Substituting the ab ov e expressions into the GOP EPDiﬀ equation (4.64) and in tegrating against a smooth test function yields the follo wing relations for time deriv atives of p i ( t ) and q i ( t ): ˙ q i =  u ( x ) − v ( x )     x = q i ( t ) (4.67) = N X j =1 p j G ( q i − q j ) − N X j,k =1 p j p k R ( q i − q j , q i − q k ) , ˙ p i = − p i  u 0 ( x ) + v 0 ( x )     x = q i ( t ) (4.68) = − p i N X j =1 p j ∂ q i G ( q i − q j ) + p i N X j,k =1 p j p k ∂ q i R ( q i − q j , q i − q k ) , The choices of H , E and µ as functionals of m determine the ensuing dynamics of the singular solutions. In particular, in the case when the v elo city u is given b y u [ m ] = δ H δ m = (1 − α 2 ∂ 2 x ) − 1 m = Z e − | x − x 0 | α m ( x 0 )d x 0 , for H = 1 2 Z m u [ m ] d x . (4.69) Equation (4.64) is a GOP version of the integrable Camassa-Holm equation with p eaked soliton solutions [CaHo1993]. Nonlinear interactions of N trav eling wa ves of this system ma y b e in vestigated b y following the approach of F ringer and Holm [F rHo2001]. CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 117 Remarks on dissipative semidirect product dynamics. The equations derived abov e consolidate the idea that any con tinuum equation in c haracteristic form, ( ∂ t + £ u ) κ = 0 , ma y b e mo diﬁed to include dissipation via the substitution u → u + v , in which v is the dissipativ e velocity term expressed in equation (4.13). This idea ma y also b e extended to the semidirect pro duct framew ork presented in [HoMaRa], in order to include compressible ﬂuid ﬂows and plasma ﬂuid mo dels such as the barotropic ﬂuid model or MHD. Instead of constructing GOP equations for such structures, chapter 5 derives the barotropic ﬂuid equations from moment dynamics in the kinetic approach. Once the structure of these equations is identiﬁed, the same can b e applied to other examples such as MHD. 4.6.2 Dissipative Vlaso v dynamics One ma y also extend the diamond dissipation framew ork to systems suc h as the Vlaso v equation in the symplectic framew ork of coordinates and momen ta as independent v ariables. This extension requires the in tro duction of the Vlaso v Lie-P oisson brack et, deﬁned for phase space densities f ( q , p ) d q ∧ d p on T ∗ R N as { F , H } ( f ) = Z Z f  δ F δ f , δ H δ f  dq ∧ dp . (4.70) where the brack et { · , · } in the integral is the canonical Poisson brac ket. At this p oint a new deﬁnition of the diamond op erator is required. This is found by the well known iden tiﬁcation of Hamiltonian vector ﬁelds and their generating functions, whic h iden tiﬁes the symplectic Lie algebra action on the Vlaso v distribution and the new kind of symple ctic diamond. This treatment is extensively presented in chapter 5 and introduces the idea of a microscopic description for Darcy’s la w. This section rep orts the ﬁnal result. Extending the previous discussions to the symplectic case, one can write the follo wing form of GOP dissipativ e Vlaso v equation, ∂ f ∂ t +  f , δ H δ f  =  f ,  µ [ f ] , δ E δ f   (4.71) where, in general, the functionals H and E are independent. This equation has the same form as the equations for a dissipative class of Vlasov plasmas in astrophysics, proposed by Kandrup [Ka1991] to mo del gravitational radiation reaction. Kandrup’s form ulation for an azim uthally symmetric particle distribution is reco vered by c ho osing a linear phase space CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 118 mobilit y µ = αf with positive constant α and taking E to b e J z [ f ] the total azimuthal angular momentum for the Vlaso v distribution f . More generally , if one c ho oses µ [ f ] = α f and E to b e the Vlaso v Hamiltonian H [ f ], the dissipative Vlasov equation (4.71) assumes the double br acket form, ∂ f ∂ t +  f , δ H δ f  = α  f ,  f , δ H δ f   . (4.72) This is also the Vlasov-P oisson equation in Blo c h et al. [BlKrMaRa1996]. Ho wev er, in con trast to the c hoices in [BlKrMaRa1996, Ka1984, Mo1984, Ka1991], the GOP form of the Vlasov equation (4.71) allo ws more general mobilities such as µ [ f ] = K ∗ f (which denotes con volution of f with a smo othing kernel K ). The GOP choice has the adv antage of reco vering the one-particle solution as its singular solution. The inv estigation of this equation and the consequen t kinetic theory is the sub ject of chapter 5, which will presen t imp ortan t connections with the theory of double br acket dissip ation and will show how a geometric form of dissipation can b e introduced for kinetic moments. 4.7 Discussion This chapter has pro vided a contribution to the GOP theory of Holm and Putk aradze (HP) [HoPu2007] for the construction of dissipative e v olutionary equations in the form (4.13) for a v ariety of diﬀeren t t yp es of geometric order parameters κ . As a result, the HP method now pro duces a plethora of fascinating singular solutions for these evolutionary GOP equations. Eac h GOP equation is expressed as a characteristic equations in a certain geometric sense. Ho wev er, the c haracteristic velocities in these equations may b e nonlo c al . That is, the c haracteristic v elo cities may dep end on the solution in the en tire domain. The equations ma y p ossess either or b oth of the follo wing structures: (i) a conserv ativ e Lie-P oisson Hamiltonian structure; (ii) a dissipative Riemannian metric structure. The tw o t yp es of evolution are com bined by simply adding the c haracteristic velocities in their Lie deriv ativ es. Similar t yp es of equations were discussed by Blo ch et al. [BlKrMaRa1996] who studied the eﬀects on the stability of equilibrium solutions of con tinuum Lie-Poisson Hamiltonian systems of adding a t ype of geometric dissipation that preserv es the coadjoin t orbits of the Hamiltonian systems. Suc h equations hav e the form dF dt = { F , H } − {{ F , H }} CHAPTER 4. GOP THEOR Y AND GEOMETRIC DISSIP A TION 119 for tw o brack et operations, one antisymmetric and Poisson ( { F , H } ) and the other symmetric and Leibnitz ( {{ F , H }} ). The GOP theory has b een sho wn to apply in a num b er of contin uum ﬂows with geometric order parameters, each allowing singular solutions. The v arious types of singular solutions include p oin t vortices, v ortex ﬁlaments and sheets, solitons and single particle solutions for Vlaso v dynamics. In some cases, the singular solutions emerge from smo oth conﬁned initial conditions [HoPu2005, HoPu2006, HoPu2007]. In other cases, suc h emergent b ehavior does not o ccur. It remains an open question to determine whether the singular solutions of a giv en geometric t yp e will emerge from smooth initial conditions. In particular, the existence of a “steep ening lemma” [CaHo1993, HoPu2005, HoPu2006] (cf. c hapter 1) for a certain class of order pa- rameters, the GOP equations w ould guarantee the emergence of singularities in ﬁnite time for some choices of energy E and mobility µ [ κ ]. F or example, one may conjecture that this prop ert y is actually v alid in the case of the dissipativ e EPDiﬀ equation (one-form densities) and the formulation of a “steep ening lemma” w ould b e necessary to prov e this conjecture. After they are created, the singular solutions ev olve with their o wn dynamics. Inv estigations of the interactions of these singular solutions and the t yp es of motions av ailable to them will b e discussed in the remainder of this work. The present c hapter has derived the dynamical equations for these singular solutions in v arious cases. Chapter 5 Geometric dissipation for kinetic equations 5.1 In tro duction Non-linear dissipation in ph ysical systems can mo deled b y the sequen tial application of t wo P oisson brack ets, just as in magnetization dynamics [Gilb ert1955]. A similar double brac ket op eration for mo deling dissipation has b een prop osed for the Vlasov equation. Namely , ∂ f ∂ t +  f , δ H δ f  = α  f ,  f , δ H δ f  , (5.1) where α > 0 is a p ositive constant, H is the Vlasov Hamiltonian and {· , ·} is the canon- ical Poisson brack et. When α → 0, this equation reduces the Vlaso v equation for colli- sionless plasmas. F or α > 0, this is the double br acket dissip ation approach for the Vlaso v-Poisson equation introduced in Kandrup [Ka1991] and developed in Blo ch et al. [BlKrMaRa1996]. This double brac ket approach for introducing dissipation in to the Vlaso v equation diﬀers from the standard F okker-Planc k linear diﬀusive approac h [F okker-Plank1931], represen ted b y the equation ∂ f ∂ t +  f , δ H δ f  = ∂ ∂ p  γ p + D ∂ ∂ p  f ! whic h adds dissipation on the right hand side as the Laplace op erator in the momentum co ordinate ∆ p f . An interesting feature of the double brack et approach is that the resulting symmetric brac ket gives rise to a metric tensor and an associated Riemannian (rather than symplectic) 120 CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 121 geometry for the solutions, as explained in c hapter 4. The v ariational approac h also preserv es the adve ctive nature of the evolution of Vlaso v phase space density , b y coadjoint motion under the action of the canonical transformations on phase space densities. As Otto [Ot2001] explained, the geometry of dissipation ma y b e understoo d as emerging from a v ariation principle. This c hapter follo ws the v ariational approac h to deriv e the fol- lo wing generalization of the double brack et structure in equation (5.1) that reco vers previous cases for particular choices of modeling quantities [HoPuT r2007-CR], ∂ f ∂ t +  f , δ H δ f  =  f ,  µ ( f ) , δ E δ f   . (5.2) Eq. (5.2) extends the double brac ket op eration in (5.1) and reduces to it when H is identical to E and µ ( f ) = α f . The form (5.2) of the Vlaso v equation with dissipation allo ws for more general mobilities than those in [BlKrMaRa1996, Ka1991, Ka1984, Mo1984]. F or example, one may choose µ [ f ] = K ∗ f (in which ∗ denotes conv olution in phase space). As in [HoPuT r2007] the smo othing operation in the deﬁnition of µ ( f ) introduces a fundamental length scale (the ﬁlter width) into the dissipation mechanism. Smo othing also has the adv an tage of endowing (5.2) with the one-particle solution as its singular solution. The generalization Eq. (5.2) ma y also be justiﬁed by using a thermo dynamic and geometric argumen ts [HoPuT r2007]. In particular, this generalization extends the classic Darcy’s law (v elo city b eing proportional to force) to allow the corresponding modeling at the microscopic statistical level. 5.1.1 History of double-brac ket dissipation Blo c h, Krishnaprasad, Marsden and Ratiu ([BlKrMaRa1996] abbreviated B KMR) observed that linear dissipativ e terms of the standard Rayleigh dissipation t yp e are inappropriate for dynamical systems undergoing coadjoin t motion. Such systems are expressed on the duals of Lie algebras and they commonly arise from v ariational principles deﬁned on tangent spaces of Lie groups. A w ell kno wn example of coadjoin t motion is provided b y Euler’s equations for an ideal incompressible ﬂuid [Ar1966]. Not unexpectedly , adding linear viscous dissipation to create the Na vier-Stokes equations breaks the coadjoin t nature of the ideal ﬂo w. Of course, ordinary viscosity do es not suﬃce to describ e dissipation in the presence of orientation- dep enden t particle in teractions. CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 122 Restriction to coadjoin t orbits requires nonlinear dissipation, whose gradient structure diﬀers from the Rayleigh dissipation approac h leading to Na vier-Stokes viscosity . As a fa- miliar example on which to build their paradigm, BKMR emphasized a form of energy dissipation (Gilbert dissipation [Gilb ert1955]) arising in models of ferromagnetic spin sys- tems that preserves the magnitude of angular momentum. In the con text of Euler-P oincar ´ e or Lie-P oisson systems, this means that coadjoint orbits remain in v ariant, but the energy decreases along the orbits. BKMR discov ered that their geometric construction of the non- linear dissipative terms summoned the double brack et equation of Brock ett [Br1988, Br1993]. In fact, the double brack et form is well adapted to the study of dissipative motion on Lie groups since it was originally constructed as a gradient system [Br1994]. While a single P oisson brac k et operation is bilinear and antisymmetric, a double brac ket op eration is a symmetric op eration. Symmetric brac kets for dissipativ e systems, partic- ularly for ﬂuids and plasmas, were considered previously by Kaufman [Ka1984, Ka1985], Grmela [Gr1984, Gr1993a, Gr1993b], Morrison [Mo1984, Mo1986], and T urski and Kaufman [T uKa1987]. The dissipative brack ets introduced in BKMR were particularly motiv ated by the double brac ket op erations in tro duced in V allis, Carnev ale, and Y oung [V aCaY o1989] for incompressible ﬂuid ﬂows. 5.1.2 The origins: selectiv e decay h yp othesis One of the motiv ations for V allis et al. [V aCaY o1989] was the sele ctive de c ay hyp othesis , whic h arose in turbulence research [MaMo1980] and is consisten t with the preserv ation of coadjoin t orbits. According to the selective decay hypothesis, energy in strongly nonequi- librium statistical systems tends to deca y muc h faster than certain other ideally conserved prop erties. In particular, energy deca ys muc h faster in such systems than those “kinematic” or “geometric” prop erties that would hav e b een preserved in the ideal nondissipative limit indep endently of the choic e of the Hamiltonian . Examples are the Casimir functions for the Lie-P oisson form ulations of v arious ideal ﬂuid mo dels [HoMaRaW e1985]. The selective decay hypothesis w as inspired b y a famous example; namely , that enstroph y deca ys m uch more slo wly than kinetic energy in 2D incompressible ﬂuid turbulence [Kr1967]. In 2D ideal incompressible ﬂuid ﬂo w the enstrophy (the L 2 norm of the vorticit y) is preserved on coadjoint orbits. That is, enstrophy is a Casimir of the Lie-Poisson brack et in the Hamiltonian formulation of the 2D Euler ﬂuid equations. V allis et al. [V aCaY o1989] c hose a form of dissipation that w as expressible as a double Lie-Poisson brack et. This c hoice of CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 123 dissipation preserv ed the enstrophy and thereby enforced the selectiv e decay hypothesis for all 2D incompressible ﬂuid solutions, laminar as well as turbulen t. Once its dramatic eﬀects were recognized in 2D turbulence, selective decay w as p osited as a go verning mechanism in other systems, particularly in statistical b ehavior of ﬂuid systems with high v ariabilit y . F or example, the slo w decay of magnetic helicit y w as p opularly in vok ed as a p ossible means of obtaining magnetically conﬁned plasmas [T a86]. Lik ewise, in geoph ysical ﬂuid ﬂows, the slo w decay of p oten tial vorticit y (PV) relativ e to kinetic energy strongly inﬂuences the dynamics of w eather and climate patterns muc h as in the in verse cascade tendency in 2D turbulence. The use of selectiv e deca y ideas for PV thinking in meteorology and atmospheric science has b ecome standard practice since the fundamen tal w ork in [HoMcRo1985, Y o1987]. A form of selectiv e deca y based on double-brac k et dissipation is also the basis of equation (5.1), proposed in astrophysics by Kandrup [Ka1991] for the purp ose of mo deling gravita- tional radiation of energy in stars. In this case, the double-brack et dissipation pro duced rapidly gro wing instabilities that again had dramatic eﬀects on the solution. The form of double-brac ket dissipation prop osed in Kandrup [Ka1991] is a strong motiv ation for the presen t w ork and it also play ed a central role in the study of instabilities in BKMR. 5.2 Double brac k et structure for kinetic equations 5.2.1 Background review This section starts by reviewing the ideas on geometric dissipative terms for conserv ation la ws formulated in chapter 4. Supp ose that on ph ysical grounds one knows that a certain quan tity κ is conserved, i.e., dκ ( x , t ) /dt = 0 on d x /dt = u , where u is the velocity of particle constituting the contin uum at the given point x . The nature of the conserv ation la w dep ends on the geometry of the conserved quan tity κ and the conserv ation law ma y b e alternativ ely written in the Lie Deriv ativ e form ∂ t κ + £ u κ = 0. The ph ysics of the problem dictates the nature of the quan tity κ . In order to close the system, an expression for u m ust b e established. In the treatment for geometric order parameters, one takes the inspiration from self-organization phenomena and pattern formation of spherical particles. In this case one relates the v elo city to densit y using the Darcy’s la w that establishes a linear dependence of the local particle velocity u and force acting on the particle ∇ δ E /δ ρ as u = µ [ ρ ] ∇ δ E /δ ρ . Here, E [ ρ ] is the total energy of the system in a given conﬁguration and δ E /δ ρ is the CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 124 p oten tial at a given point. In mathematical terms, the generalization to any geometric order parameter arises from the action of a Lie algebra g on some vector space V . A frequent example of suc h an action is the Lie deriv ative, that is the basis for any order parameter equation on conﬁguration space. Giv en a tensor κ on the conﬁguration space Q and an element ξ ∈ X of the Lie algebra X of vector ﬁelds, the action of ξ on κ is deﬁned as ξ κ := £ ξ κ . The imp ortance of the Lie deriv ativ e in conﬁguration space is given b y the fact that any geometric quantit y ev olves along the integral curves of some velocity v ector ﬁeld whose explicit expression dep ends only on the ph ysics of the problem. At this p oint the diamond op eration is deﬁned as the dual op erator to Lie deriv ativ e. More precisely , given a tensor ζ dual to κ , one deﬁnes h κ  ζ , ξ i := h κ , − £ ξ ζ i , so that κ  ζ ∈ X ∗ . Once this operation has b een deﬁned, the general equation for an order parameter κ is written as ∂ t κ + £ u κ , where u is called Dar cy’s velo city and is given b y u = ( µ  δ E /δ ρ ) ] . This chapter aims to mo del dissipation in Vlasov kinetic systems through a suitable generalization of Darcy’s law. Indeed, it is reasonable to b eliev e that the basic ideas of Darcy’s Law in conﬁguration space can b e transferred to a phase space treatmen t giving rise to the kinetic description of self-organizing collisionless m ultiparticle systems. The main issue here is to accurately consider not only the geometry of particle distribution, but also the structure of the phase space itself. As is well known, the prop erties of the phase space (momen tum and p osition) are completely diﬀeren t from the conﬁguration space (p osition only) b ecause of the symplectic relation betw een the momentum and p osition. This structure of the phase space w arrants a suitable modiﬁcation of the diamond operator. The follo wing sections will construct kinetic equations for geometric order parameters that resp ect the symplectic nature of the phase space by considering the Lie algebra of generating functions of canonical transformations (symplectomorphisms). 5.2.2 A new m ultiscale dissipative kinetic equation The ﬁrst step is to establish ho w a geometric quantit y evolv es on phase space, so that the symplectic nature is preserved. F or this, one regards the action of the symplectic algebra as an action of the generating functions h on κ , rather then an action of vector ﬁelds. Here κ CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 125 is a tensor ﬁeld ov er the phase space. The action is formally expressed as h κ = £ X h κ . The dual op eration of the action (here denoted by ? ) is then deﬁned as h κ ? ζ , h i = h κ, − £ X h ζ i . Here X h ( q , p ) is the Hamiltonian v ector ﬁeld generated by a Hamiltonian function h ( q , p ) through the deﬁnition X h ω := dh . Notice that the star op eration tak es v alues in the space F ∗ of phase space densities κ ? ζ ∈ F ∗ . In the particular case of in terest here, κ is the phase space densit y κ = f dq ∧ dp and ζ = g , a function on phase space. In this case, the star op eration is simply the canonical Poisson brac ket, κ ? g = { f , g } dq ∧ dp . It it p ossible to employ these considerations to ﬁnd the pur ely dissip ative part of the kinetic equation for a particle densit y on phase space. T o this purp ose, one c ho oses v ariations of the form δ f = − £ X h ( φ ) µ ( f ) = − { µ ( f ) , h ( φ ) } with h ( φ ) = ( f ? φ ) ] = { f , φ } where ( · ) ] in ( f ? φ ) ] transforms a phase space densit y to a scalar function. The op eration ( · ) ] will b e understo o d in the pairing b elow. One then follows the steps:  φ, ∂ f ∂ t  =  δ E δ f , δ f  =  δ E δ f , −  µ ( f ) , h ( φ )  = *  µ ( f ) , δ E δ f  ,  f , φ  + = − * φ,  f ,  µ ( f ) , δ E δ f  + . Therefore, a functional F ( f ) satisﬁes the following ev olution equation in brack et nota- tion [HoPuT r2007-CR], dF dt =  ∂ f ∂ t , δ F δ f  = − *  µ ( f ) , δ E δ f  ,  f , δ F δ f  + =: { { E , F } } . (5.3) The mobility µ and dissipation energy functional E app earing in (5.3) are mo deling c hoices and must b e selected based on the additional input from ph ysics. The brac ket (5.3) reduces to the dissipative brack et in Blo c h et al. [BlKrMaRa1996] for the mo deling choice of µ ( f ) = α f with some α > 0. In this case the dissipation energy E was taken to be the Vlasov Hamiltonian (see b elow), but in the present approach it also can be taken as a CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 126 mo deling c hoice. This extra freedom allows for more ph ysical in terpretation and treatment of the dissipation. Prop osition 35 Ther e exist choic es of mobility µ [ f ] for which the br acket (5.3) dissip ates ener gy E . Pro of. The dissipativ e brac ket in equation (5.3) yields ˙ E = { { E , E } } whic h is negativ e when µ [ f ] is chosen appropriately . F or example, µ [ f ] = f M [ f ], where M [ f ] ≥ 0 is a non- negativ e scalar functional of f . (That is, M [ f ] is a num b er.) Remark 36 The dissip ative br acket (5.3) satisﬁes the L eibnitz rule for the derivative of a pr o duct of functionals. In addition, it al lows one to r eformulate the e quation (5.2) in terms of ﬂow on a Riemannian manifold with a metric deﬁne d thr ough the dissip ation br acket, as discusse d in mor e detail in chapter 4. 5.3 Prop erties and consequences of the mo del 5.3.1 GOP theory and double brac k et dissipation: bac kground The previous section has sho wn how the GOP theory can b e applied to kinetic equations if one considers the symplectic structure of Vlasov dynamics. The result is a kinetic equation in double brack et form. At this p oin t one may wonder what is meant by “double brack et” in rigorous mathematical terms. The preceding discussion has presented the double brack et as simply the comp osition of t wo Poisson brack ets and the reason is that this comp osition alw ays yields a quantit y of deﬁnite sign, so that the energy functional can b e taken to decrease monotonically in time. Ho wev er, the double brack et structure has deep geometric ro ots, in particular for dissipative systems whose ideal limit can b e written in Lie-Poisson form. Chapter 4 presented the GOP brack et and presen ted its application to several cases, but some of them turn out to b e more sp ecial then others. Indeed, the ideal limit of the GOP equations for v orticity and one-form densities reduce to w ell kno wn Lie-P oisson systems: the Euler and EPDiﬀ equations. On the other hand, for densities and diﬀerential forms, the GOP equations cannot b e reduced to non-dissipative cases without obtaining trivial dynamics κ t = 0. In order to b etter understand the geometric structure of a Lie-Poisson double brack et equation, one starts with GOP theory and observ es that the Lie algebra action on the Lie CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 127 algebra g itself is alwa ys given b y ξ η = ad ξ η so that the corresp ondent diamond operation is given b y h µ  η , ξ i = h µ, − ad ξ η i = h µ, ad η ξ i =  ad ∗ η µ, ξ  that is, the diamond op eration is giv en by the inﬁnitesimal coadjoin t op erator ad ∗ . Inserting this result in the GOP brack et yields for the geometric order parameter κ ∈ g ∗ dF dt = { { E , F } } : = − *  µ ( κ )  δ E δ κ  ] , κ  δ F δ κ + = − *  ad ∗ δ E δ κ µ ( κ )  ] , ad ∗ δ F δ κ κ + = − * κ, " δ F δ κ ,  ad ∗ δ E δ κ µ ( κ )  ] # + . This is the Lie-Poisson double br acket structur e and one easily recognizes the Lie- P oisson form, which becomes evident b y taking a Hamiltonian functional H [ κ ] suc h that δ H δ κ =  ad ∗ δ E δ κ µ ( κ )  ] . Of course, the Hamiltonian H is not the energy of the dissipativ e system under consideration, whic h is instead giv en b y the energy functional E . Rather it is the conserved energy of another system that has ph ysically nothing to do with the original one. (It should b e noticed that the existence of such Hamiltonian is not certain: it is p ossible that this do es not even exist.) Also, the new Lie-Poisson system is alwa ys left-inv ariant. In fact, the sign in the brack et do es not dep end on whether the Lie algebra action is left or right, since the signs cancel b ecause of the product of t wo ad ∗ terms in the brack et. The sign dep ends only on the requirement that the original system dissipates energy: in verting the sign yields a monotonic increase of the functional E . Remark 37 (Double brack et form ulation of T oda lattice) It is worth noticing that the T o da lattic e also has a double br acket formulation on the sp e cial line ar algebr a sl ( R , n ) of r e al matric es [BlBrR a92]. In this c ase, the double br acket is explicitly given by {{ E , F }} = − T r  A, δ E δ A  T  A, δ F δ A  ! wher e the op er ator [ · , · ] denotes the c ommutator of two matric es. CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 128 This structur e has b e en extende d at the c ontinuum level in [BlFlR a95]. In this c ase the double br acket is formal ly the same as (5.3), although the c anonic al Poisson br acket is c alculate d on new c o or dinates ( z , θ ) ∈ [0 , 1] × [0 , 2 π [ that r epr esent the c o or dinates on the annulus. In this sense, this structur e b e c omes r elate d to the ar e a pr eserving diﬀe omorphisms of the annulus. Remark 38 (Double brack et and complex maps) The double br acket structur e also app e ars in the study of c omplex maps. Inde e d, let f : M → C , with M a symple ctic manifold. Then the fol lowing e quation app e ars [Donaldson1999] in minimizing the norm E = || f || 2 = R | f | 2 d x d y ∂ f ∂ t = − 1 2 { f , { f , f ∗ }} wher e {· , ·} is now the c anonic al Poisson br acket in ( x, y ) . 5.3.2 A ﬁrst consequence: conserv ation of en tropy F rom the arguments in the previous section is no w clear that any Double brac k et Lie-P oisson system can b e written as ∂ κ ∂ t + ad ∗ δ H δ κ κ = ad ∗ „ ad ∗ δE δκ µ ( κ ) « ] κ or, in more compact form ∂ κ ∂ t + ad ∗ Γ[ κ ] κ = 0 with Γ[ κ ] := δ H δ κ −  ad ∗ δ E δ κ µ ( κ )  ] ∈ g . F or the Vlasov equation, this b ecomes ∂ f ∂ t + n f , Γ[ f ] o = 0 with Γ[ f ] := δ H δ f −  µ ( f ) , δ E δ f  . No w, from the Lie-Poisson theory of the Vlaso v equation suc h an equation is known to p ossess the following property Prop osition 39 (Casimir functionals) F or an arbitr ary smo oth function Φ the functional C Φ = R Φ( f ) is pr eserve d for any ener gy functional E . CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 129 Pro of. It suﬃces to calculate the brack et dC Φ dt = {{ C Φ , E }} : = − *  µ ( f ) , δ E δ f  ,  f , δ C Φ δ f  + (5.4) = − *  µ ( f ) , δ E δ f  ,  f , Φ 0 ( f )  + = 0 . (5.5) An imp ortan t corollary follows, concerning the en tropy functional [HoPuT r2007-CR]: Corollary 40 The entr opy functional S = R f log f is pr eserve d by the dynamics in e quation (5.2) for any ener gy functional E . This result can appear surprising b ecause the ma jor part of dissipative contin uum sys- tems in volv e an increase of en tropy , basically connected with the Bro wnian motion of the particles that constitute the system. This Brownian motion yields diﬀusion pro cesses and con tinuous particle tra jectories, whic h are far from b eing diﬀeren tiable. This is the rea- son why the single particle tra jectory cannot b e a solution of the contin uum description. Moreo ver, in the mathematical description, Brownian motion is related via the Langevin sto c hastic equation to a source of noise that represen ts a loss of information in the system. Basically , one in tro duces a Langevin force in the single particle tra jectory that ﬁnally leads to the Laplace op erator. The microscopic noise is the reason why the entrop y functional is monotonically increasing in time and therefore the information on particle paths is deﬁnitely lost. Ho wev er, the double brack et Vlaso v equation is not related with Bro wnian motion and it is constructed in a completely deterministic fashion, so that no diﬀusion process is in volv ed in the kinetic description. T o see this, it suﬃces to write the double brack et Vlasov ﬂo w as coadjoin t motion in the form f ( t ) = Ad ∗ g − 1 ( t ) f (0) with g ( t ) = e t Γ[ f ] . This relation well enlightens the geometric nature of the motion, whic h is purely giv en b y the group action of the symplectic group on its (dual) Lie algebra. Ho wev er, not only is this of mathematical importance, but it also has important ph ysical implications. In fact, this form of dissipation yields a completely r eversible dynamics and it is clear how in verting the group element at eac h time gives the rev ersed time ev olution. In this sense, the rev ersibility CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 130 of dynamics yields the conserv ation of the en tropy functional. Imp ortan tly , this fact is not related with the single particle paths, which ma y or may not b e a solution of the equation. F or example, the preserv ation of entrop y is shared by Kandrup’s dynamics ( µ ( f ) = α f ). How ever, the ev olution under Kandrup’s equation do es not allo w single particle solutions. The absence of the single particle solution might appear as a common element b et ween Kandrup’s equation and the usual diﬀusiv e F okker-Plank approac h. How ever it is not p ossible to establish such a relation, since diﬀusiv e pro cesses destro y the geometric nature of the dynamical v ariable and the microscopic physics under- lying the t wo approaches is very diﬀerent. Also, the existence of single particle paths as a solution of the equation may alwa ys b e allo wed in the double brac ket equation b y in tro- ducing the mobilit y on phase space, which is nothing but a smo othed version of the Vlaso v distribution. This smo othing pro cess yields the singular δ -like solutions representing the single particle tra jectories, as it is shown in the next section. Prop osition 41 (cf. [HoPuT r2007-CR]) V ariations of the form δ f = − £ X h ( φ ) f = − [ f , h ( φ )] with h ( φ ) = µ ( f ) ? φ = [ µ ( f ) , φ ] in (5.3) yield the dissip ative double br acket dF dt = − *  f , δ E δ f  ,  µ ( f ) , δ F δ f  + =: { { E , F } } . with µ ( f ) ↔ f switche d in the c orr esp onding entries with r esp e ct to (5.3). This br acket yelds entr opy dynamics of the form dS dt = {{ S, E }} = − * µ ( f ) f , ( f ,  f , δ E δ f  )+ 6 = 0 . Pro of. One rep eats the calculation for deriving (5.3) and insert the new v ariation δ f = − £ X h ( φ ) f = − [ f , h ( φ )] to obtain  φ, ∂ f ∂ t  =  δ E δ f , δ f  =  δ E δ f , −  f , h ( φ )  = *  f , δ E δ f  ,  µ ( f ) , φ  + = − * φ,  µ ( f ) ,  f , δ E δ f  + . The evolution of the entrop y functional is obtained by direct substitution of its expression S = R f log f as follows dS dt = {{ S, E }} = − *  f , δ E δ f  ,  µ ( f ) , log f  + = − * µ ( f ) f , ( f ,  f , δ E δ f  )+ . CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 131 Remark 42 F or entr opy incr e ase, this alternative variational appr o ach would r e quir e µ ( f ) and E ( f ) to satisfy an additional c ondition (e.g., µ ( f ) /f and δ E /δ f functional ly r elate d). However, the Vlasov dissip ation induc e d in this c ase would not al low the r eversible single- p article solutions, c onsistently with the loss of information asso ciate d with entr opy incr e ase. 5.3.3 A result on the single-particle solution The discussion from the previous sections produces an interesting opp ortunity for the ad- dition of dissipation to kinetic equations. This opp ortunity arises from noticing that the dissipativ e brack et deriv ed here could just as well be used with any t yp e of ev olution op er- ator. In particular, one may consider in tro ducing a double brack et to modify Hamiltonian dynamics as in the approach b y Kaufman [Ka1984] and Morrison [Mo1984]. In particu- lar, the dissipated energy may naturally b e asso ciated with the Hamiltonian arising from the corresp onding Lie-Poisson theory for the evolution of a particle distribution function f . Therefore, it is p ossible to write the total dynamics generated b y any functional F ( f ) as ˙ F = { F , H } + {{ F, E }} where {· , ·} represents the Hamiltonian part of the dynamics. This gives the dissip ative Vlasov e quation of the form (5.2) with E = H , where H ( f ) is the Vlasov Hamiltonian. T o illustrate these ideas it is worth while to compute the singu- lar (measure-v alued) solution of equation (5.2), which represen ts the rev ersible motion of a single particle [HoPuT r2007-CR]. Theorem 43 T aking µ ( f ) to b e an arbitr ary function of the smo othe d distribution ¯ f = K ∗ f for some kernel K al lows for single p article solutions f = P N i =1 w i δ ( q − Q i ( t )) δ ( p − P i ( t )) . The single p article dynamics is governe d by c anonic al e quations with Hamiltonian given by H =  δ H δ f −  µ ( f ) , δ H δ f  ( q ,p )=( Q i ( t ) ,P i ( t )) Pro of. One writes the equation of motion (5.2) in the following compact form ∂ f ∂ t = − { f , H } , with H :=  δ H δ f −  µ ( f ) , δ H δ f  and substitute the single particle solution ansatz f ( q , p, t ) = P i w i δ ( q − Q i ( t )) δ ( p − P i ( t )). No w tak e the pairing with a phase space function φ and write h φ, ˙ f i = − h { φ, H } , f i . Ev aluating on the delta functions prov es the theorem. Remark 44 The quantity −{ µ ( f ) , δ H /δ f } plays the r ole of a Hamiltonian for the ad- ve ctive dissip ation pr o c ess by c o adjoint motion. This Hamiltonian is c onstructe d fr om the CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 132 momentum map J deﬁne d by the ? op er ation (Poisson br acket). That is, J h ( f , g ) = h g , − £ X h f i = h g , { h, f }i = h h, { f , g }i = h h, f ? g i . 5.4 Geometric dissipation for kinetic moments This section shows how Eq. (5.2) leads very naturally to a nonlo cal form of Darcy’s la w. In order to sho w how this equation is reco vered, one ﬁrst reviews the Kup ershmidt-Manin structure for kinetic moments. The discussion pro ceeds by considering a one-dimensional conﬁguration space; an extension to higher dimensions w ould also be p ossible b y considering the treatment in c hapter 2. 5.4.1 Review of the momen t brac ket Chapter 2 has shown how the equations for the moments of the Vlasov equation are a Lie-P oisson system [Gi1981, GiHoT r05, GiHoT r2007]. The n -th moment is deﬁned as A n ( q ) := Z p n f ( q , p ) dp . and the dynamics of these quantities is regulated by the Kupershmidt-Manin structure { F , G } =  A m + n − 1 ,  δ F δ A n , δ G δ A m  , where summation ov er rep eated indices is omitted and the Lie brack et [ · , · ] is deﬁned as [ α m , β n ] = n β n ( q ) α 0 m ( q ) − m α m ( q ) β 0 n ( q ) =: ad α m β n The moment equations are ˙ A n = − ad ∗ β n A m + n − 1 = − ( n + m ) A n + m − 1 ∂ β n ∂ q − n β n ∂ A n + m − 1 ∂ q , where β n = δ H /δ A n and the ad ∗ op erator is deﬁned by h ad ∗ β n A k , α k − n +1 i := h A k , ad β n α k − n +1 i . 5.4.2 A multiscale dissipativ e momen t hierarch y A t this point one can consider the following Lie algebra action on Vlaso v densities [HoPuT r2007-CR] β n f := £ X p n β n f =  f , p n β n  (no sum) CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 133 whic h is ob viously given by the action of the Hamiltonian function h ( q , p ) = p n β n ( q ). Now, the dual action is given b y D f ? n g , β n E := D f , β n g E = D f ? g , p n β n ( q ) E =  Z { f , g } p n dp , β n  (5.6) and the dissipative brac ket for the momen ts (5.3) is written in this notation as { { E , F } } = − * Z p n  µ [ f ] , δ E δ f  dp, Z p n  f , δ F δ f  dp + = − D ad ∗ β k e µ k + n − 1 ,  ad ∗ α m A m + n − 1  ] E where one substitutes δ E δ f = p k β k , δ F δ f = p m α m , e µ s ( q ) := Z p s µ [ f ] dp . Th us the purely dissipative momen t equations are [HoPuT r2007-CR] ∂ A n ∂ t = ad ∗ γ m A m + n − 1 with γ m :=  ad ∗ δ E δ A k e µ k + m − 1  ] (5.7) whic h arise from the subsequent application of tw o moment Lie-Poisson brack ets, as it can b e seen by the nested ad ∗ op erator. Th us the dissipative momen t equations reﬂect the double-brac ket construction of the ﬂow. Remark 45 The explicit expr ession of e µ n may involve al l the moments. In or der to se e this, it is ne c essary to c onsider the smo othe d distribution µ [ f ] , whose e µ n is the n -th moment. One c an write its functional derivative as δ µ δ f = X s p s ν s ( q ) so that µ [ f ] = Z Z H ( q , p, q 0 , p 0 ) f ( q 0 , p 0 ) d q 0 d p 0 ⇒ H ( q , p, q 0 , p 0 ) = X s p 0 s e H s ( q , p, q 0 ) and thus µ [ f ] = X s Z e H s ( q , p, q 0 )  Z p 0 s f ( q 0 , p 0 )d p 0  d q 0 = X s Z e H s ( q , p, q 0 ) A s ( q 0 ) d q 0 CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 134 A t this p oint the moment e µ i is written as Z p i µ ( f ) d p = X s Z  Z e H s ( q , p, q 0 ) p i d p  A s ( q 0 ) d q 0 = X s G si ∗ A s := e µ i wher e one deﬁnes G si ( q , q 0 ) := Z e H s ( q , p, q 0 ) p i d p Conse quently, the smo othe d moments e µ n c an dep end on al l the moments, although one c an cho ose e µ n = e µ n [ A n ] for simplicity. 5.5 Prop erties of the dissipativ e momen t hierarc h y 5.5.1 A ﬁrst result: reco vering Darcy’s la w If one now writes the equation for ρ := A 0 and consider only γ 0 and γ 1 , then one recov ers the following form of Darcy’s la w [HoPu2005, HoPu2006, HoPuT r2007-CR] ˙ ρ = ad ∗ γ 1 ρ = ∂ ∂ q  ρ µ [ ρ ] ∂ ∂ q δ E δ ρ  (5.8) where one chooses E = E [ ρ ] and e µ 0 = µ [ ρ ], so that γ 1 = e µ 0 ∂ q β 0 . Sp ecial cases. Two in teresting cases ma y be considered at this p oint. In the ﬁrst case one makes Kandrup’s choice in (5.1) for the mobility at the kinetic level µ [ f ] = f , so that Darcy’s law is written as ˙ ρ = ∂ ∂ q  ρ 2 ∂ ∂ q δ E δ ρ  . Kandrup’s case applies to the dissipativ ely induced instabilit y of galactic dynamics [Ka1991]. The previous equation is Darcy’s la w description of this type of instabilit y . In the second case, one considers the mobility µ [ ρ ] as a functional of ρ (a n um b er), leading to the equation ˙ ρ = µ [ ρ ] ∂ ∂ q  ρ ∂ ∂ q δ E δ ρ  , whic h leads to the classic energy dissipation equation, dE dt = − Z ρ µ [ ρ ]     ∂ ∂ q δ E δ ρ     2 d q . CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 135 5.5.2 A new dissipativ e ﬂuid mo del and its prop erties The dissipative brack et on the moments provides an answer to the question form ulated in section 4.6.1. In particular, it formulates the dissipativ e equation of a ﬂuid undergoing Darcy dissipativ e dynamics. As already mentioned in section 4.6.1, one would exp ect that these ﬂuid equations require the substitution u → u + v , where v is Darcy’s v elo city . Ho wev er, the whole discussion in c hapter 4 considers an energy functional E dep ending only on the densit y ρ , so that v = µ [ ρ ] ∂ q δ E /δ ρ . In general, one can consider an energy functional also dep ending on the ﬂuid momentum m , for example in the case E = H , where H is the ﬂuid Hamiltonian. This section form ulates this mo del, by taking into accoun t this dep endence on m . One starts with the moment equations ˙ A n = ad ∗ γ m A m + n − 1 with γ m :=  ad ∗ β k e µ k + m − 1  ] and expand γ 1 =  ad ∗ β k e µ k  ] =  ad ∗ β 0 e µ 0  ] +  ad ∗ β 1 e µ 1  ] = µ 0 ∂ β 0 ∂ q + 2 µ 1 ∂ β 1 ∂ q + β 1 ∂ µ 1 ∂ q γ 0 =  ad ∗ β 1 µ 0  ] = ∂ ∂ q ( µ 0 β 1 ) . By changing notation β 1 = δ E δ m , β 0 = δ E δ ρ , γ 1 = − v , µ 0 = µ ρ [ ρ ] , µ 1 = µ m [ m ] one writes the expression of Darcy’s velocity v = − µ ρ ∂ ∂ q δ E δ ρ − 2 µ m ∂ ∂ q δ E δ m − δ E δ m ∂ µ m ∂ q =  µ ρ  δ E δ ρ  ] +  µ m  δ E δ m  ] Also one ﬁnds γ 0 =  £ δ E δ m µ ρ  ] (5.9) and the ﬂuid equations are ∂ ρ ∂ t + £ v ρ = 0 ∂ m ∂ t + £ v m = − ρ   £ δ E δ m µ ρ  ] CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 136 No w, if one wan ts to incorp orate the Hamiltonian part with v elo city u = δ H /δ m , then this yields ∂ ρ ∂ t + £ u + v ρ = 0 ∂ m ∂ t + £ u + v m = ρ  δ H δ ρ −  £ δ E δ m µ ρ  ] ! (5.10) Th us, these equations show that the total ﬂuid v elo city is indeed u + v . Ho wev er no w Darcy’s v elo city v also dep ends on the ﬂuid momentum m and its smo othed version µ m . Moreov er the diamond term on the righ t hand side is also mo diﬁed b y a dissipative term, so that the contribution of pressure (right hand side in the second equation) is itself “dissipated”, consisten tly with the double brack et structure. This is a particular example of how the kinetic moments are p ow erful in deriving macro- scopic con tinuum mo dels from microscopic kinetic treatments. This section has deriv ed the dissipativ e moment equations b y simply implemen ting the moment double brac ket without w orrying ab out the semidirect pro duct structure of the equations with no dissipation. And still, the semidirect pro duct structur e eviden tly app ears in the dissipative moment equations, that hav e the same form as the non-dissipativ e case. The simplest case of Darcy ﬂuid is the one dimensional case. F or simplicity , the Hamilto- nian part may b e omitted and one can consider the purely dissipativ e ﬂuid equations, which are written as ˙ ρ + ∂ ∂ q  ρ µ ρ ∂ ∂ q δ E δ ρ + ρ µ m ∂ ∂ q δ E δ m + 2 ρ µ 0 m δ E δ m  = 0 ˙ m + m ∂ ∂ q  µ ρ ∂ ∂ q δ E δ ρ + µ m ∂ ∂ q δ E δ m + 2 µ 0 m δ E δ m  + 2 m 0  µ ρ ∂ ∂ q δ E δ ρ + µ m ∂ ∂ q δ E δ m + 2 µ 0 m δ E δ m  = ρ ∂ 2 ∂ q 2  µ ρ δ E δ m  where the prime stands for deriv ation. It is interesting to notice that the right hand side of the second equation do es not prev ent the existence of singular solutions. Indeed, the substitution of the single particle solution ansatz ( ρ, m )( q , t ) = ( w , P )( t ) δ ( q − Q ( t )) in to the equation does not generate second-order deriv ativ es of delta functions, pro vided CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 137 the v ector ﬁeld δ E /δ ρ is suﬃciently smo oth (it is useful to recall that the energy functional E do es not need to coincide with the Hamiltonian of the non-dissipative case). Th us the existence of singular solutions is allo wed also for the Darcy ﬂuid and one can address the question whether these solutions app ear sp on taneously , for example, in the case of a purely quadratic energy functional of the form E [ ρ, m ] = 1 2 Z Z ρ ( q ) G ρ ( q − q 0 ) ρ ( q 0 ) d q d q 0 + 1 2 Z Z m ( q ) G m ( q − q 0 ) m ( q 0 ) d q d q 0 whic h coincides with the truncation of the quadratic moment Hamiltonian (3.8) in Chap- ter 3 to only A 0 and A 1 . In this sense, these equations represent a dissipative v ersion of the EPSymp ﬂuid equations (3.29), which preserve their geometric nature and allo w for singular solutions. F uture research will study the b eha vior of singularities under competi- tion of length-scales in volv ed in the smo othed quan tities µ ρ and µ m , in the same spirit of [HoOnT r07]. F or example, since this system contains Darcy’s law for aggregation and self- assem bly (the ﬁrst t wo terms in the equation for ρ ) and it has the same geometric structure, one may seek conditions for the same aggregation phenomena in the dynamics of singular solutions. 5.6 F urther generalizations 5.6.1 A double brac ket structure for the b -equation. The dissipative Kup ershmidt-Manin brac k et pro vides a hin t to the possibility of inserting the double brac ket moment structure into the b -equation (developed in [HoSt03] and treated in Chapter 2). In general, the b -equation is a c haracteristic equation for a cov ariant symmetric tensor (densit y) along a smooth nonlo cal vector ﬁeld. The analogy betw een this equation and the momen t hierarch y arises b ecause of the imp ortan t prop erty that ad ∗ β n = £ β n iﬀ n = 1. Thus the Kup ersmhmidt-Manin operator en ters naturally in this problem, since it establishes a Lie algebra structure in the space of symmetric contra v arian t tensors (dual to the symmetric cov arian t tensor-densities). One could b e tempted to call “dissipative” the double brack et term in the equation; how ev er, since the b -equation is not Hamiltonian (at least under the Kupershmidt-Manin structure), one should b e careful when talking about dissipation in this context: it is not clear a priori what should be dissipated. Rather, the momen t double brack et is a wa y of preserving the geometric nature of the equations and in the case of the b -equation it can b e interesting to see how the action of diﬀeomorphisms CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 138 b eha ves under this construction. In order to formulate a double brac ket version of the b -equation, one may proceed by writing the c haracteristic equation for the momen t A n and separating the simple adv ection term from the double brack et term. ∂ A n ∂ t + £ β 1 A n = £ γ 1 A n Up on recalling that ad ∗ β n A k is a symmetric co v ariant ( k − n + 1)-tensor-density , one notes that ad ∗ β n A n is a one form-density , so that  ad ∗ β k A k  ] is a v ector ﬁeld for an y in teger k . In particular, up on substituting A k → µ k [ A k ] and summing o ver k one obtains the dissipative v ector ﬁeld γ 1 = P k  ad ∗ β k µ k  ] from the last section, whic h is needed in the righ t hand side of the equation abov e in order to c onstruct the double brac ket term. The resulting equation is ∂ A n ∂ t + £ β 1 A n = £ “ P k ad ∗ β k µ k ” ] A n (5.11) Since one w ants β 1 to regulate the momen t dynamics, it is p ossible to ﬁx k = 1, so that one writes ∂ A n ∂ t + £ β 1 A n = £ “ ad ∗ β 1 µ 1 ” ] A n If k = 1, one recalls that the ad ∗ β 1 coincides with Lie deriv ative and the equation reduces to ∂ A n ∂ t + £ β 1 A n = £ “ £ β 1 µ 1 ” ] A n A t this p oint one p erforms the choice β 1 = G ∗ A n (as in the b -equation) and lets the smo othed moment µ 1 = µ [ A n ] dep end only on A n (instead of A 1 ), since one recalls that in the general case µ n can dep end on any sequence of momen ts. The result is the equation ∂ A n ∂ t + £ β 1 A n = £ “ £ β 1 µ [ A n ] ” ] A n with β 1 = G ∗ A n (5.12) where µ [ A n ] is some ﬁltered version of the n -th moment µ [ A n ] = H ∗ A n . In the particular case n = 1, one reco vers the dissipative EPDiﬀ equation in tro duced in Chapter 4, exactly as it happ ens in the ordinary case with simple advection. In this sense, the double brack et brac ket can b e understo o d as “double advection”, since it inv olves a sequen tial application of tw o Lie deriv atives. CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 139 5.6.2 A GOP equation for the momen ts The dev elopment of the double brack et form of the b -equation provides an interesting hint to formulate a GOP form of the moment equations. In order to see this, one can consider again the equation (5.11) and discard simple advection to obtain ∂ A n ∂ t = £ “ P k ad ∗ β k µ k ” ] A n Since one ma y w ant to a void the presence of moments diﬀerent from A n , it is possible to ﬁx k = n . Also, one recalls that ad ∗ β n µ n = n β n µ 0 n + ( n + 1) µ n β 0 n . It is interesting to notice that  ad ∗ β n A n , α 1  = − h A n , ad α 1 β n i = − h A n , £ α 1 β n i := h A n  β n , α 1 i whic h allo ws to write the equation in the GOP form as ∂ A n ∂ t = £ “ µ n  δE δA n ” ] A n (5.13) In this case the quantit y µ n is chosen to depend only on A n as µ n [ A n ] = H ∗ A n and the energy functional E = E [ A n ] is left as a mo deling c hoice. Two particular cases are n = 0 and n = 1. In the ﬁrst case, the GOP equation reduces to Darcy’s law, while in the second case, the equation reduces to the pur ely dissipative EPDiﬀ equation. Thus the fact that this equation reduces to such in teresting cases promises w ell for future research. Remark 46 The pr op erty ad ∗ β n A n = A n  β n le ads to another way of writing the e quation for the ﬁrst-or der moment A 1 in the Hamiltonian hier ar chy ˙ A n = − P m ad ∗ β m A m + n − 1 . Inde e d, it is evident how the e quation for A 1 may b e written in the form ˙ A 1 = − P m A m  β m . This p articular form of the diamond op er ator b etwe en symmetric tensors was known to Schouten, sinc e it arises natur al ly fr om his symmetric br acket [BlAs79, DuMi95] (also cf. chapter 2), and it is c al le d “L agr angian Schouten c onc omitant” [Ki82]. CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 140 5.7 Discussion and op en questions This c hapter has developed a new symplectic v ariational approac h for mo deling dissipation in kinetic equations that yielded a double br acket structur e in phase sp ac e . This approac h has b een fo cused on the Vlaso v example and it yielded the existence of single- particle solutions. In general, it is p ossible to extend the present theory to the evolution of an arbitrary geometric quantit y deﬁned on any smo oth manifold [HoPuT r2007]. F or example, the restriction of the geometric formalism for symplectic motion considered here to cotangen t lifts of diﬀeomorphisms reco vers the corresp onding results for ﬂuid momen tum. The last section has pro vided a consistent derivation of Dar cy’s law from ﬁrst princi- ples in kinetic theory , obtained b y inserting dissipativ e terms in to the Vlaso v equation whic h resp ect the geometric nature of the system. This form of the Darcy’s law has b een studied and analyzed in [HoPu2005, HoPu2006], where it has b een sho wn to p ossess emergent sin- gular solutions ( clump ons ), which form sp ontaneously and collapse together in a ﬁnite time, from any smooth conﬁned initial condition. Also, the last section has formulated the dissipative version of compressible ﬂuids by follo wing the moment double br acket appr o ach . These ﬂuid equations (5.10) can b e called “Darcy ﬂuid” and it is an interesting question whether these equations p ossess emer- gen t singularities , like in the case of EPDiﬀ. In order to establish whether the singular solutions app ear sp on taneously in ﬁnite time, one needs to pro ve a “steep ening Lemma” [CaHo1993] (cf. chapter 1) for the equations of the EPSymp ﬂuid. F or example, a similar result has b een found for Darcy’s law b y Holm and Putk aradze [HoPu2005, HoPu2006]. F urther speculations ha ve in volv ed the double br acket form of the b -e quation (5.12). This has tw o relev an t prope rties, one of which is that it reduces to the dissipative EPDiﬀ equation for the ﬁrst order moment. The second and more interesting property is that this equation allo ws for the existence of singular solutions for any in teger n . It would b e in teresting to chec k whether these solutions emerge from an y conﬁned initial distribution for some v alues of n , as it happ ens for the ordinary b -family . This is a p ossible road for further analysis. Similar considerations also apply to the GOP e quations for the moments (5.13). One ma y also extend the presen t phase space treatmen t and the corresp onding momen t brac ket to include an additional set of dimensions corresp onding to internal degrees of free- dom (order parameters, or orientation dep endence) carried by the microscopic particles, rather than requiring them to b e simple point particles. This is a standard approac h in CHAPTER 5. GEOMETRIC DISSIP A TION FOR KINETIC EQUA TIONS 141 condensed matter theory , for exam ple in liquid crystals, see, e.g., [Ch1992, deGePr1993]. These questions are pursued in the next chapter, whic h is the main chapter and con tains only new results. Chapter 6 Anisotropic in teractions: a new mo del 6.1 In tro duction and bac kground 6.1.1 Geometric mo dels of dissipation in ph ysical systems This c hapter explains how the geometry of double-brac ket dissipation mak es its wa y from the microscopic (kinetic theory) lev el to the macroscopic (con tinuum) level, when the particles in the microscopic description carry an in ternal v ariable that is orien tation dep endent. Without orien tation dep endence, the moment equations deriv ed previously yield a nonlocal v ariant of the famous Darcy’s la w [Darcy1856]. When orientation is included, the resulting Lie-Darcy momen t equations identify the macroscopic parameters of the con tinuum description and go vern their ev olution. In previous w ork, Gibb ons, Holm and Kup ershmidt [GiHoKu1982, GiHoKu1983] (ab- breviated GHK) show ed that the pro cess of taking momen ts of the Vlaso v equation for such particles is a Poisson map. GHK used this prop erty to deriv e the equations of chr omohy- dr o dynamics . These are the equations of a ﬂuid plasma consisting of particles carrying Y ang-Mills charges and in teracting self-consistently via a Y ang-Mills ﬁeld. The GHK P oisson map for c hromohydrodynamics provides the guidelines for an extension of the Kup ershmidt- Manin (KM) brac ket [KuMa1978] for the momen ts. GHK considered only Hamiltonian mo- tion and did not consider the corresp onding double-brack et P oisson structure of dissipation. 142 CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 143 That is the sub ject of the present c hapter. 6.1.2 Goal and presen t approach The goal of the present w ork is to determine the macroscopic implications of introducing nonlinear double-brac ket dissipation at the microscopic lev el, so as to resp ect the coadjoint orbits of canonical transformations for dynamics that dep ends up on particle orien tation. The presen t approach in tro duces this orientation dep endence in to the microscopic description by augmen ting the canonical P oisson brack et in position q and momentum p so as to include the Lie-P oisson part for orien tation g taking v alues in the dual g ∗ of the Lie algebra g , with ev entually g = so (3) for physical orientation. Thus this chapter mak es use of the total Poisson br acket fr om GHK , n f , h o 1 := n f , h o +  g ,  ∂ f ∂ g , ∂ h ∂ g  , (6.1) where [ · , · ] : g × g → g is the Lie algebra brack et and h · , · i : g ∗ × g → R is the pairing b et ween the Lie algebra g and its dual g ∗ . F or rotations, g = so (3) and the brack et [ · , · ] b ecomes the cross pro duct of v ectors in R 3 . Correspondingly , the pairing h · , · i b ecomes the dot pro duct of vectors in R 3 . This chapter considers the double-brac ket dynamics of f ( q , p, g , t ) resulting from replacing the canonical Poisson brac kets in Eq. (5.2) by the direct sum of canonical and Lie-Poisson brack ets { · , · } 1 in Eq. (6.1). One then takes moments of the resulting dynamics of f ( q , p, g , t ) with respect to momen tum p and orientation g , to ob- tain the dynamics of the macroscopic description. The momen ts with res pect to momen tum p alone provide an in termediate set of dynamical equations for the p − moments, A n ( q , g , t ) := Z p n f ( q , p, g , t ) dp . These intermediate dynamics are reminiscent of the Smolucho wsi equation for the probability A 0 ( q , g , t ). How ev er, the intermediate dynamics of the p − moments cannot b e identical to the Smoluc howsi equation even for the probabilit y A 0 ( q , g , t ), because the kinetic double-br acket dissip ation is deterministic, not sto chastic . This chapter presen ts closed sets of equations for the intermediate dynamics of A 0 ( q , g , t ) and A 1 ( q , g , t ). A closed set of contin uum equations for the ( p, g ) moments is also found. The ﬁnal closure provides the macroscopic contin uum dynamics for the set of moments of the double-brack et kinetic equations (5.2) under the replacemen t { · , · } → { · , · } 1 with resp ect to { 1 , p, g , p 2 , pg , g 2 } . This macroscopic contin- uum closure inherits the geometric prop erties of the double brack et, b ecause the pro cess of taking these moments is a P oisson map, as observed in GHK. CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 144 6.2 Geometric dissipation for anisotropic interactions 6.2.1 A dissipative v ersion of the GHK-Vlaso v equation F ollowing GHK, one introduces a particle distribution which dep ends not only on the p o- sition and momentum co ordinates q and p , but also on an extra co ordinate g asso ciated with orientation . The co ordinate g b elongs to the dual of a certain Lie algebra g , whic h for anisotropic interactions w ould b e g = so (3). Ho wev er, this chapter will formulate the problem in a more general context and analyze the case of rotations separately . In the non- dissipativ e case, the Vlasov equation is written in terms of a P oisson brack et, which is the direct sum of the canonical ( pq )-brack et and the Lie-Poisson brack et on the Lie algebra g . Explicitly , this Poisson brac ket is written as n f , h o 1 := n f , h o +  g ,  ∂ f ∂ g , ∂ h ∂ g  . (6.2) The non-dissipative Vlaso v equation now becomes ∂ f ∂ t = −  f , δ H ∂ f  1 = − b X δH δf ( f ) , where one deﬁnes the vector ﬁeld b X h asso ciated with the Hamiltonian function h as b X h := ∂ h ∂ p ∂ ∂ q − ∂ h ∂ q ∂ ∂ p +  ad ∗ ∂ h ∂ g g , ∂ ∂ g  = X h +  ad ∗ ∂ h ∂ g g , ∂ ∂ g  . The Vlasov equation is thus a char acteristic e quation for the evolution go verned by the ﬂo w of the vector ﬁeld b X δ H /δ f , determined by the action of this vector ﬁeld on the densit y f . One can iden tify b X h with h and deﬁne an action h · f := b X h ( f ), so that its dual op eration denoted by ( ? ) is deﬁned b y  f ? k , h  =  k , − h · f  =  k ,  h, f  1  =  k ,  h, f   −  k ,  g ,  ∂ f ∂ g , ∂ h ∂ g   = −  k ,  f , h   − Z  k ad ∗ ∂ f ∂ g g , ∂ h ∂ g  d q d p d g =   f , k  , h  + Z h ∂ ∂ g ·  k ad ∗ ∂ f ∂ g g  d q d p d g =   f , k  , h  + Z h  ad ∗ ∂ f ∂ g g , ∂ k ∂ g  d q d p d g =   f , k  , h  + Z h  g ,  ∂ f ∂ g , ∂ k ∂ g  d q d p d g =   f , k  1 , h  . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 145 where in the 5th line one uses the following argumen t ∂ ∂ g · ad ∗ ∂ f ∂ g g = ∂ ∂ g c  g a C a bc ∂ f ∂ g b  = b g bc ∂ 2 f ∂ g c ∂ g b = 0 , with b g bc := g a C a bc = − b g cb . This is justiﬁed by the an tisymmetry of C a bc and by the symmetry of ∂ g c ∂ g b . Thus, f ? k = { f , k } 1 . Up on applying the same argumen ts as in the previous c hapter and making use of the general theory of the double brack et dissipation, one ﬁnds the purely dissipativ e Vlasov equation in double-brack et form, ∂ f ∂ t =  f ,  µ [ f ] , δ E ∂ f  1  1 . (6.3) where the deriv ative δ E /δ f is the energy of the single particle (the following discussion treats the energy E [ f ] and the Hamiltonian H [ f ] indiﬀerently). This equation has exactly the same form as in (5.2), but now one substitutes the direct sum Poisson brack et {· , ·} 1 in (6.2) instead of the canonical Poisson brack et {· , ·} . This form ulation can now be used to derive the double-brac ket dissipative v ersion of the Vlasov equation for particles undergoing anisotropic interaction. 6.2.2 Dissipative momen t dynamics: a new anisotropic mo del T o deriv e the momen t dynamics with orien tation dep endence, one follo ws the same steps as in the previous section, b eginning by in tro ducing the quantities A n ( q , g ) := Z p n f ( q , p, g ) dp with g ∈ g ∗ . One may ﬁnd the entire hierarch y of equations for these moment quantities and then inte- grate ov er g in order to ﬁnd the equations for the mass density ρ ( q ) := R A 0 ( q , g ) dg and the contin uum charge density G ( q ) = R g A 0 ( q , g ) dg . Without the in tegration o ver g , such an approach would yield the Smolucho wski approximation for the density A 0 ( q , g ), usually denoted by ρ ( q, g ). This approach is follo wed in the Sec. 6.5, where the dynamics of ρ ( q , g ) is presented explicitly . This section extends the Kup ershidt-Manin approac h as in GHK to generate the dynam- ics of moments with resp ect to b oth momentum p and charge g . The main complication is that the Lie algebras of physical interest (such as so (3)) are not one-dimensional and in CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 146 general are not Ab elian. Th us, in the general case one needs to use a multi-index notation as in [Ku1987, Ku2005, GiHoT r05]. One can introduce multi-indices σ := ( σ 1 , σ 2 , . . . , σ N ), with σ i ≥ 0, and deﬁne g σ := g σ 1 1 . . . g σ N N , where N = dim( g ). Then, the moments are expressed as A n σ ( q ) := Z p n g σ f ( q , p, g ) dp dg . This multi-dimensional treatment leads to cumbersome calculations. F or the purp oses of this section, one is primarily in terested in the equations for ρ and G , so one restricts to consider only moments of the form A n,ν = Z p n g ν f ( q , p, g ) dp dg ν = 0 , 1 , . . . , N . Here one deﬁnes g 0 = 1 and g a = h g , e a i where e a is a basis of the Lie algebra and h g b e b , e a i = g a ∈ R represen ts the result of the pairing h · , · i b et ween an elemen t of the Lie algebra basis and an element of the dual Lie algebra. One writes the single particle Hamiltonian as h = δ H /δ f = p n g ν δ H /δ A n,ν =: p n g ν β ν n ( q ), which means that one emplo ys the following assumption. Assumption 47 The single-p article Hamiltonian h = δ H /δ f is line ar in g and c an b e expr esse d as h ( q , p, g ) = p n ψ n ( q ) + p n  g , ψ n ( q )  , wher e ψ n ( q ) ∈ R is a r e al sc alar function and ψ n ( q ) ∈ R ⊗ g is a r e al Lie-algebr a-value d function. This assumption wil l b e use d thr oughout the pr esent chapter, exc ept in Se ction 6.5. Dual Lie algebra action. The action of β ν n on f is deﬁned as β ν n · f =  p n g ν β ν n , f  1 (no sum) . The dual of this action is denoted b y ( ? n,ν ). It ma y be computed analogously to the equation (5.6) in the previous chapter and found to b e f ? n,ν k = Z Z p n g ν  f , k  1 d p d g = Z g ν g σ ad ∗ α σ m A m + n − 1 d g + Z g ν  g ,  ∂ A m + n ∂ g , ∂ ( g σ α σ m ) ∂ g  d g = ad ∗ α σ m Z g ν g σ A m + n − 1 d g + Z g ν  g ,  ∂ A m + n ∂ g , ∂ ( g a α a m ) ∂ g  d g . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 147 Here, k = p m g σ α σ m ( q ) and one uses the deﬁnition of the moment A n ( q , g ) = Z p n f ( q , p, g ) dp . Ev olution equation. Having characterized the dual Lie algebra action, one may write the evolution equation for an arbitrary functional F in terms of the dissipative brac ket as follo ws: ˙ F = { { F , E } } = − * *  µ [ f ] ? n,ν δ E ∂ f  ] , f ? n,ν δ F ∂ f + + (6.4) where the pairing h h · , · i i is giv en by in tegration ov er the spatial co ordinate q . Now we ﬁx m = 0, n = 1. The equation for the evolution of F = A 0 ,λ := R g λ A 0 d g d p is found from (6.4) to b e ∂ A 0 ,λ ∂ t = ad ∗ γ ] 1 ,ν Z g ν g λ A 0 d g + Z g λ * g , " ∂ A 1 ∂ g , ∂ ( g σ γ ] 1 ,σ ) ∂ g # + " ∂ A 0 ∂ g , ∂ ( g σ γ ] 0 ,σ ) ∂ g # !+ d g = ∂ ∂ q  γ ] 1 ,ν Z g ν g λ A 0 d g  + Z g λ * g , " ∂ A 1 ∂ g , ∂ ( g a γ ] 1 ,a ) ∂ g # + " ∂ A 0 ∂ g , ∂ ( g a γ ] 0 ,a ) ∂ g # !+ d g , (6.5) where one deﬁnes the analogues of Darcy’s velocities: γ 0 ,ν := µ [ f ] ? 0 ,ν δ E δ f = Z g ν  g ,  ∂ e µ k ∂ g , ∂ ( g a β a k ) ∂ g  d g = Z g ν  g ,  ∂ e µ 0 ∂ g , ∂ ( g a β a 0 ) ∂ g  d g and γ 1 ,ν := µ [ f ] ? 1 ,ν δ E δ f = ad ∗ β σ k Z g ν g σ e µ k d g + Z g ν  g ,  ∂ e µ k +1 ∂ g , ∂ ( g a β a k ) ∂ g  d g = ∂ β σ 0 ∂ q Z g ν g σ e µ 0 d g + Z g ν  g ,  ∂ e µ 1 ∂ g , ∂ ( g a β a 0 ) ∂ g  d g . Here one assumes that the energy functional E dep ends only on A 0 ,λ (recall that β λ n := δ E /δ A n,λ ), so that it is possible to ﬁx k = 0 in the second line. These equations ab o ve will b e treated as a higher level of approximation in section 6.4. No w, one further simpliﬁes the treatmen t by discarding all terms in γ 1 ,a , that is one truncates the summations in equation (6.5) to consider only terms in γ 0 , 0 , γ 0 ,a and γ 1 , 0 . This is equiv alent to consider a single- particle Hamiltonian of the form h ( q , p, g ) = ψ 0 ( q ) + h g , ¯ ψ 0 ( q ) i + p ψ 1 ( q ) . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 148 With this simpliﬁcation the equation (6.5) b ecomes ∂ A 0 ,λ ∂ t = ad ∗ γ 1 , 0 Z g λ A 0 d g + Z g λ * g , " ∂ A 0 ∂ g , ∂ ( g σ γ ] 0 ,σ ) ∂ g # !+ d g = ∂ ∂ q  γ 1 , 0 Z g λ A 0 d g  + Z g λ * g , " ∂ A 0 ∂ g , ∂ ( g a γ ] 0 ,a ) ∂ g # !+ d g , (6.6) and the expression for γ 1 , 0 is γ 1 , 0 : = µ [ f ] ? 1 , 0 δ E δ f = ad ∗ β σ k Z g σ e µ k d g + Z  g ,  ∂ e µ k +1 ∂ g , ∂ ( g a β a k ) ∂ g  d g = ∂ β σ 0 ∂ q Z g σ e µ 0 d g . A t this p oin t it is conv enient to simplify the notation by deﬁning the following dynamic quan tities ρ = Z f d g d p , G = Z g f d g d p . Lik ewise, one deﬁnes the mobilities as µ ρ = Z µ [ f ] d g d p , µ G = Z g µ [ f ] d g d p . In terms of these quantities, it is p ossible to write the following. Theorem 48 The moment e quations for ρ and G ar e given by ∂ ρ ∂ t = ∂ ∂ q ρ  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ! (6.7) ∂ G ∂ t = ∂ ∂ q G  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ! + ad ∗ „ ad ∗ δE δG µ G « ] G . (6.8) Remark 49 Equations in this family (c al le d Ge ometric Or der Par ameter e quations) wer e derive d via a diﬀer ent appr o ach in [HoPu2007]. 6.2.3 A ﬁrst prop erty: singular solutions Equations (6.7) and (6.8) admit singular solutions. This means that the tra jectory of a single ﬂuid particle is a solution of the problem and all the microscopic information ab out the particles is preserved. One can prov e the following. CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 149 Theorem 50 Equations (6.7) and (6.8) admit solutions of the form ρ ( q , t ) = w ρ ( t ) δ ( q − Q ( t )) G ( q , t ) = w G ( t ) δ ( q − Q ( t )) (6.9) wher e w ρ , w G and Q under go the fol lowing dynamics ˙ w ρ = 0 ˙ w G = ad ∗ “ ad ∗ δE /δ G µ G ” ] q = Q w G ˙ Q = −  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  q = Q Pro of. After deﬁning the quantities γ 1 := γ 1 , 0 = µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  γ 0 := γ ] 0 ,a e a =  ad ∗ δE δG µ G  ] one pairs equations (6.7) and (6.8) respectively with φ ρ ( q ) and φ G ( q ). This yields the follo wing results, Z ˙ ρ φ ρ d q = Z φ ρ ∂ ∂ q  ρ γ 1  d q = − Z ∂ φ ρ ∂ q ρ γ 1 d q Z D ˙ G, φ G E d q = Z  ∂ ∂ q  G γ 1  + ad ∗ γ 0 G, φ G  d q = − Z  G, γ 1 ∂ φ ρ ∂ q  d q + Z D G, h γ 0 , φ ρ iE d q Up on substituting the singular solution ansatz (6.9), one calculates ˙ w ρ φ ρ ( Q ) + w ρ ˙ Q ∂ φ ρ ∂ q     q = Q = − w ρ γ 1 ( Q ) ∂ φ ρ ∂ q     q = Q D ˙ w G , φ G ( Q ) E + ˙ Q  w G , ∂ φ G ∂ q      q = Q = − γ 1 ( Q )  w G , ∂ φ G ∂ q      q = Q +  ad ∗ γ 0 w G , φ G ( Q )  so that identiﬁcation of corresponding co eﬃcients yields ˙ w ρ = 0 , ˙ w G = ad ∗ γ 0 w G , ˙ Q = − γ 1 ( Q ) and the thesis is prov en. CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 150 Remark 51 A similar r esult applies for the Ge ometric Or der Par ameter (GOP) e qua- tions investigate d in [HoPu2007]. Inde e d, the e quations (6.7) and (6.8) r e duc e to those in [HoPu2007] when one c onsiders a c ommutative Lie algebr a. 6.3 An application: ro d-lik e particles on the line 6.3.1 Moment equations In this case G = m ( x ), ad v w = v × w and ad ∗ v w = − v × w , and the Lie algebra pairing is represen ted b y the dot pro duct of vectors in R 3 . Therefore the equations are ∂ ρ ∂ t = ∂ ∂ x ρ  µ ρ ∂ ∂ x δ E δ ρ + µ m · ∂ ∂ x δ E δ m  ! (6.10) and ∂ m ∂ t = ∂ ∂ x m  µ ρ ∂ ∂ x δ E δ ρ + µ m · ∂ ∂ x δ E δ m  ! + m × µ m × δ E δ m (6.11) Note that equations for density ρ and orientation m hav e conserv ative parts (coming from the divergence of a ﬂux). In addition, when µ m = a m for a constant a , the orien tation m has precisely the dissipation term m × m × δ E /δ m introduced by Gilb ert [Gilb ert1955]. Th us, this section has deriv ed the Gilb ert dissipation term at the macroscopic level, starting from double-brack et dissipativ e terms in the kinetic theory description. 6.3.2 More results: emergence and in teraction of singularities When considering the rotation algebra so (3), n umerical exp erimen ts ha v e sho wn [HoOnT r07] that under certain conditions, the singular solutions in section 6.2.3 emerge sp ontaneously from any conﬁned initial distribution. This result was already kno wn in the case of isotropic in teraction for which a rigorous pro of is also a v ailable [HoPu2005, HoPu2006]. In that con- text the singular solutions w ere called clump ons . The numerics shows that the anisotropic nature of the self interaction preserves this b eha vior. In particular the exp erimen ts hav e CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 151 0 10 20 0 50 100 H*Density 0 10 2 0 0 50 100 H*mx 0 10 20 0 50 100 H*my 0 10 20 0 50 100 H*mz Figure 6.1: L eft: Emerging singularities. Plots of the smo othed density ¯ ρ = H ∗ ρ and orien tation m = H ∗ m (three comp onents), where the smoothing kernel is the Helmholtzian e −| x | . The ﬁgures sho w ho w the singular solution emerges from a Gaussian initial condition for the energy in (6.12). Smo othed quantities are chosen to av oid the necessity to represent δ -functions. Right: Orienton formation in a d = 1 dimensional simulation. The color-co de on the cylinder denotes lo cal aver age d density: blac k is maximum density while white is ρ = 0. Purple lines denote the three-dimensional vector m = H ∗ m . The formation of sharp peaks in av eraged quan tities corresponds to the formation of δ -functions. (Figures b y V. Putk aradze) sho wn that not only these solutions form for the density v ariable ρ , but also for the orien- tation densit y m . Such a situation represen t a state in whic h the particles are concen trated in only one p oint in space and are all aligned tow ards only one direction (ﬁg. 6.3.2). This section studies the in teraction of t wo singular solutions of the equations (6.10) and (6.11). Each delta function has the in terpretation of a single particle, whose w eights and p ositions satisfy a ﬁnite set of ordinary diﬀerential equations. In particular one wan ts to in vestigate the t wo-particle case analytically . It is p ossible to ﬁnd conditions for which the particles tend to merge and align. F rom section 6.2.3, upon renaming the v ariable w G with the simpler notation λ (so that w G = λ ), one writes ˙ x i ( t ) = V ( x i ( t )) ˙ λ i ( t ) = λ i ( t ) × Φ ( x i ( t )) CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 152 where λ i is the orientation (or magnetic momen t) of the i -th particle and V ( x i ) = −  µ ρ ∂ ∂ x δ E δ ρ + µ m · ∂ ∂ x δ E δ m  x = x i Φ ( x i ) =  µ m × δ E δ m  x = x i In order to sp ecify the physical system one has to choose an energy and the quan tities µ ρ and µ m . This section presents the nonlocal purely quadratic case E [ ρ, m ] = 1 2 Z ρ G ρ ∗ ρ d x + 1 2 Z m G m ∗ m d x (6.12) where * denotes con volution and G ρ , G m are the kernels of some symmetric in vertible op erators (later c hosen to b e all equal to the Helmholtz op erator). Analogously , one can tak e µ ρ = H ρ ∗ ρ and µ m = H m ∗ m . Under these circumstances, one writes V ( x, t ) = − H ρ ∗ ρ ∂ x G ρ ∗ ρ − H m ∗ m · ∂ x G m ∗ m Φ ( x, t ) = H m ∗ m × G m ∗ m . Substituting the multi-particle solution ρ ( x, t ) = X i w i ( t ) δ ( x − x i ( t )) m ( x, t ) = X j λ j ( t ) δ ( x − x j ( t )) yields V ( x, t ) = − X j,k w j w k H ρ ( x j − x ) ∂ x G ρ ( x k − x ) − X j,k λ j · λ k H m ( x j − x ) ∂ x G m ( x h − x ) Φ ( x, t ) = X j,k H m ( x j − x ) G m ( x k − x ) λ j × λ k . where all kernels are now assumed to b e Helmholtz kernels (so that H (0) = K (0) = 1). One now considers a system of tw o identical clump ons ( j, k = 1 , 2 and w 1 = w 2 = 1) and ev aluate V ( x 1 , t ) = −  1 + H ρ ( x 2 − x 1 )  ∂ x 1 G ρ ( x 2 − x 1 ) −  λ 1 + λ 2 H m ( x 2 − x 1 )  · λ 2 ∂ x 1 G m ( x 2 − x 1 ) Φ ( x 1 , t ) =  G m ( x 2 − x 1 ) − H m ( x 2 − x 1 )  λ 1 × λ 2 . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 153 and analogously V ( x 2 , t ) = −  1 + H ρ ( x 2 − x 1 )  ∂ x 2 G ρ ( x 2 − x 1 ) −  λ 1 + λ 2 H m ( x 2 − x 1 )  · λ 2 ∂ x 2 G m ( x 2 − x 1 ) Φ ( x 2 , t ) = − Φ ( x 1 , t ) . The equations of motion are then ˙ x 1 =  1 + H ρ ( x 2 − x 1 )  ∂ x 1 G ρ ( x 2 − x 1 ) +  λ 1 + λ 2 H m ( x 2 − x 1 )  · λ 2 ∂ x 1 G m ( x 2 − x 1 ) ˙ x 2 =  1 + H ρ ( x 2 − x 1 )  ∂ x 2 G ρ ( x 2 − x 1 ) +  λ 2 + λ 1 H m ( x 2 − x 1 )  · λ 1 ∂ x 2 G m ( x 2 − x 1 ) ˙ λ 1 =  G m ( x 2 − x 1 ) − H m ( x 2 − x 1 )  λ 1 × λ 1 × λ 2 ˙ λ 2 =  G m ( x 2 − x 1 ) − H m ( x 2 − x 1 )  λ 2 × λ 2 × λ 1 No w calculate d dt | x 1 − x 2 | = sign ( x 2 − x 1 ) d dt ( x 2 − x 1 ) = − sign 2 ( x 2 − x 1 )  2 α ρ  1 + H ρ ( x 2 − x 1 )  G ρ ( x 2 − x 1 ) + 1 α m  2 λ 2 · λ 1 +  k λ 2 k 2 + k λ 1 k 2  H m ( x 2 − x 1 )  G m ( x 2 − x 1 )  (6.13) where one uses the fact that ∂ x 1 G ρ ( x 2 − x 1 ) = − ∂ x 2 G ρ ( x 2 − x 1 ). It is easy to notice that the tw o particles mov e together after merging. T o inv estigate the asymptotic dynamics of alignment of λ 1 and λ 2 , one calculates d dt ( λ 1 · λ 2 ) = ˙ λ 1 · λ 2 + λ 1 · ˙ λ 2 = 2  H m ( x 2 − x 1 ) − G m ( x 2 − x 1 )  k λ 1 × λ 2 k 2 (6.14) where the equations for ˙ λ 1 and ˙ λ 2 ha ve b een substitute in the second step. One notices that the dynamics of λ 1 · λ 2 is nontrivial only if the particles hav e not clump ed yet. Indeed, after the particles merge, the angle b etw een the λ ’s remains constant in time. The following discussion considers the case when the time b efore merging is suﬃcien tly long for the λ ’s to reac h their asymptotic equilibrium state. Already at this stage one can conclude from equation (6.14) that CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 154 Theorem 52 The two clump ons always tend to a ﬁnal state, which is either alignment or anti-alignment. If H m < G m ( H narr ower than G ), then λ 1 · λ 2 tends to its minimum value λ 1 · λ 2 → − k λ 1 k k λ 2 k , so that clump ons tend to anti-align . Alternatively, if H m > G m (for G is narr ower than H ), then λ 1 · λ 2 → k λ 1 k k λ 2 k and the clump ons tend to align . This alignment pr o c ess lasts as long as the two clump ons ar e sep ar ate d by a nonzer o distanc e. Pro of. One notices that the expression in (6.14) has a deﬁnite sign, positive or negativ e dep ending on w ether H m > G m or H m < G m resp ectiv ely . Thus the pro duct λ 1 · λ 2 tend to grow or deca y in the t wo diﬀerent cases. How ever one has max {| λ 1 · λ 2 |} = k λ 1 k k λ 2 k , which means that lim t → + ∞ λ 1 · λ 2 = ± k λ 1 k k λ 2 k On the other hand, when H m − G m = 0, then λ 1 · λ 2 remains constan t. In particular x 2 − x 1 = 0 ⇒ H m = G m = 0 and this prov es the last part of the theorem. Th us, the comp etition betw een the length scales of the smo othing functions G and H de- termines the alignment in the asymptotic state. Tw o more relev an t results are the following Corollary 53 In the p articular c ase H m > G m ( G narr ower than H ) and λ 1 · λ 2 > 0 at t = 0 , then the p articles wil l align and clump asymptotic al ly in time. Pro of. Up on using the vector iden tity k λ 1 × λ 2 k 2 = k λ 1 k 2 k λ 2 k 2 − ( λ 1 · λ 2 ) 2 one can write d dt ( λ 1 · λ 2 ) = 2  H m ( x 2 − x 1 ) − G m ( x 2 − x 1 )   k λ 1 k 2 k λ 2 k 2 − ( λ 1 · λ 2 ) 2  where k λ 1 k 2 and k λ 2 k 2 are constants. So the evolution equation is d ( λ 1 · λ 2 ) ( λ 1 · λ 2 ) 2 − k λ 1 k 2 k λ 2 k 2 = − 2  H m ( x 2 − x 1 ) − G m ( x 2 − x 1 )  d t CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 155 Up on assuming that initially λ 1 (0) · λ 2 (0) = 0, one writes the following solution λ 1 · λ 2 = k λ 1 k k λ 2 k tanh  2 k λ 1 k k λ 2 k Z t 0  H m ( x 2 − x 1 ) − G m ( x 2 − x 1 )  d t 0  so that λ 1 · λ 2 has a deﬁnite p ositiv e sign (if λ 1 · λ 2 > 0 at t = 0). Consequently the expression (6.13) has also a p ositive deﬁnite sign and thus if G and H are such that H m > G m , then the tw o particles indeﬁnitely approach eac h other and align. Corollary 54 When H m < G m , in the p articular c ase G m = G ρ and k λ 1 k k λ 2 k ≤ 1 , then the clump ons tend to clump and anti-align asymptotic al ly in time. Pro of. One has 2  1 + H ρ ( x 2 − x 1 )  + 2 λ 2 · λ 1 +  k λ 2 k 2 + k λ 1 k 2  H m ( x 2 − x 1 ) > 2 (1 + λ 2 · λ 1 ) ≥ 2 (1 − k λ 1 k k λ 2 k ) whic h is p ositiv e by hypothesis. By comparing with equation (6.13), one ﬁnds that the expression in (6.13) is negative deﬁnite in sign, so that k λ 1 k k λ 2 k ≤ 1 becomes a suﬃcien t condition for clumping and the thesis is prov en. 6.3.3 Higher dimensional treatmen t Although this c hapter formulates a one dimensional treatmen t, the p ossibilit y of going to higher dimensions is pretty clear b y lo oking at the momen t equations, b y remem b ering that ad ∗ γ 1 , 0 = £ γ 1 , 0 . If one specializes to the case of so (3), then it is p ossible to write the moment equations as ∂ ρ ∂ t = div ρ  µ ρ ∇ δ E δ ρ + µ m · ∇ δ E δ m  ! ∂ m ∂ t = div m ⊗  µ ρ ∇ δ E δ ρ + µ m · ∇ δ E δ m  ! + m × µ m × δ E δ m CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 156 Figure 6.2: Evolution of a ﬂat magnetization ﬁeld and a sinusoidally-v arying density . Sub- ﬁgure (a) sho ws the evolution of ¯ ρ = H ∗ ρ for t ∈ [0 . 5 , 1]; (b) sho ws the evolution of ¯ m x . The proﬁles of ¯ m y and ¯ m z are similar. At t = 0 . 5, the initial data hav e formed eight equally spaced, iden tical clumpons, corresp onding to the eight density maxima in the initial conﬁg- uration. By impulsively shifting the clump on at x = 0 by a small amount, the equilibrium is disrupted and the clump ons merge rep eatedly until only one clump on remains. (Figures b y L. ´ O N´ araigh) These equations also p ossess singular solutions of the form [HoPuT r08] ρ ( q , t ) = Z δ ( q − Q ( s, t )) d s m ( q , t ) = Z w m ( s, t ) δ ( q − Q ( s, t )) d s where s is a co ordinate on a submanifold of R 3 . If s is a one-dimensional curvilinear co- ordinate, then this solution represents an orientation ﬁlament (see ﬁg. 6.3.3), while for s b elonging to a t wo-dimensional surface one obtains an orientation she et . The simple case of particle-lik e solutions (6.9) is still p ossible in higher dimensions and ﬁg. 6.3.3 sho ws their sp on taneous emergence in tw o dimensions. The p g -moment brac ket in more dimensions. Up on following the tensorial interpre- tation for the momen ts established in Chapter 2, it is p ossible to calculate the higher dimen- sional version for the momen t brac ket. In addition the tensorial in terpretation also pro vides the hin t to calculate momen ts of the general type A m,k = R p m g k f ( q , p , g , t ) d N q d N p d N g , where p o wers ha ve to b e in tended as tensor p ow ers, so that A n,k ( q , t ) = Z D ⊗ n p ⊗ k g f ( q , p , g , t ) d 3 q ∧ d 3 p ∧ d 3 g (6.15) CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 157 Figure 6.3: Spontaneous emergence of clump ed states in tw o dimensions. Random ini- tial conditions break up into dots. The expression for the energy interaction is E = 1 / 2 R H ( x − y ) ( ρ ( x ) ρ ( y ) + m ( x ) · m ( y )), where H ( x − y ) = e −| x − y | . The left plot shows the smo othed densit y ¯ ρ , while righ t plot sho ws the mo dulus | m | (Figure by V. Putk aradze). CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 158 Figure 6.4: An example of t wo oriented ﬁlaments (red and green) attracting each other and un winding at the same time. The blue v ectors illustrate the vector m at each p oin t on the curv e. Time scale is arbitrary . (Figure by V. Putk aradze) CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 159 where D = T ∗ q Q × g . This can b e written in terms of the basis as A n,k ( q , t ) = Z D  p i d q i  n ( g a e a ) k f ( q , p , g , t ) d 3 q ∧ d 3 p ∧ d 3 g = Z D p i 1 . . . p i n d q i 1 . . . d q i n g a 1 . . . g a k e a 1 . . . e a k f ( q , p , g , t ) d 3 q ∧ d 3 p ∧ d 3 g = ( A n,k ( q , t )) i 1 ,...,i n , a 1 ,...,a k d q i 1 . . . d q i n e a 1 . . . e a k d 3 q In order to ﬁnd the higher dimensional momen t Lie-Poisson brack et one can follow the same steps as in Chapter 2 { G , H } = Z Z Z f n α m,h ( q ) p m ⊗ g h , β n,k ( q ) p n ⊗ g k o 1 d 3 q ∧ d 3 p ∧ d 3 g = Z Z Z f g k + h  p i 1 . . . p i m ∂ ( α m ) i 1 ,...,i m ∂ q k ∂ p j 1 . . . p j n ∂ p k ( β n ) j 1 ,...,j n − p j 1 . . . p j n ∂ ( β n ) j 1 ,...,j n ∂ q h ∂ p i 1 . . . p i m ∂ p h ( α m ) i 1 ,...,i m  d 3 q ∧ d 3 p ∧ d 3 g + Z Z Z f  g ,  ∂ ∂ g  α m,h ( q ) p m ⊗ g h  , ∂ ∂ g  β n,k ( q ) p n ⊗ g k   d 3 q ∧ d 3 p ∧ d 3 g = Z A m + n − 1 ,k + h h α m,h , β n,k i d 3 q + Z Z Z f p m + n α m,h ( q ) β n,k ( q )  g d C d bc ∂ g a 1 . . . g a h ∂ g b ∂ g l 1 . . . g l k ∂ g c  d 3 q ∧ d 3 p ∧ d 3 g = Z A m + n − 1 ,k + h h α m,h , β n,k i d 3 q + Z Z Z f p m + n ( α m,h ) a 1 ,...,a h − 1 ,b ( β n,k ) l 1 ,...,l k − 1 ,c  hk g d C d bc g a 1 . . . g a h − 1 g l 1 . . . g l k − 1  d 3 q ∧ d 3 p ∧ d 3 g = Z A m + n − 1 ,k + h h α m,h , β n,k i d 3 q + h k Z Z Z f p m + n  g a 1 . . . g a h + k − 1  ( α m,h ) a 1 ,...,a h C a h + k − 1 a h a h + k ( β n,k ) a h +1 ,...,a h + k d 3 q ∧ d 3 p ∧ d 3 g =  A m + n − 1 ,k + h , h α m,h , β n,k i  + h k D A m + n, h + k − 1 , C α m,h ⊗ β n,k E = :  A m + n − 1 ,k + h , h α m,h , β n,k i  + D A m + n, h + k − 1 , h α m,h , β n,k i g E In conclusion, one summarizes in the following CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 160 Prop osition 55 The moments deﬁne d in e quation (6.15) ar e symmetric c ontr avariant n + k -tensors deﬁne d on ⊗ n T q Q ⊗ k g . These quantities under go Lie-Poisson dynamics, whose Poisson br acket is given by the fol lowing expr ession { F , G } =  A m + n − 1 ,h + k ,  n  δ E δ A n,k ∇  δ F δ A m,h − m  δ F δ A m,h ∇  δ E δ A n,k  +  A m + n, h + k − 1 , C  h δ F δ A m,h ⊗ k δ E δ A n,k  wher e C is the structur e tensor of g and summation over al l indexes is intende d. It is easy to see that if one considers only ( m, h ) , ( n, k ) ∈ { (0 , 0) , (0 , 1) , (1 , 0) } , then one reco vers the GHK brack et for chromoh ydro dynamics [GiHoKu1982, GiHoKu1983]. 6.4 A higher order of approximation This section extends the previous moment equations to include also higher order moments, whic h represent higher accuracy in the model. In this treatment the equations do not close exactly and one needs to formulate a suitable closure, suc h as the cold-plasma closure. Remark ably , this closure uniquely determines the moment dynamics and do es not require an y other hypothesis on the mo del. This in tro duction of a momentum dynamical v ariable arises in a natural wa y and its equation can b e written in terms of the other moments without introducing further higher terms. 6.4.1 Moment dynamics The starting equation is (6.5) ∂ A 0 ,λ ∂ t = ad ∗ γ ] 1 ,ν g ν g λ A 0 d g + Z g λ * g , " ∂ A 1 ∂ g , ∂ ( g σ γ ] 1 ,σ ) ∂ g # + " ∂ A 0 ∂ g , ∂ ( g σ γ ] 0 ,σ ) ∂ g # !+ d g = ∂ ∂ q  γ ] 1 ,ν Z g ν g λ A 0 d g  + Z g λ * g , " ∂ A 1 ∂ g , ∂ ( g a γ ] 1 ,a ) ∂ g # + " ∂ A 0 ∂ g , ∂ ( g a γ ] 0 ,a ) ∂ g # !+ d g , (6.16) CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 161 where one deﬁnes the analogues of Darcy’s velocities: γ 0 ,ν := µ [ f ] ? 0 ,ν δ E δ f = Z g ν  g ,  ∂ e µ k ∂ g , ∂ ( g a β a k ) ∂ g  d g = Z g ν  g ,  ∂ e µ 0 ∂ g , ∂ ( g a β a 0 ) ∂ g  d g and γ 1 ,ν := µ [ f ] ? 1 ,ν δ E δ f = ad ∗ β σ k Z g ν g σ e µ k d g + Z g ν  g ,  ∂ e µ k +1 ∂ g , ∂ ( g a β a k ) ∂ g  d g = ∂ β σ 0 ∂ q Z g ν g σ e µ 0 d g + Z g ν  g ,  ∂ e µ 1 ∂ g , ∂ ( g a β a 0 ) ∂ g  d g . Here the only assumption is that the energy functional E dep ends only on A 0 ,λ (recall that β λ n := δ E /δ A n,λ ), so that it is p ossible to ﬁx k = 0 in the ﬁrst line of the equation ab o ve. It is now con venien t to introduce the following notation ρ = Z f d g d p , G = Z g f d g d p , J = Z p g f d g d p , ¯ T = Z g g f d g d p . and analogously for the mobilities µ ρ = Z µ [ f ] d g d p , µ G = Z g µ [ f ] d g d p , µ J = Z p g µ [ f ] d g d p , ¯ K = Z g g µ [ f ] d g d p . where g g := g a g b e a ⊗ e b and ¯ K := ¯ K ab e a ⊗ e b . In terms of these quantities, one ma y write the following. Theorem 56 The moment e quations for ρ and G ar e given by ∂ ρ ∂ t = ∂ ∂ q ρ  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ! + ∂ ∂ q * G, µ ] G ∂ ∂ q δ E δ ρ +  ad ∗ δE δG µ J  ] +  ¯ K · ∂ ∂ q δ E δ G  ] + (6.17) and ∂ G ∂ t = ∂ ∂ q G  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ! + ∂ ∂ q ¯ T ·  µ ] G ∂ ∂ q δ E δ ρ +  ¯ K · ∂ ∂ q δ E δ G  ] +  ad ∗ δE δG µ J  ]  ! + ad ∗ „ µ G ∂ ∂ q δE δρ + ¯ K · ∂ ∂ q δE δG + ad ∗ δE δG µ J « ] J + ad ∗ „ ad ∗ δE δG µ G « ] G (6.18) wher e the symb ol ( · ) stands for c ontr action in the Lie algebr a, for example ( ¯ T · Γ) a := ¯ T ab Γ b . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 162 The pro of of this theorem is given in Section 6.4.3. Remark 57 In e quation (6.18), the tensor ¯ T plays the r ole of a Lie algebra-pressure tensor which gener ates the se c ond adve ction term in the e quation for G , exactly as it happ ens for the or dinary pr essur e tensor in the motion of c ompr essible ﬂuids. Mor e over, one c an se e that the last term in the e quation for G is a dissip ative term, which involves only quantities in the (dual) Lie algebr a and do es not intr o duc e any further adve ction term in sp ac e. This term gener alizes the L andau-Lifschitz dissip ation in so (3) to any Lie algebr a g . These equations need a suitable closure, obtained, for example, by expressing the un- kno wn quantities ¯ T , ¯ K , and J in terms of the dynamical v ariables ρ and G . This can b e done b y using the c old plasma formulation . (Other closures would also be p ossible, but these are not considered here). In this wa y , one can easily ﬁnd a closure for ¯ T and ¯ K . Ho wev er, this is not enough for the closure of the ﬂux J , which instead will b e determined b y the evolution equation for the ﬁrst order moments. 6.4.2 Cold plasma form ulation and momen t closure The cold-plasma solution of the Vlasov equation is given by the following pro duct of delta functions in momentum and orien tation, f ( q , p, t ) = ρ ( q , t ) δ ( p − ¯ p ( q , t )) δ ( g − ¯ g ( q , t )) , (6.19) so that (with indices suppressed) G = ρ ¯ g , J = G ¯ p , ¯ T = 1 ρ GG , It remains to mo del the phase space mobility µ [ f ] appropriately . One p ossibility w ould b e to take µ [ f ] = µ ρ ( q , t ) δ ( p − µ ¯ p ( q , t )) δ ( g − µ ¯ g ( q , t )) so that µ G = µ ρ µ ¯ g , µ J = µ G µ ¯ p , ¯ K = 1 µ ρ µ G µ G . In this case the equation for ρ is ∂ ρ ∂ t = ∂ ∂ q ρ  1 +  G ρ , µ ] G µ ρ   µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  +  G,  ad ∗ δE δG µ J  ]  ! (6.20) Similarly , one ﬁnds the following equation for the macroscopic orientation G : ∂ G ∂ t = ∂ ∂ q G  1 +  G ρ , µ ] G µ ρ   µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  + G ρ  G,  ad ∗ δE δG µ J  ]  ! +  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ad ∗ µ ] G µ ρ J + ad ∗ „ ad ∗ δE δG µ J « ] J + ad ∗ „ ad ∗ δE δG µ G « ] G (6.21) CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 163 These equations comprise the cold-plasma closure of the exact (but incomplete) equations (6.17,6.18). T o complete the pro cess, one needs to ﬁnd a closure for the Lie algebra-v alued ﬂux J . This closure arises very naturally from the momen t equation for A 1 ,λ . Remark 58 It is worth to emphasize that the c old plasma appr oximation and the line arity Assumption 47 ar e suﬃcient for c omplete closur e of the system. No additional assumptions wil l b e ne e de d. F rom (6.2) one deduces that ∂ A 1 ,λ ∂ t = ad ∗ γ ] 0 ,ν Z g ν g λ A 0 d g + ad ∗ γ ] 1 ,ν Z g ν g λ A 1 d g + Z g λ * g , " ∂ A 2 ∂ g , ∂ ( g a γ ] 1 ,a ) ∂ g # + " ∂ A 1 ∂ g , ∂ ( g a γ ] 0 ,a ) ∂ g #!+ d g . In the particular case λ = 0, the equation is written as ∂ A 1 , 0 ∂ t = ad ∗ γ ] 0 ,ν A 0 ,ν + ad ∗ γ ] 1 ,ν A 1 ,ν = A 0 ,ν ∂ γ ] 0 ,ν ∂ q + £ γ ] 1 ,ν A 1 ,ν = A 0 ,ν  γ ] 0 ,ν + £ γ ] 1 ,ν A 1 ,ν No w, from the cold plasma approximation (6.19) one obtains A 1 , 0 = Z p ρ δ ( p − ¯ p ) δ ( g − ¯ g ) d p d g = ρ ¯ p . Ph ysically , the quantit y A 1 , 0 = ρ ¯ p =: M is the macroscopic momen tum. Since γ 0 , 0 = 0, then the evolution equation for M is ∂ M ∂ t = A 0 ,a ∂ γ ] 0 ,a ∂ q + £ γ ] 1 , 0 M + £ γ ] 1 ,a J a =  G, ∂ ∂ q  ad ∗ δE δG µ G  ]  + γ ∂ M ∂ q + 2 M ∂ γ ∂ q +  ∂ J ∂ q , ¯ γ  + 2  J, ∂ ¯ γ ∂ q  where the following notation is in tro duced γ : = γ ] 1 , 0 = µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ¯ γ : = µ ] G µ ρ  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  +  ad ∗ δE δG µ J  ] so that ¯ γ a = γ ] 1 ,a . Now, by the cold plasma approximation (6.19), the ﬂux J and its corresp onding generalized mobility µ J ma y b e written as J = 1 ρ G ⊗ M , µ J = 1 µ ρ µ G ⊗ µ M . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 164 Th us, the ﬂux of orientation J is asso ciated with an induced mean momentum M . The ﬁnal equation for M can b e written as ∂ M ∂ t =  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ∂ ∂ q  M ρ  ρ +  G, µ ] G µ ρ  + M ρ  G, ∂ ∂ q µ ] G µ ρ  ! + 2 M ρ  ρ +  G, µ ] G µ ρ  ∂ ∂ q  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  +  ∂ ∂ q  M ρ G  + 2 M ρ G,  ad ∗ δE δG µ G  ]  +  G, ∂ ∂ q  ad ∗ δE δG µ G  ]  . (6.22) This equation for the ﬂuid momentum provides the necessary closure of the system. The corresp onding equations for the density ρ and orientation densit y G b ecome ∂ ρ ∂ t = ∂ ∂ q ρ  1 +  G ρ , µ ] G µ ρ   µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  + µ M µ ρ  G,  ad ∗ δE δG µ G  ]  ! (6.23) ∂ G ∂ t = ∂ ∂ q G  1 +  G ρ , µ ] G µ ρ   µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  + G ρ µ M µ ρ  G,  ad ∗ δE δG µ G  ]  ! + M ρ  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ad ∗ µ ] G µ ρ G +  1 + M ρ µ M µ ρ  ad ∗ „ ad ∗ δE δG µ G « ] G (6.24) Remark 59 The last term in e quation (6.22) is a sour c e of momentum M which must vanish for M = 0 to b e a ste ady solution. 6.4.3 Pro of of Theorem 56 The moment equations in Theorem 56 for particles with anisotropic interactions are derived as follows. One starts with equation (6.16) ∂ A 0 ,λ ∂ t = ∂ ∂ q  γ ] 1 ,ν Z g ν g λ A 0 d g  + Z g λ * g , " ∂ A 1 ∂ g , ∂ ( g a γ ] 1 ,a ) ∂ g # + " ∂ A 0 ∂ g , ∂ ( g a γ ] 0 ,a ) ∂ g # !+ d g with velocities given b y γ 0 ,ν = Z g ν  g ,  ∂ e µ 0 ∂ g , ∂ ( g a β a 0 ) ∂ g  d g γ 1 ,ν = ∂ β σ 0 ∂ q Z g ν g σ e µ 0 d g + Z g ν  g ,  ∂ e µ 1 ∂ g , ∂ ( g a β a 0 ) ∂ g  d g . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 165 No w, ﬁx λ = 0 in equation (6.16), so that the equation for ρ := A 0 , 0 is ∂ ρ ∂ t = ∂ ∂ q  γ ] 1 , 0 Z A 0 d g  + ∂ ∂ q  γ ] 1 ,a Z g a A 0 d g  = ∂ ∂ q ( γ ρ ) + ∂ ∂ q h G, γ i where the other terms in (6.16) cancel by integration by parts and one deﬁnes γ := γ 1 , 0 and γ a := γ ] 1 ,a . F or γ 1 ,ν one writes γ 1 ,ν = ad ∗ δE δρ Z g ν µ 0 d g + Z g ν g a ad ∗ δE δG a µ 0 d g + Z g ν  g ,  ∂ µ 1 ∂ g , δ E δ G  d g where µ n := R p n µ [ f ] dpdg and one remem b ers that β λ n := δ E /δ A n,λ . Therefore γ := γ 1 , 0 = ad ∗ δE δρ µ ρ + ad ∗ δE δG a Z g a µ 0 dpdg = µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  . In what follows the follo wing Lemma will b e useful. Lemma 60 L et g b e a ﬁnite-dimensional Lie algebr a. Given η ∈ g and a function f ( g ) on g ∗ , the fol lowing holds Z g  g ,  ∂ f ∂ g , η  d g = ad ∗ η G wher e G := R g f ( g ) d g and g ∈ g ∗ . Pro of. One calculates Z g  g ,  ∂ f ∂ g , η  d g = − Z g  ad ∗ η g , ∂ f ∂ g  d g = − Z g ∂ ∂ g ·  f ad ∗ η g  d g = Z f ad ∗ η g d g where w e hav e used resp ectively the deﬁnition of ad and ad ∗ , the Leibnitz rule and the in tegration by parts. The thesis follo ws immediately . By using the Lemma ab ov e one ﬁnds γ γ := µ ] G ∂ ∂ q δ E δ ρ +  ad ∗ δE δG µ J  ] +  ¯ K · ∂ ∂ q δ E δ G  ] . Substituting these expressions into the equation for ρ yields the explicit moment equation for the mass density ∂ ρ ∂ t = ∂ ∂ q ρ  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ! + ∂ ∂ q * G, µ ] G ∂ ∂ q δ E δ ρ +  ad ∗ δE δG µ J  ] +  ¯ K · ∂ ∂ q δ E δ G  ] + . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 166 No w let λ = a in (6.16). The equation b ecomes ∂ G a ∂ t = ad ∗ γ ] 1 , 0 G a + ad ∗ γ ] 1 ,b Z g b g a A 0 d g + Z g a * g , " ∂ A 1 ∂ g , ∂ ( g b γ ] 1 ,b ) ∂ g # + " ∂ A 0 ∂ g , ∂ ( g b γ ] 0 ,b ) ∂ g # !+ d g whic h ma y b e written more compactly as ∂ G ∂ t = ∂ ∂ q  γ G + ¯ T · γ  + ad ∗ γ J + ad ∗ Γ G where one uses again the Lemma 60 and introduces Γ a := γ ] 0 ,a . On the other hand, one has (again by Lemma 60) Γ =  ad ∗ δE δG µ G  ] and substituting into the equation for G , one has ∂ G ∂ t = ∂ ∂ q G  µ ρ ∂ ∂ q δ E δ ρ +  µ G , ∂ ∂ q δ E δ G  ! + ∂ ∂ q ¯ T ·  µ ] G ∂ ∂ q δ E δ ρ +  ¯ K · ∂ ∂ q δ E δ G  ] +  ad ∗ δE δG µ J  ]  ! + ad ∗ „ µ G ∂ ∂ q δE δρ + ¯ K · ∂ ∂ q δE δG + ad ∗ δE δG µ J « ] J + ad ∗ „ ad ∗ δE δG µ G « ] G This ﬁnishes the deriv ation of the moment equations in Theorem 56 for particles with anisotropic interactions. The present treatment has shown how the kinetic moments in Vlasov dynamics can b e extended to include anisotropic interactions. Their Lie-Poisson dynamics has b een found explicitly and diﬀerent levels of appro ximations hav e b een presented for the dynamics of the ﬁrst moments. The simplest mo del extends Darcy’s law to anisotropic in teractions and allo ws for singular solutions in an y spatial dimension. A second level of approximation is giv en by a truncation of the moment Lie algebra and determines a non trivial dynamics of the ﬂuid momen tum v ariable. The next section fo cuses on another kind of momen ts, whic h do not dep end only on the spatial co ordinate, but also on the orien tation.These momen ts are commonly known as “Smoluc howski momen ts”. 6.5 Smoluc ho wski approac h to momen t dynamics This section considers the Smolucho wski approac h to the description of the interaction of anisotropic particles. Usually , these particles are assumed to b e ro d-lik e, so their orien tation CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 167 can b e describ ed b y a p oin t on a tw o-dimensional sphere S 2 [DoEd1988]. Ho wev er, this section considers particles of arbitrary shap e, for which one needs the full S O (3) to deﬁne their orientation. The next section presents an example of Smolucho wski approach for the corresp onding Lie algebra so (3), while the later sections deal with a general ﬁnite- dimensional Lie algebra g . 6.5.1 A new GOP-Smoluc howski equation A ﬁrst example of a Smoluc howski approach can b e given in geometric terms as follows. An equation can b e form ulated for a distribution function on the ( x , m )-space spanned b y p osition and orientation. The evolution equation for the distribution ϕ ( x , m , t ) may be written as a conserv ation form along the velocity U = ( U x , U m ) on the ( x , m )-space. F or x ∈ R 3 one writes ∂ ϕ ∂ t = − div ( x , m ) ( ϕ U ) = − ∇ x · ( ϕ U x ) − ∂ ∂ m · ( ϕ U m ) . A t this p oint one has to choose appropriate v elo cities in order to resp ect the nature of Darcy’s law (or, more mathematically , GOP theory). A p ossibilit y is to introduce Darcy v elo city U x = ˙ x = µ [ ϕ ] ∇ δ E δ ϕ while a suitable choice for U m is given b y the rigid b o dy dynamics on so (3) U m = ˙ m = m × ∂ ∂ m δ E δ ϕ so that the ﬁnal equation can b e written as ∂ ϕ ∂ t = div  ϕ µ [ ϕ ] ∇ δ E δ ϕ  +  ϕ,  µ [ ϕ ] , δ E δ ϕ  (6.25) where {· , ·} denotes the rigid b o dy brac ket { g , h } := m · ∂ m g × ∂ m h . Theorem 61 The e quation (6.25) is a GOP e quation with r esp e ct to the dir e ct sum Lie algebr a X ( R 3 ) ⊕ X can ( so ∗ (3)) . Pro of. Consider the action of a vector ﬁeld v ∈ X ( R 3 ) on the densit y v ariable ϕ ∈ Den( R 3 ): v · ϕ = £ v ϕ = div ( v ϕ ) CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 168 and consider the action of the Hamiltonian function h ∈ X can ( so ∗ (3)) on ϕ h · ϕ = ad ∗ ∂ h/∂ m m · ∂ ϕ ∂ m = { h, ϕ } No w consider the action of the direct sum on the densities on the ( x , m )-space ( v ⊕ h ) · ϕ = div ( ϕ v ) + ad ∗ ∂ h/∂ m m · ∂ ϕ ∂ m This is the action of the Lie algebra X ( R 3 ) ⊕ X can ( so ∗ (3)). Deﬁne the dual action h ϕ  k , v ⊕ h i := h k , ( v ⊕ h ) · ϕ i The GOP equation is deﬁned as ˙ ϕ =  µ [ ϕ ]  δ E δ ϕ  ] · ϕ By integration b y parts, it is easy to see that ( ϕ  k ) ] = ϕ ∇ k · ∂ ∂ x + ad ∗ ∂ { ϕ, k } /∂ m m · ∂ ∂ m so that the GOP equation is ∂ ϕ ∂ t = div  ϕ µ [ ϕ ] ∇ δ E δ ϕ  +  ϕ,  µ [ ϕ ] , δ E δ ϕ  and the thesis is prov en. Consequen tly this equation expresses a geometric dissipativ e ﬂow in the con text of GOP- double brack et theory . By the usual arguments, one shows that when δ E /δ ϕ is suﬃciently smo oth, this equation allows for singular solutions of the form ϕ ( x , m , t ) = X i w i Z δ ( x − Q i ( s, t )) δ ( m − Λ i ( s, t )) d s where w i denotes the weigh t of the i -th particle and s is a coordinate of a submanifold of R 3 × so (3) ' R 6 . The next sections sho w ho w a more complete Som ulcho wski approac h can b e deriv ed from the dissipative Vlaso v equation, which is preferable to the present form ulation obtained by ad ho c arguments. 6.5.2 Systematic deriv ation of moment equations In the Smolucho wski approach, momen ts are deﬁned as A n ( q , g ) := Z p n f ( q , p, g ) dp . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 169 F rom the general theory , these moments are dual to β n ( q , g ), whic h are introduced by expanding the Hamiltonian function h ( q , p, g ) as h ( q , p, g ) = p n β n ( q , g ). The quantities β n ha ve a Lie algebr a br acket giv en b y [ [ β n , α m ] ] 1 = [ [ β n , α m ] ] + h g , [ β 0 n , α 0 m ] i , where prime denotes partial deriv ative with resp ect to g and [[ · , · ]] denotes the moment Lie brac ket. The Lie algebr a action is given b y β n · f = £ b X p n β n f where the vector ﬁeld b X h w as deﬁned in Section 6.2. The dual action is given b y D f ? n k , β n E := D f , β n k E = D f ? k , p n β n ( q , g ) E =  Z p n { f , k } 1 dp , β n  , and the star op er ator is deﬁned explicitly for k = p m α m as f ? n k = ad ∗ α m A m + n − 1 +  g ,  ∂ A m + n ∂ g , ∂ α m ∂ g  . The c o adjoint action op er ator ad ∗ is the Kup ershmidt-Manin operator deﬁned section 2.2. One introduces the dissip ative br acket by ˙ F = {{ F, E }} = −  µ [ f ] ? n δ E ∂ f , f ? n δ F ∂ f  . (6.26) By using this evolution equation for an arbitrary functional F , the rate of c hange for zero-th momen t A 0 is found to b e ∂ A 0 ∂ t = ad ∗ γ n A n − 1 + { A n , γ n } where {· , ·} stands for the Lie-Poisson brac k et on g { A n , γ n } :=  g ,  ∂ A n ∂ g , ∂ γ n ∂ g  . As usual, summation o ver repeated indices is assumed, n ≥ 0.One truncates this sum, b y taking n ≤ 1 so that ∂ A 0 ∂ t = ∂ ∂ q ( γ 1 A 0 ) + { A 0 , γ 0 } + { A 1 , γ 1 } (6.27) where the Darcy velocities are given b y γ n := Z p n  µ [ f ] , δ E δ f  1 d p . In particular, one ﬁnds γ 0 = { µ 0 , β 0 } , γ 1 = ad ∗ β 0 µ 0 + { µ 1 , β 0 } . CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 170 6.5.3 A cold plasma-lik e closure T o close the system for A 0 , it is necessary to ﬁnd an ev olution equation for the ﬁrst moment A 1 . Again, one uses the dissipative brac ket (6.26), and truncates the sum in the brack et to include A 0 , A 1 and A 2 terms. Con tinuing this pro cedure to write an equation for A k , w ould require including A 0 , A 1 , . . . , A k + m . Suc h extensions are p ossible, but they lead to v ery cumbersome calculations and there is no clear physical wa y of justifying the closure. The equation for A 1 is the following: ∂ A 1 ∂ t = ad ∗ γ 0 A 0 + ad ∗ γ 1 A 1 + { A 1 , γ 0 } + { A 2 , γ 1 } = A 0 ∂ γ 0 ∂ q + £ γ 1 A 1 + { A 1 , γ 0 } + { A 2 , γ 1 } where A 1 is a one-form densit y in the p osition space (from the momen t theory), and the Lie deriv ativ e has to b e computed accordingly . One introduces the cold-plasma approximation (cf. equation (6.19)) f ( q , p, g ) = A 0 ( q , g ) δ  p − A 1 ( q , g ) A 0 ( q , g )  so that A 2 = A 2 1 A 0 and the equation for A 1 closes to b ecome ∂ A 1 ∂ t = A 0 ∂ γ 0 ∂ q + £ γ 1 A 1 + n A 1 , γ 0 o +  A 2 1 A 0 , γ 1  . The ﬁnal brack et form of the moment equations is thus ∂ A 0 ∂ t = ∂ ∂ q  A 0  µ 0 ∂ β 0 ∂ q + n µ 1 , β 0 o  + n A 0 , n µ 0 , β 0 oo +  A 1 ,  µ 0 ∂ β 0 ∂ q + n µ 1 , β 0 o  (6.28) and ∂ A 1 ∂ t = A 0 ∂ ∂ q n µ 0 , β 0 o +  µ 0 ∂ β 0 ∂ q + n µ 1 , β 0 o  ∂ A 1 ∂ q + 2 A 1 ∂ ∂ q  µ 0 ∂ β 0 ∂ q + n µ 1 , β 0 o  + n A 1 , n µ 0 , β 0 oo +  A 2 1 A 0 ,  µ 0 ∂ β 0 ∂ q + n µ 1 , β 0 o  (6.29) These equations contain spatial gradients combined with b oth single and double Poisson brac kets. By deﬁning a ﬂux F 01 = µ 0 ∂ β 0 ∂ q + n µ 1 , β 0 o (6.30) CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 171 the previous equations may be written compactly as ∂ A 0 ∂ t = ∂ ∂ q  A 0 F 01  + n A 0 , n µ 0 , β 0 oo + n A 1 , F 01 o (6.31) and ∂ A 1 ∂ t = ∂ ∂ q  A 1 F 01  + A 0 ∂ ∂ q n µ 1 , β 0 o + A 1 ∂ F 01 ∂ q + n A 1 , n µ 0 , β 0 oo +  A 2 1 A 0 , F 01  (6.32) 6.5.4 Some results on sp ecializations and truncations An in teresting feature of the Smolucho wski momen t equations is that they reco ver b oth the well know Landau-Lifshitz equation and the GOP-Smolucho wski equation (6.25) as particular cases. First, one sees that upon considering only γ 0 in the equation (6.27) this equation b ecomes ∂ A 0 ∂ t +  µ 0 , δ E δ A 0  , A 0  = 0 . whic h is an equation in double brac ket form. Now if one considers the linear moment G ( q , t ) = R g A 0 ( q , g , t ) d g and rep eats the same treatment as in the previous sections for the moment equations, then it is p ossible to express the equation for G as ∂ G ∂ t = ad ∗ „ ad δE δG µ G « ] G . Sp ecializing to the case G = m ∈ so (3) yields the purely dissipative Landau-Lifshitz equa- tion ∂ m ∂ t = m × µ m × δ E δ m . Another sp ecialization is to neglect the ﬁrst-order moments A 1 , µ 1 in equation (6.28). It is easy to see that this yields ∂ A 0 ∂ t = ∂ ∂ q  A 0  µ 0 ∂ β 0 ∂ q  + n A 0 , n µ 0 , β 0 oo whic h is exactly the equation (6.25) for β 0 = δ E /δ A 0 and A 0 = ϕ . Th us diﬀeren t specializations in the Smolucho wski moment equations yield diﬀeren t order of appro ximations. Indeed one can see, that the diﬀerence b etw een the dissipative Landau- Lifshitz equation and the GOP Smoluc howski equation (6.25) diﬀer in that the latter allows for particle motion with a v elo city which is prop ortional to the collective force (Darcy’s v elo city), while the ﬁrst tak es in to accoun t only magnetization eﬀects without considering particle displacement. CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 172 6.5.5 A divergence form for the momen t equations A t this p oin t it is conv enien t to introduce the following Lemma 62 Given any two functions h and f on the Lie algebr a g , the fol lowing r elation holds n h, f o :=  g ,  ∂ h ∂ g , ∂ f ∂ g  = − ∂ ∂ g ·  h ∂ ∂ g · ( f b g )  with g ∈ g wher e the antisymmetric tensor b g is deﬁne d in terms of the structur e c onstants C a bc as b g bc := g a C a bc Pro of. By the Leibnitz rule one has  g ,  ∂ h ∂ g , ∂ f ∂ g  = − ∂ ∂ g ·  h ad ∗ ∂ f ∂ g g  + h  ∂ ∂ g · ad ∗ ∂ f ∂ g g  . Also, one calculates, by the Leibnitz rule again and the an tisymmetry of the structure constan ts that ad ∗ ∂ f ∂ g g = g a C a bc ∂ f ∂ g b e c = ∂ ∂ g b  f g a C a bc  e c = ∂ ∂ g ·  f b g  ∂ ∂ g · ad ∗ ∂ f ∂ g g = ∂ ∂ g c  g a C a bc ∂ f ∂ g b  = g a C a bc ∂ 2 f ∂ g c ∂ g b = b g : ∂ ∂ g ⊗ ∂ ∂ g f = 0 where the symbol : stands for con traction of all indices. The result in the second line is justiﬁed by symmetry , as it inv olves a con traction of an antisymmetric tensor b g with the symmetric tensor ∂ g ⊗ ∂ g . This completes the pro of. By making use of this Lemma, one can rearrange equations (6.28-6.29) in to the following form ∂ A 0 ∂ t = ∂ ∂ q  A 0  µ 0 ∂ β 0 ∂ q − ∂ ∂ g ·  µ 1 ∂ ∂ g · ( β 0 b g )    + ∂ ∂ g ·  A 0 ∂ ∂ g ·  b g ∂ ∂ g ·  µ 0 ∂ ∂ g · ( β 0 b g )    + ∂ ∂ g ·  A 1 ∂ ∂ g ·  b g ∂ ∂ g ·  µ 1 ∂ ∂ g · ( β 0 b g )    − ∂ ∂ g ·  A 1 ∂ ∂ g ·  b g µ 0 ∂ β 0 ∂ q  CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 173 and ∂ A 1 ∂ t = − A 0 ∂ ∂ q  ∂ ∂ g ·  µ 0 ∂ ∂ g · ( β 0 b g )  +  µ 0 ∂ β 0 ∂ q − ∂ ∂ g ·  µ 1 ∂ ∂ g · ( β 0 b g )  ∂ A 1 ∂ q + 2 A 1 ∂ ∂ q  µ 0 ∂ β 0 ∂ q − ∂ ∂ g ·  µ 1 ∂ ∂ g · ( β 0 b g )  + ∂ ∂ g ·  A 1 ∂ ∂ g ·  b g ∂ ∂ g ·  µ 0 ∂ ∂ g · ( β 0 b g )    + ∂ ∂ g ·  A 2 1 A 0 ∂ ∂ g ·  b g ∂ ∂ g ·  µ 1 ∂ ∂ g · ( β 0 b g )    − ∂ ∂ g ·  A 2 1 A 0 ∂ ∂ g ·  b g µ 0 ∂ β 0 ∂ q  If one inserts the notation λ 0 ( q , g ) = ∂ ∂ g ·  µ 0 ∂ ∂ g · ( β 0 b g )  = ∂ ∂ g ·  µ 0 ad ∗ ∂ β 0 ∂ g g  = − n µ 0 , β 0 o (6.33) and similarly , λ 1 ( q , g ) = − { µ 1 , β 0 } , then it is p ossible to can write the ( A 0 , A 1 ) dynamics more compactly as ∂ A 0 ∂ t = ∂ ∂ q  A 0 F 01  + ∂ ∂ g ·  A 0 ad ∗ ∂ λ 0 ∂ g g − A 1 ad ∗ ∂ F 01 ∂ g g  (6.34) and ∂ A 1 ∂ t = ∂ ∂ q  A 1 F 01  − A 0 ∂ λ 1 ∂ q + A 1 ∂ ∂ q F 01 + ∂ ∂ g ·  A 1 ad ∗ ∂ λ 0 ∂ g g − A 2 1 A 0 ad ∗ ∂ F 01 ∂ g g  . These equations may also b e written in slightly more familiar form by writing the ad ∗ op erations explicitly in terms of deriv atives on the Lie algebra, ∂ A 0 ∂ t = ∂ ∂ q  A 0 F 01  + ∂ ∂ g ·  A 0 ∂ ∂ g · ( b g λ 0 ) − A 1 ∂ ∂ g · b g F 01  (6.35) ∂ A 1 ∂ t = ∂ ∂ q  A 1 F 01  − A 0 ∂ λ 1 ∂ q + A 1 ∂ ∂ q F 01 + ∂ ∂ g ·  A 1 ∂ ∂ g · ( b g λ 0 ) − A 2 1 A 0 ∂ ∂ g · ( b g F 01 )  (6.36) Remark 63 (Relation to Smoluc howski equations) A c onne ction may exist with the nonline ar “diﬀusion ” term div g ( G f ) in e quation (6) in [Co2005], wher e subscript g denotes CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 174 the metric on S 2 and G = ∇ g U + W for some sc alar U and a ve ctor ﬁeld W on S 2 . In the pr esent formulation, g is an element of Lie algebr a g , not of the Lie gr oup, the diver genc e terms ar e of the typ e div g  A 0 div g ¯ F  , wher e ¯ F is a (0 , 2) antisymmetric tensor over the Lie algebr a g . It is not p ossible for this tensor to b e diagonal. In p articular, if one c onsiders the c ase of the GOP Smoluchowski e quation in the diver genc e form ∂ A 0 ∂ t = ∂ ∂ q  A 0  µ 0 ∂ ∂ q δ E δ A 0   + ∂ ∂ g ·  A 0 ∂ ∂ g ·  b g λ 0   the p ossibility of a c onne ction app e ars mor e explicitly. In addition, classic al Smoluchowski e quations in [Co2005] do not have the A 1 c ontri- bution of the inher ent p article momentum. Inste ad, they c ouple the evolution of A 0 to the ambient ﬂuid motion describ e d by a variant of the Navier-Stokes e quations. In the pr esent appr o ach, no ambient ﬂuid motion is imp ose d, r ather the c ontinuum ﬂow is induc e d by the dynamics of orientation, le ading to the induc e d momentum A 1 . The pr esenc e of A 1 is an- other diﬀer enc e b etwe en the physic al interpr etation of the pr esent appr o ach and the classic al Smoluchowski tr e atment. The me aning of these diﬀer enc es b etwe en the r esults obtaine d her e and the Smoluchowski appr o ach [Co2005] wil l b e pursue d further in futur e work. 6.6 Summary and outlo ok The double-brac ket Vlaso v moment dynamics discussed here pro vides an alternative to both the v ariational-geometric approach of [HoPu2007] and the Smoluc howski treatment reviewed in [Co2005]. These are early da ys in this study of the beneﬁts aﬀorded b y the double-brac ket approac h to Vlaso v momen t dynamics. Ho wev er, the deriv ations of Darcy’s la w in (5.8) and the Gilbert dissipation term in (6.11) by this approach lends hop e that this direction will pro vide the systematic deriv ations needed for mo dern tec hnology of macroscopic mo dels for microscopic pro cesses in volving interactions of particles that dep end on their relative orien tations. Although some of these formulas ma y lo ok daun ting, they p ossess an internal consistency and systematic deriv ation that might b e worth pursuing further. Possible next steps will b e the following: • Extend the theory of straigh t ﬁlament consisten t of ro d-like particles to deformable media, • Perform the analysis of the mobility functionals in kinetic space µ [ f ] as well as the mobilities for each particular geometric quan tity µ ρ , µ G etc. CHAPTER 6. ANISOTR OPIC INTERACTIONS: A NEW MODEL 175 • Study the conditions for the emergence of weak solutions (singularities) in the macro- scopic (av eraged) equations. • Add more ph ysics to the moment approach. F or example, it could b e worth while to in vestigate the b eha vior of singularities in a relativistic version of the nonlo cal Darcy’s la w (5.8). This might provide some insight into galaxy clustering in the Universe, esp ecially if the sp on taneous emergence of singularities p ersists in the relativistic ap- proac h. Chapter 7 Conclusions and p ersp ectiv es This thesis has developed a geometric basis for modeling contin uum dynamics using double brac kets. It has established the geometric approach, prov en its eﬀectiveness and used it to rev eal new p ersp ectiv es for mo deling dissipative structures. It has dev elop ed new t yp es of in tegral-PDE systems that are av ailable in this approach with a sp ecial fo cus on emergen t singular solutions. The result is a framework and vista for p ossible applications for the new science of geo- metric momen t equations. These equations address physical and tec hnological very promis- ing phenomena whose mo deling description lies at the b oundary b et ween contin uum me- c hanics and kinetic theory . The particles that aggregate and form patterns are allo wed to b e anisotropic. The in- ternal degrees of freedom of such particles (including, for example, the nano-ro ds dev elop ed recen tly for exploring shap e and orien tation eﬀects in nanotechnology) inﬂuence their ag- gregation into patterns. The deriv ation of a wide v ariet y of these new mo dels shows the ric hness of the mo deling approac h developed here. F uture inv estigations will seek the ap- propriate applications of this new geometric approac h for deriving moment equations that p ossess singular solutions. This chapter summarizes the results obtained and outlines a plan for future research. 7.1 The Sc houten concomitan t and momen t dynamics This work has used the geometric formulation of moment dynamics to obtain macroscopic con tinuum description from the microscopic kinetic theory . The k ey idea is that the op era- 176 CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 177 tion of taking the moments is a Poisson map leading to the Kup ershmidt-Manin structure. The ﬁrst result is a geometric interpretation of the momen ts in terms of symmetric tensors on the conﬁguration space [GiHoT r2008]. This idea provides the identiﬁcation of the moment Lie brac ket with the symmetric Schouten br acket (or “concomitan t”) [BlAs79, Ki82, DuMi95], which is diﬀeren t from the Lie brack et presented b y Kup ershmidt [Ku1987, Ku2005] in terms of multi-indexes. This fact relates moment dynamics with the theory of inv arian t diﬀerential op erators [Ki82]. In formulas, the Schouten form of the momen t brac ket is giv en b y [GiHoT r2008] { F , G } = ∞ X n,m =0  A m + n − 1 ,  n  δ E δ A n ∇  δ F δ A m − m  δ F δ A m ∇  δ E δ A n  where β n ∇ := β i 1 , ... , i n n ∂ i n denotes the usual tensor contraction of indexes. The symmetry prop erty of the momen ts relates their geometry with the symmetric group S n in volving p ermutations of the n comp onents of the n -th moment A n . After all, this is not surprising, since the symmetric group S n is already in volv ed in other kinds of moments in kinetic theory , i.e. the statistical Vlaso v moments [HoLySc1990] and the BBGKY mo- men ts [MaMoW e1984] of the Liouville equation for the phase space distribution of a discrete n umber of particles. F or the kinetic moments treated here, the role of the symmetric group is not clear since it is related with the nature of coadjoin t motion, which is not known y et. Questions concerning the nature of the coadjoin t motion for the momen ts pro vide an in teresting topic for future research. After showing how diﬀeomorphisms act on the momen ts yielding the equations of ﬂuid dynamics, this work has reviewed some of the physical applications where moments play a cen tral role. In particular a new result concerns b e am dynamics in p article ac c eler ators [GiHoT r2007]: the dynamics of coasting b eams [V en turini] is gov erned b y the integrable Ben- ney equation [Be1973, Gi1981] and this explains the observ ation of nonlinear coherent struc- tures [ScF e2000] in several exp erimen ts [KoHaLi2001, CoDaHoMa04, BlBr&Al., MoBa&Al.]. So far these nonlinear excitations ha v e b een explained in terms of soliton behavior , while the fact that the Benney equation is disp ersionless suggests that solitons are unlikely to app ear in this con text. Rather these are coherent structures that cannot b e studied through simple p erturbativ e approaches. CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 178 7.2 Geo desic momen t equations and EPSymp The main ob jec tiv e of the ﬁrst part of this work (c hapters 2 and 3) is the study of ge o desic motion on the momen ts. This in vestigation has pro vided [GiHoT r05, GiHoT r2007] a clear explanation of this dynamics in terms of ge o desic ﬂow on the symple ctomorphisms Symp( T ∗ Q ) of the cotangent bundle, whic h is the natural extension of the geo desic ﬂo w on the diﬀeomorphisms Diﬀ( Q ) of the conﬁguration space, known as EPDiﬀ [HoMa2004]. (By analogy the geo desic ﬂow on the symplectic group has been called EPSymp.) Surprising similarities of this system hav e b een shown with the in tegrable Blo c h-Iserles system [BlIs, BlIsMaRa05], which is again a geodesic motion on the linear symplectomorphisms Sp( T ∗ Q ), i.e. the group of symplectic matrices. This direction provides an interesting topic to b e pursued in the next future. F or example, one wonders what relation holds b etw een the Blo c h-Iserles system and EPSymp. Do in tegrability issues arise for the latter? Also, singular solutions ha ve been analyzed for the geo desic momen t equations and they coincide with the single particle tra jectory [GiHoT r05, GiHoT r2007]. The fact that a tensor p o wer app ears in the singular solution A n ( q , t ) = Z ⊗ n P ( s ) δ ( q − Q ( s, t )) d s is not only justiﬁed by the single particle nature, but also by the fact that the p ow er is the only function that alw ays restricts these contra v arian t tensors to b e fully symmetric. The last observ ation provides an interpretation of these solutions in terms of momentum map [MaRa99] deﬁned on the cotangent bundle of the em b eddings Q : s 7→ x ∈ R 3 . The ev aluation of the momentum map at the p oin t ( Q , P ) alwa ys yields a sequence of con trav ariant symmetric tensors (the symmetry is guaran teed b y the p ow er ⊗ n P ), that is a kinetic moment . By following the same treatmen t in [HoMa2004], one writes the momen tum map as J : ( Q , P ) 7→ Z ⊗ n P ( s ) δ ( q − Q ( s, t )) d s . The geo desic moment equations hav e b een shown to p ossess r emarkable sp e cializa- tions , whose ﬁrst example is the integrable Camassa-Holm equation [CaHo1993, HoMa2004] (obtained for Hamiltonians dep ending only on A 1 ). When considering b oth moments A 0 and A 1 , the geodesic momen t equations yield the t wo comp onent Camassa-Holm equation [ChLiZh2005, F alqui06, Ku2007], which is again an integrable system. The geo desic momen t equations hav e also b een extended [GiHoT r2007] to include aniso- tr opic inter action by following the treatmen t in [GiHoKu1983]. Singular solutions hav e CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 179 b een analyzed as w ell as their m utual in teraction, yielding the problem of the interaction of t wo rigid b o dies. 7.3 Geometric dissipation The second part of this w ork (starting with chapter 4) presented a form of geometric ﬂow for geometric order parameters (GOP equations). This ﬂow arose in the work of Holm and Putk aradze [HoPu2007] during their eﬀorts to establish a geometric in terpretation of Darcy’s law [HoPu2005, HoPu2006]. Darcy’s la w (4.1) is also known as the “p orous media equation” and is used to mo del self-aggregation phenomena in physical applications. The fact that these phenomena can be mo delled by Darcy’s law mak es this equation an inter- esting opp ortunit y for its mathematical insp ection. Indeed Darcy’s la w turns out to hav e a geometric structure that suggests its applicabilit y to any quantit y belonging to any vec- tor space V acted on by a Lie algebra g (i.e. a g -mo dule V ). Nevertheless, the geometric structure of Darcy’s law presents an ambiguit y (cf. c hapter 4), which makes it not suﬃcien t for the extension to generic order parameters. Chapter 4 has shown ho w the requiremen t of singular solutions uniquely determines a general geometric structure, thereby generating what has been called GOP e quation [HoPu2007, HoPuT r2007]. This is is a dissipativ e ﬂo w [HoPu2007], whic h is generated b y the Lie group G corresp onding to the Lie algebra g = T e G and is completely justiﬁed b y thermo dynamic arguments [HoPuT r2007]. In formulas, when g = T e Diﬀ , the GOP equation for an order parameter κ ∈ V is given b y ∂ κ ∂ t + £ ` µ [ κ ]  δE δκ ´ ] κ = 0 The mathematical geometric structure of GOP equations can b e in terpreted in terms of an in v ariant Riemannian metric deﬁned on V ∗ [HoPu2007]. The symmetric nature of the metric is the mathematical reason for dissipation, in agreemen t with the w ork of Kaufmann and Morrison [Ka1984, Mo1984, Mo1986]. The main result in this w ork concerning the GOP equations is the existenc e of sin- gular solutions , which is made p ossible by the appropriate introduction of a “mobility functional”, that is a smoothed version of the dynamical v ariable itself. The smo othing pro cess yields an equation which is nonlo cal. In the case when the dynamical v ariable is acted on by diﬀeomorphisms (Lie deriv ative), the GOP equation is a characteristic equation along a smo oth vector ﬁeld, whic h includes the nonlo cal eﬀects. CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 180 Applications of this ﬂow hav e been proposed for diﬀer ential forms , which are cases of in terest in ph ysical applications (e.g. the magnetic ﬁeld in magnetized plasmas [HoMaRa]). In the case of exact forms, it has b een sho wn that singular solutions are allow ed for both the forms themselves and their p otentials and these solutions are diﬀerent in the tw o cases [HoPuT r2007]. 7.4 Dissipativ e equation for ﬂuid v orticit y A sp ecial case of dissipative dynamics is provided by the v orticity exact t wo–form in ﬂuid dynamics [MaW e83]. Indeed, it has b een shown how the GOP equation for the vorticit y in section 4.5 yields a double br acket dissip ation for p erfe ct inc ompr essible ﬂuids , thereb y reco vering the results in [BlKrMaRa1996] previously in tro duced in [V aCaY o1989]. The dissipative equation for the v orticity ∂ ω ∂ t + curl  ω × curl  δ H δ ω − curl  µ [ ω ] × curl δ E δ ω   = 0 has been sho wn to preserve many prop erties of the ideal case, suc h as Ertel’s theorem, Kelvin circulation theorem and the conserv ation of helicit y [HoPuT r2007]. The main diﬀerence from the vorticit y equation in the ideal case is the presence of a mo diﬁe d velo city , such that the c haracteristic velocity of the equation is giv en by the sum of the ideal v elo city and (min us) the “Darcy v elo city”, whic h tak es in to accoun t the dissipation and “slo ws down” the ﬂuid particles while preserving the coadjoint orbits as in the theory of double brack et dissipation. The tw o–dimensional case has als o b een presented to p ossess the same structure of the ideal case, but with a velocity suitably decreased in time by the dissipativ e eﬀects. The p oint vortex solution has b een analyzed [HoPuT r2007]. Another application has b een presented to the case of one form–densities , inv olving the Camassa-Holm equation [CaHo1993]. In this case the p eakon solutions undergo dissipa- tiv e dynamics [HoPuT r2007] and the equations of the p eakon lattice ha ve been presented. GOP equations hav e b een shown to reduce to double brack et equations when applied to v ariables whose Hamiltonian dynamics is given in Lie-P oisson form. The cases of the v orticity equation and the Camassa–Holm equation are clear examples of this situation. This fact constitutes one of the mathematical motiv ations for the remaining discussions in this work. CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 181 7.5 Geometric dissipation in kinetic theory The dissipativ e ﬂo w for geometric order parameters provides a basis for deriving a dissipativ e kinetic ﬂo w in terms of Vlasov equation. The fact that kinetic moments ar e a Poisson map [Gi1981] yields the corresp onding dissip ative ﬂow for the moments . The idea of a GOP equation for the Vlasov distribution directly in volv es the action of symplectomorphisms on the phase space densities. At the Lie algebra lev el, the Lie deriv ative is written as Poisson brack et and the fact that the GOP theory reduces to double brack et determines the dissipative Vlaso v equation [HoPuT r2007-CR] ∂ f ∂ t +  f , δ H δ f  =  f ,  µ [ f ] , δ E δ f  where E is a suitable energy functional, which is p ossibly diﬀerent from the Hamiltonian H and it is usually chosen to b e the collective potential. The case E = H and µ [ f ] ∝ f reduces to the equation presented in [BlKrMaRa1996] and the choice E = J (alwa ys with µ [ f ] ∝ f ) is presented in the w ork of Kandrup [Ka1991], who ﬁrst introduced this form of dissipative Vlaso v equation for applications in astrophysics. The ﬁrst consequence of this equation is that the evolution of f o ccurs in the form of coadjoin t motion and thus it allows all the Casimirs of the Hamiltonian case and more imp ortan tly the en tropy functional S = R f log f is also conserved [HoPuT r2007-CR]. The conserv ation of en tropy can b e physically interpreted in terms of r eversibility of the dynamics . Indeed, b eing a form of coadjoint motion, the evolution is giv en by the action of canonical transformations that are alwa ys inv ertible, b y deﬁnition of Lie group. Th us the ev olution of f can alwa ys b e in verted without any loss of information and this fact is the key to understand the preserv ation of en tropy . This case diﬀers from the conv en tional F okker–Planc k approac h, whic h is based on the h yp othesis of brownian motion through the Langevin sto c hastic equation. Ho wev er, in principle it is p ossible to recov er the increasing en tropy b y adding a diﬀusion term to the equation. This pro cess w ould stil l b e diﬀeren t from the Langevin approach that in volv es a linear dissipation in the microscopic equation, but w ould recov er the br ownian prop erty and th us would increase en tropy . The com bination of sto c hastic eﬀects with the deterministic eﬀects discussed here is a sub ject for future researc h. The single p article solution has been sho wn to b e consisten t with the t wo–dimensional v orticity equation. The existence of the single particle solution is an imp ortan t prop ert y , whic h is not shared with any other dissipative kinetic equation. Besides its absence in the F okker-Planc k theory , it is w orth mentioning that even the equations presented by Kandrup CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 182 [Ka1991] and Blo ch et al. [BlKrMaRa1996] do not p ossess the single particle solution. Moreo ver it is also imp ortant to notice that the existence of this solution has nothing to do with the preserv ation of entrop y , whic h is instead shar e d with the theory of Kandrup [Ka1991] and Blo ch et al. [BlKrMaRa1996]. 7.6 Double brac k et equations for the momen ts Once the dissipative Vlaso v equation has b een established, the present w ork has inv estigated the corresp onding moment dynamics. Making use of the double brack et theory , one can ﬁnd the dissipativ e double br acket form of moment dynamics [HoPuT r2007-CR], whose full expression (5.7) is rather complicated. Ho wev er it has b een shown ho w it is p ossible to construct diﬀerent closures of this hierarc hy by considering truncations at the zero–th or ﬁrst order [HoPuT r2007-CR]. The simplest example is Darcy’s la w: it has b een sho wn ho w the simplest trunc ation of the hierarch y in volving only the zero–th order moment coincides with Darcy’s la w. The im- p ortance of this result is that Darcy’s la w can no w b e provided with a complete justiﬁc ation in terms of kinetic the ory and this is the ﬁrst time this result has b een accomplished. There ha ve b een imp ortan t results concerning this p oint inv olving the F okk er-Planck treat- men t [Chav anis04]; how ever they require µ [ ρ ] = const for the mobilit y functional and the diﬀusion cannot b e neglected as done in the present treatmen t. Another simpliﬁcation of the moment hierarch y is what has b een called “Darcy ﬂuid”, i.e. the closure of the hierarc hy giv en by considering only the zero-th and the ﬁrst moments. The resulting equations ∂ ρ ∂ t + £ “ µ ρ  δE δρ + µ m  δE δm ” ] ρ = 0 ∂ m ∂ t + £ “ µ ρ  δE δρ + µ m  δE δm ” ] m = − ρ   £ δE δm µ ρ  ] are rather complicated, although each single term can b e identiﬁed both ph ysically and mathematically , thereby sho wing clearly the underlying geometric structure and its ph ysical meaning. The importance of this example is that it sho ws once again ho w the momen t brac ket is an extremely p ow erful to ol to obtain macroscopic ﬂuid mo dels starting from microscopic kinetic equations. The Dar cy ﬂuid e quations still allo w for the single particle solution (in the purely dissipativ e case) and for suitable c hoices of the energy may mo del the dissipativ e version of the t wo–component Camassa-Holm equation [ChLiZh2005, F alqui06, CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 183 Ku2007]. This point can be an interesting opp ortunit y for pursuing this direction further in terms of dissipative dynamics on semidirect product group of the type G s H , where H is an appropriate G -mo dule (in ﬂuid dynamics this is Diﬀ s Den, where Den denotes density v ariables). F or example, one may w onder what kind of p eakon dynamics corresp onds to this kind of ﬂow and the peakon-peakon in teraction would b e a p ossible case of study . F urther moment equations hav e b een sho wn to possess interesting b eha vior. F or example, the dissipative moment brack et has been used to form ulate a double br acket form of the b -e quation (5.12), whic h em bo dies to the dissipativ e Camassa-Holm equation [HoPuT r2007] as a sp ecial case. Also the GOP e quation for the moments (5.13) has been form ulated, reco vering b oth Darcy’s law and the dissipativ e Camassa-Holm equation as sp ecial cases. The b ehavior of singular solutions in these cases is also a p ossible direction to b e pursued further. 7.7 Anisotropic in teractions The previous eﬀorts to geometrize Darcy’s law hav e yielded its microscopic justiﬁcation in terms of kinetic theory . This can pro vide important insigh t in to self-aggregation phenomena, esp ecially at nano-scales, whic h is a wide open area of physical researc h. Ho wev er, many of the collective interactions of in terest in this area are anisotropic and most of the time the in teractions b et ween tw o particles dep end on their mutual orien tation. The orientation of a nano-p article can b e interpreted in terms of rigid-bo dy dynamics, so that each single particle is not a p oint particle, but rather it carries a momen t of inertia and th us it has a non-zero spatial length. A p ossible approac h for such systems has b een formulated by Gibb ons, Holm and Ku- p ershmidt (GHK) [GiHoKu1982, GiHoKu1983]. Thus the extension of Dar cy’s law to anisotr opic inter actions [HoPuT r2007-Poisson] has been sho wn to arise from the momen t equations of the double brack et form of the Vlaso v-GHK equation. Indeed, since the Vlaso v- GHK equation is in Lie-P oisson form, all the double brac ket theory can be transferred to this case. In this treatment the Vlaso v distribution f dep ends on p osition, momentum and orien tation [GiHoKu1982, GiHoKu1983] f = f ( q , p , g , t ) . CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 184 Once the double brack et equation [HoPuT r2007-Poisson] ∂ f ∂ t =  f ,  µ [ f ] , δ E δ f  1  1 is established, kinetic moments are in tro duced in the form A n,k ( q , t ) = Z p n g k f ( q , p , g , t ) d p d g and the moment theory can b e transferred to these p g -moments. The moment equations are again rather complicated but the main result of this work concerns the truncation of the hierarch y to consider the sp ecial case n = 0 , k = 0 , 1. The resulting equations are [HoPuT r08] ∂ ρ ∂ t = div ρ  µ ρ ∇ δ E δ ρ + µ m · ∇ δ E δ m  ! ∂ m ∂ t = div m ⊗  µ ρ ∇ δ E δ ρ + µ m · ∇ δ E δ m  ! + m × µ m × δ E δ m where ρ := A 0 , 0 , m := A 0 , 1 and µ ρ , µ m are ﬁltered versions of ρ and m resp ectiv ely . As one can easily see, the right hand side of the second equations r e c overs the L andau- Lifshitz-Gilb ert dissip ative dynamics for the magnetization in ferromagnetic media and this constitutes one of the main results of this w ork: the dissipative magnetization dynamics of Landau, Lifshitz and Gilb ert has b een recov ered from microscopic argumen ts in kinetic theory , by following a double brack et approach for the Vlasov equation. T o the author’s kno wledge this is the ﬁrst time that the Landau-Lifshitz-Gilbert dynamics is derive d fr om a micr osc opic kinetic tr e atment . This term is recov ered at all levels of approximation, since the double brack et preserv es the geometric structure of the dynamics, as explained in c hapter 5. The singular solutions allow ed by this mo del hav e b een extensiv ely analyzed in the presen t work in the one-dimensional case. How ever, imp ortant questions concern their b eha vior in three dimensions, when the tw o v ariables are supported on submanifolds of the Euclidean space (ﬁlaments and sheets), eac h following its own dynamics. F or example, in one dimension the singularities hav e b een sho wn to emerge sp on taneously , but do es this feature p ersist in more dimensions? How do the orien tation ﬁlaments in teract? All these questions need to b e answered in future research. Possible applications are suggested in protein folding and other issues in nano-sciences. As a further step, a higher order of approximation has b een introduced in the truncation of the moment equations, whic h takes into account the evolution of the ﬂuid momentum CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 185 A 1 , 0 as w ell as of the p olarization ﬂux A 1 , 1 . Ho wev er these equations are complicated and they do not allow for singular solutions. 7.8 The Smoluc ho wski approac h The last part of this work has presen ted what is known as Smoluc howski kinetic approach. In this context, the moments are still p -moments and they depend on both p osition and orien tation. Two p ossibilities hav e b een presented. The ﬁrst is simpler in construction and it leads to the GOP-Smoluchowski e quation [HoPuT r08] ∂ ϕ ∂ t = div  ϕ µ [ ϕ ] ∇ δ E δ ϕ  +  ϕ,  µ [ ϕ ] , δ E δ ϕ  (7.1) where {· , ·} denotes here the rigid b o dy brac ket { g , h } := m · ∂ m g × ∂ m h . The interesting feature of this equation is that it leads naturally to the Landau-Lifshitz equation for the magnetization m = R m ϕ ( q , m , t ) d m , when δ E /δ ϕ is constan t in q (otherwise it also leads to the previous equations for ρ and m ). This equation is not rigorously derived from the dissipativ e Vlasov equation; rather it is established as a GOP contin uity equation in the ( q , m )-space. The second approach follo ws the process of taking the p -moments of the dissipativ e Vlaso v equation. The resulting equations are complicated and the truncation to the ﬁrst momen t requires a cold plasma-like closure that does not allo w for singular solutions. Ho w- ev er it has b een sho wn ho w t wo particular truncations are p ossible, whose simplest one is iden tical to the Landau-Lifshitz-Gilbert equation. The GOP equation for ϕ is also obtained as the second sp ecialization. P ossible roads for future research inv olve the analysis of this hierarch y and in particular it is not clear how the app earance of the GOP equation can b e rigorously justiﬁed b y considering the geometric structure of the whole hierarch y . Also, the singular solutions allo wed b y the GOP equation may deserv e further study . 7.9 F uture ob jectiv es in geometric moment dynamics The study presen ted in this thesis raises new open questions, which are sk etched in this section. The following scheme presents a plan of ob jectiv es that is divided in tw o main topics, i.e. Hamiltonian and dissipativ e momen t ﬂows. The ﬁnal part is dev oted to the question of coadjoint momen t dynamics. CHAPTER 7. CONCLUSIONS AND PERSPECTIVES 186 Geo desic motion on Vlasov moments Singular solutions. Study of singular solutions for geo desic moment equations and their closures. Analysis of their spontaneous emergence. Study of the geometric properties of the ﬂuid closure (dual pairs and plasma-to-ﬂuid momentum map). Analysis of ﬁlaments and sheets in higher dimensions. Extension to oriente d nano-p articles. Study of singular solutions in the anisotropic case for nano-particles. Analysis of their interaction in higher dimensions, “orien tation ﬁlamen ts” and sheets. Conne ctions to inte gr able PDE’s. Developmen t of further connections with in tegrable systems, in particular the Blo ch-Iserles equation, which has the same geometric nature as the geo desic moment equations. Double-brac ket dissipation for momen t dynamics Singular solutions. Study of singular solutions of the dissipative moment equations, whic h hav e a v ery diﬀerent b ehavior from the Hamiltonian geo desic case. Analysis of the dissipativ e ﬂuid closure both in terms of singular solutions and its geometric prop erties (e.g. dual pair analogues for the dissipative EPDiﬀ equation). Study of singular solutions in higher dimensions. A nisotr opic inter actions. Study of singular solutions in the anisotropic case (oriented nano-particles), esp ecially in higher dimensions (some results on the interaction of tw o ori- en ted ﬁlaments hav e just b een published [HoPuT r08]). 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Geometric dynamics of Vlasov kinetic theory and its moments

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