Lorentz-boosted diffusion: initial value formulation and exact solutions

Loren tz-b o osted diffusion: initial v alue form ulation and exact solutions L. Ga v assino Dep artment of App lie d Mathematics and The or etic al Physics, University of Cambridge, Wilb erfor c e R o ad, Cambridge CB3 0W A, Unite d Kingdom It is w ell kno wn that the diffusion equation, when treated as a stand-alone p artial differen tial equation, exhibits exponential instabilities in b oosted frames, whic h render the corresp onding initial - v alue problem ill-posed. Recen tly , how ev er, it w as shown that Fick-t ype diffusion arises as the exact hydrodynamic sector of relativistic F okker-Planc k kinetic theory . In this work, we exploit this kinetic embedding to form ulate a mo dified initial -v alue problem for one-dimension al Lorentz-bo osted diffusion. W e show that the resulting dynamics are well p osed both forw ard and bac kw ard in time, pro vided the b o osted densit y profiles admit a kinetic-theory realization. Such profiles form a space of band-limited functions, within which the evo lution can be expressed as a discrete sup erp osition of spatially sampled initial data, weigh ted by a Shannon-Whittak er-t yp e Green function defined on the full Minko wski plane. The Green function is obtained in closed analytic form. I. INTR ODUCTION Let n ( t, x ) denote the densit y of an ensemble of relativistic particles constrained to mov e in one spatial dimension. Suc h particles are assumed to propagate through a medium at rest, whic h induces sto chastic scattering and gives rise to diffusive transport. In the long-wa v elength limit, the densit y field is exp ected to ob ey Fick’s la w [1, § 9.3], ∂ t n = ∂ 2 x n , (1) written here in natural units. A fundamen tal question in relativistic hydrodynamics [2 – 4] is how such a diffusive description should be form ulated in a reference frame ( ˜ t, ˜ x ) in whic h the medium mov es with constan t velocity v , i.e. ( ˜ t = γ ( t + v x ) , ˜ x = γ ( x + v t ) , (2) with γ = (1 − v 2 ) − 1 / 2 . Below, we briefly summarize the most natural (and common) strategies, and the difficulties asso ciated with each of them. A first p ossibilit y is to solv e (1) for prescribed initial data at t = 0, and then to apply a Loren tz bo ost to the resulting solution. This pro cedure has the limitation that, for generic initial data, solutions of (1) cannot be extended to negative times [5, § 2.1]. As a consequence, the b o osted density profiles are t ypically defined only in the region ˜ t > v ˜ x (corresponding to t > 0). In particular, there exists no time ˜ t at which the densit y profile is defined for all spatial p oin ts ˜ x . One may attempt to circum ven t this issue by restricting the space of admissible initial data so that the corresp onding solutions n ( t, x ) exist for all t ∈ R , but such a restriction app ears ph ysically unmotiv ated. F or instance, it would exclude compactly supp orted initial data, which are otherwise admissible. A second p ossibility is to b o ost equation (1) itself, ( ∂ ˜ t + v ∂ ˜ x ) n = γ ( ∂ ˜ x + v ∂ ˜ t ) 2 n , (3) and to solv e it for initial data at ˜ t = 0. This approach encoun ters a more sev ere obstruction: the resulting initial-v alue problem is ill-p osed in the sense of Hadamard, in that solutions fail to exist for generic Sob olev initial data [3]. The origin of this pathology lies in the presence of exponential instabilities (e.g., n = e ˜ t/ ( γ v 2 ) is a solution), whose gro wth rate diverges in the short-w a velength limit. F or this reason, (3) is commonly regarded as devoid of predictive conten t. A third line of attack, widely explored in the literature, consists in mo difying Fic k’s la w itself, so as to restore compatibilit y with relativistic requirements. In particular, guaranteeing stabilit y in all reference frames necessit ates rendering (1) causal [6], which can b e ac hieved by the inclusion of a second-order time deriv ative [7 – 10], as in ∂ t n = ( ∂ 2 x − ∂ 2 t ) n . The resulting mo del, commonly referred to as Cattaneo theory [11, § 6.5.1], can indeed b e shown (see Appendix A) to reproduce the exact density dynamics in a specific microscopic setting, namely one in which the particles are massless, and their v elo city ( V = ± 1) undergo es random sign flips with a momentum-independent rate [4]. The dra wbac k of this construction is that Cattaneo’s theory lac ks the univ ersal character of Fick’s la w [12–14]. Motiv ated by this limitation, a diff erent strategy has recently b een prop osed [4], in which (1) is left unchanged, but the Lorentz transformation is implemented only approximately . Thi s pro cedure preserv es the long-wa v elength, low- frequency behavior of the theory , while reducing the temp oral order of (3) so as to guarantee well-posedness, leading to ( ∂ ˜ t + v ∂ ˜ x ) n = γ − 3 ∂ 2 ˜ x n . The price paid in this approach is the explicit loss of Loren tz cov ariance: differen t observers no longer construct solutions that are related to one another by Loren tz transformations [15]. 2 In this article, w e explore yet another approac h. W e will start from the recent realization [16, 17] that, when the microscopic dynamics are go vern ed b y F okker-Planc k momentum diffusion, the densit y n ob eys Fic k’s la w (1) exactly within the h ydro dynamic sector, understo o d as the branc h of mo des con tinuously conn ected to the origin in frequency space. As a consequence, bo osting relativistic F okk er-Planck kinetic theory (whic h is itself cov arian tly stable [18]) results in a hydrodynamic sector gov erned by (3). Importantly , how ev er, not every solution of (3) admits a realization within the kinetic theory , as exp onen tially unstable solutions suc h as n = e ˜ t/ ( γ v 2 ) are automatically excluded. This observ ation prov ides a ph ysically motiv ated criterion for selecting admissible initial data for b o osted diffusion. W e exploit this criterion to deriv e an initial-v alue formulation of (3) restricted to a function space that is substan tially smaller than the Sob olev spaces conv en tionally emplo ye d in the theory of partial differen tial equations [19]. W e then find that such initial-v alue problem is w ell-p osed b oth forward and bac kward in time, and that its solutions admit an explicit represen tation in terms of a discrete Green-function expansion. Throughout the article, we w ork in natural units, c = ℏ = k B = Diffusivit y = 1. II. BOOSTED DIFFUSION FROM FOKKER-PLANCK KINETIC THEOR Y In this section, we show how the embedding of the diffusion equation in to relativistic F okke r-Planck kinetic theory , in tro duced in [16, 17], can be exploited to iden tify a ph ysically motiv ated reduced space of functions withi n which diffusion admits a well-posed form ulation. A. Diffusion as a sector of relativistic F okker-Planc k theory: brief recap Let p denote the momentum of an individual particle, ε ( p ) its energy , and V = dε/dp its velocity , all defined in the global rest frame of the external medium. The state of an ensem ble of particles is describ ed by the kinetic distribution function f ( t, x, p ) [20, § 3.1]. In our units, the relativistic Vlasov-F okk er-Planc k equation takes the form [21, 22] ( ∂ t + V ∂ x ) f = β − 2 ∂ p  e − β ε ∂ p  e β ε f  , (4) where β > 0 denotes the inv erse temperature of the external medium (which is assumed constan t). Within this framew ork, the hydrodynamic sector is spanned by modes of the form f = e − β ε + ik ( x − β p ) − iω t , (5) with k , ω ∈ C . In fact, substituting the ansatz (5) into (4), one finds that the kinetic equation is satisfied if and only if ω = − ik 2 , whic h coincides with the disp ersion relation of the diffusion equation (1). It follo ws immediately that the asso ciated density w a ve (recall that n = R dp 2 π f [23]), n = e ikx − k 2 t Z R dp 2 π e − β ε − ikβ p , (6) pro vides an exact solution of (1). By forming contin uous sup erp ositions of these mo des, one may therefore generate a broad class of solutions to the diffusion equation. There is, how ever , an imp ortant cav eat. F or a kinetic stat e to be physically admissible (and for n to b e defined), the momentum integral in (6) m ust con verge. In a Newtonian setting, where ε = p 2 / (2 m ), this integral is Gaussian and con verges for all k (since β > 0). In the relativistic case, the situation is qualitativ ely differen t: at large momenta, the energy grows linearly , ε ( p ) ∼ | p | . As a consequence, conv ergence requires that | Im k | < 1 . (7) This condition constitutes the fundamental consistency requiremen t imp osed by the underlying kinetic theory . 1 The b ound (7) automatically ensures that the analysis is restricted to a class of mo des for which diffusion is co v ariantly stable, in the sense that Im ω ≤ | Im k | [25, 26]. Indeed, since Im ω = Im ( − ik 2 ) = ( Im k ) 2 − ( Re k ) 2 , one has ( Im k ) 2 − ( Re k ) 2 ≤ ( Im k ) 2 ≤ | Im k | . Nevertheless, it is important to stress that the condition (7) is stronger than the mere requiremen t of cov arian t stability . F or instance, the mode with k = 3 + 2 i is excluded by equation (7), while it still satisfies the cov ariant stabilit y condition, since Im ω = − 5 < | Im k | = 2. The b ound (7) therefore represen ts a genuine ph ysical admissibility criterion, rather than a mere reformulation of stabilit y for equation (1). 1 In [17], an additional requirement was imp osed, namely that the information current [24] be finite (equiv alently , that f belong to the Hilbert space defined b y Onsager’s inner pro duct [18]), leading to the more restrictive bound | Im k | ≤ 1 / 2. In the presen t work, we only require finiteness of the density n , although the analysis can b e straightforw ardly adapted to incorporate stronger conditions. 3 B. W a ven um b er b ounds on bo osted diffusion Consider a densit y profile n ( t, x ) constructed as a linear superp osition of the modes (6), and suppose that it can b e writ ten as a spatially lo calized perturbation δ n ( t, x ) around a homogeneous background n 0 . If the p erturbation deca ys sufficien tly rapidly at spatial infinity and is sufficiently regular at t = 0, then it belongs to a standard class of localizable functions (e.g. the Sc h w artz class, or L 2 , or H s ). In this case, δ n ( t, x ) admits a represen tation as a con tin uous sup erposition of F ourier mo des , namely of excitations of the form (6) with k ∈ R . All suc h modes automatically satisfy the bound (7). Consequen tly , aside from the usual con v ergence requiremen ts associated with F ourier in tegrals [16], this construction allo ws one to span all lo calized smo oth initial conditions δ n (0 , x ). Let us now rep eat the same reasoni ng in a reference frame in which the background medium mo ves with velocity v > 0. W e again assume that, in this bo osted frame, the densit y p erturbation δn ( ˜ t, ˜ x ) is w ell defined ev erywhere at ˜ t = 0, so that a meaningfu l initial-v alue problem can be posed in this frame. W e also assume that the initial state δ n (0 , ˜ x ) is spatially lo calized and sufficien tly regular, so that it ma y b e expanded in modes that are of F ourier type in the b o oste d fr ame , namely excitations of the form (6) with ˜ k ∈ R [3]. Here, the b o osted w a vev ector ( ˜ ω, ˜ k ) is related to the original wa v evector ( ω, k ) app earing in the ansatz (5) via e i ˜ k ˜ x − i ˜ ω ˜ t ≡ e ikx − iω t , giving ( ω = γ ( ˜ ω − v ˜ k ) , k = γ ( ˜ k − v ˜ ω ) . (8) In contrast to the unbo osted case, ho w ev er, the kinetic admissibilit y condition (7) now imp oses non trivial restrictions on the set of allo w ed F ourier mo des. In fact, substituting the second relation in ( 8) in to (7), and usi ng the assumptions ˜ k ∈ R and v > 0, one finds the constraint | Im ˜ ω | < 1 γ v . (9) This restriction has tw o immediate and physically significant consequences, illustrated in figure 1. First, all exponen- tially gro wing solutions of the b o osted diffusion equation (3) (such as e ˜ t/ ( γ v 2 ) ) are ruled out a priori . This follows directly from t he ki netic admissibility condition (7), whic h enforces co v arian t stability of the allo wed solutions. Second, there is a wa v en umber cutoff in the stable branc h, meaning that all mo des with | ˜ k | ≥ Λ are excluded. T o determine the explicit v alue of the cutoff scale Λ, one substitutes the bo ost relations (8) into the diffusion dispersion relation ω = − ik 2 , and solves for ˜ k under the saturation condition Im ˜ ω = − 1 / ( γ v ). This yields (see App endix B) Λ = 1 + 2 v p v (1 − v ) . (10) As expected, the cutoff Λ diverges in the limit v → 0, reflecting the fact that, in the rest frame of the medium, all w av enum b ers are physically admissible. Interestingly , Λ also dive rges as v → 1. As a result, the cutoff scale exhibits a minimum at v = 1 / 4 (see figure 2), where Λ = 2 √ 3 ≈ 3 . 464. - 6 - 4 - 2 0 2 4 6 - 2 0 2 4 6 k ~ Im ω ~ - 6 - 4 - 2 0 2 4 6 - 20 - 10 0 10 20 k ~ Re ω ~ FIG. 1. Imaginary part (left) and real part (right) of the b o osted disp ersion relations ˜ ω ( ˜ k ) : R → C , obtained by applying a Loren tz b o ost to Fick’s law of diffusion ω = − ik 2 . The b o ost velocity is c hosen as v = 1 / 2, but the qualitativ e features are the same for all v > 0. The blue branc h is stable, while the red branch is unstable. The shaded region in the left panel indicates the kinetically inadmissible domain, whic h mus t b e excluded on ph ysical grounds (the dashed lines correspond to Im ω = ± 1 / ( γ v )). As expected, the unstable branch lies entirely within this region. In addition, the kinetic b ound excludes the high-wa v en umber portion of the stable branc h, leading to an upp er cutoff in ˜ k (for v = 1 / 2, the cutoff w a ven um ber is Λ = 4). 4 0.0 0.2 0.4 0.6 0.8 1.0 4 6 8 10 12 v Λ FIG. 2. W av enum ber cutoff Λ as a function of the b o ost velocity v , as determined by (10). F or | ˜ k | > Λ, solutions of the b o osted diffusion equation (3) no longer admit a realization within the underlying F okker-Planc k kinetic theory , b ecause the kinetic in tegral in (6) div erges. C. Boosted initial v alue problem and well-posedness The abov e analysis allows us to single out a natural class of physic al ly admissible initial data for mo ving observers. The b oosted diffusion equation (3) is thus endo wed with the following ph ysically-motiv ated initial-v alue form ulation. First, lo calized p erturbations δ n at ˜ t = 0 are required to b elong to a space of band-limited functions. Concretely , the initial profile is assumed to admit a F ourier represen tation of the form δ n (0 , ˜ x ) = Z Λ − Λ d ˜ k 2 π φ ( ˜ k ) e i ˜ k ˜ x , φ ∈ L 2 ([ − Λ , Λ]) , (11) where the cutoff wa ve num ber Λ is given b y (10), and enforces the kinetic admissibilit y condition. Second, the time evolution of each F ourier mo de should be gov erned b y the stable disp ersion relation ˜ ω ( ˜ k ) = ˜ k v + i 2 γ v 2   1 − s 1 − 4 iv ˜ k γ   , (12) whic h corresp onds to the blue line in figure 1. Under these assumptions, the following prop erties hold. • Since L 2 ([ − Λ , Λ]) ⊂ L 1 ([ − Λ , Λ]), the F ourier integral in (11) conv erges absolutely . As a consequence, the initial profile δ n (0 , ˜ x ) is well -defined p oint wise for all ˜ x ∈ R . • The set of admissible ini tial data defined by (11) coincides with the Paley-Wiener space P W Λ [27, 28], namely the closed subspace of L 2 ( R ) consisting of functions whose F ourier transform is supported in [ − Λ , Λ]. In particular, P W Λ is a Hilb ert space when endow ed with the L 2 norm. • F or ev ery fixed ˜ t ∈ R , the ev olution factor e − i ˜ ω ( ˜ k ) ˜ t satisfies | e − i ˜ ω ( ˜ k ) ˜ t | ≤ e | ˜ t | / ( γ v ) . It follo ws that the ev olved profile δ n ( ˜ t, ˜ x ) is well defined for all times, remains in P W Λ , and satisfies the estimate ∥ δ n ( ˜ t ) ∥ L 2 ≤ e | ˜ t | / ( γ v ) ∥ δ n (0) ∥ L 2 . (13) Putting ev erything together, we conclude that the b o osted diffusion equation (3), when restri cted to suc h kinetically admissible initial data and ev olv ed along the stable branch, defines a well-posed initial v alue problem [29, § 3.10]. The dynamics dep ends contin uously on the initial data, and is well defined b oth forward and bac kward in time. Let us also remark that applying an y differential op erator ∂ ˜ x a ∂ ˜ t b ( a, b ∈ N ) to the F ourier representation of δ n amoun ts to multiplying the in tegrand by ( i ˜ k ) a ( − i ˜ ω ) b . But since ˜ k and ˜ ω ( ˜ k ) are b oth b ounded on the interv al [ − Λ , Λ], the resulting F ourier co efficients still belong to L 2 ([ − Λ , Λ]). It follows that all deriv ativ es can b e tak en under the in tegral sign, and that δ n ∈ C ∞ ( R 2 ). In particular, δ n ( ˜ t, ˜ x ) solves the b oosted diffusion equation (3) in the ordinary (i.e. non-distributional) sense globally . 5 II I. GENERAL PR OPER TIES OF THE INITIAL DA T A In the previous section, we show ed that kinetic admissibility restricts lo calized p erturbations to the P aley-Wiener space P W Λ at all times. This is a strong constraint, whic h severely limits the class of admissible initial data [28]. In this section, we briefly examine its main consequences. A. Regularit y Consider again the represen tation (11), and expand the exp onen tial e i ˜ k ˜ x in p ow ers of ˜ k ˜ x . Assuming that the resulting series may be in terchanged with the integral, one obtains δ n (0 , ˜ x ) = ∞ X a =0 ( i ˜ x ) a a ! Z Λ − Λ d ˜ k 2 π φ ( ˜ k ) ˜ k a . (14) Since the magnitude of the integrals app earing in (14) is bounded b y 1 2 π ∥ φ ∥ L 1 Λ a , the series con v erges absolutely . This justifies the exchange of summation and integration. It follows that δ n (0 , ˜ x ) is not only C ∞ , but can b e analytically con tinued to an en tire function on the complex plane. Hence, it inherits all the standard prop erties of real-analytic functions. F or example, it cannot b e compactly supported: if it v anishes on an interv al, it must v anish everywhere. T o illustrate the non trivial implications of this result, consider the following example. Let G ( t, x ) denote the retarded Green function of the (un bo osted) diffusion equation (1), defined by ( ∂ t − ∂ 2 x ) G = δ ( t ) δ ( x ) and G (0 − , x ) = 0, with explicit form G ( t, x ) = e − x 2 / (4 t ) / √ 4 π t [30, § 7.4]. Applying a Lorentz bo ost yields [6] G ( ˜ t, ˜ x ) = Θ( ˜ t − v ˜ x ) p 4 π γ ( ˜ t − v ˜ x ) exp  − γ ( ˜ x − v ˜ t ) 2 4( ˜ t − v ˜ x )  . (15) This function solves the b o osted diffusion equation (3) ev erywhere except at the p oin t ( ˜ t, ˜ x ) = (0 , 0) [29, § 1.7, Probl.3]. In particular, if we restrict attention to the region ˜ t ≥ 1, it provid es an exact solution of (3) that is smo oth (indeed C ∞ ), rapidly decaying at spatial infinity , and supp orted exclusively on the stable branc h of mo des 2 . At first sight, this migh t appear to define a p erfectly admissible physical solution of our initial-v alue problem, but that is not the case. The function (15) v anishes identically in the region ˜ x > ˜ t/v , and therefore cannot b e entire in ˜ x . As a conseq uence, it do es not b elong to the Paley-Wiener space P W Λ , meaning that its F ourier transform contains contributions from arbitrarily large | ˜ k | , including mo des with | ˜ k | > Λ (as explicitly v erified in App endix C). A simple wa y to see this is to ev olv e the solution bac kw ard in time: as ˜ t → 0 + , a singularity is encoun tered at ˜ x = 0, demonstrating that G (1 , ˜ x ) do es not b elong to a function space for whic h the initial-v alue problem is well posed backw ard in time. Despite the abov e mathematical discussion , it still may not b e entirely obvious (from a physics persp ective) why the restriction of G to ˜ t ≥ 1 cannot consisten tly arise as the densit y associated with a solution of F okk er-Planck kinetic theory . Below, we sho w that the obstruction has a direct physical origin, whic h can be traced back to causality at the lev el of the kinetic equation. Let us interpret the retarded Green function G ( t, x ) as describing the follo wing process: for t < 0 the system is in equilibrium, at t = 0 a unit of c harge is injected at x = 0, and for t > 0 the excess densit y spreads diffusiv ely . In kinetic-theory language, this corresp onds to an equilibrium distribution function f for t < 0, follo w ed by the insertion of a p erturbation at t = 0, which then relaxes under the F okker-Planc k dynamics. Now, the key subtlety is that, since the relativistic F okker-Planc k equation (4) is causal, a perturbation of the distribution function f that is lo calized at x = 0 cannot instant aneously generate supp ort outside the lightcone. In particular, the supp ort of δ f at later times m ust remain confined within the causal domain determined by the microscopic dynamics. By cont rast, the Green function G ( t, x ) exhibits infinitely long spatial tails for all x at an y t > 0. This implies that, ev en though the densit y p erturbation δ n is lo calized at x = 0 at t = 0, the corresp onding p erturbation δ f must already b e nonlo cal in space at that time 3 . In other words, constructing the diffusion Green function G at a kinetic theory level requires that the F okker-Planc k equation (4) b e supplemented by a source term δ ( t ) σ ( x ) that is instan taneous in time, but infinitely extended in space, i.e. σ ( x ) ∝ δ ( x ). As a consequence, when one p erforms a Loren tz b o ost and restricts atten tion to the region ˜ t ≥ 1, the resulting densit y profile solves the b o osted diffusion equation (3) ev erywhere in that region. Ho w ever, its asso ciated distribution function fails to satisfy the kinetic equation (4) along the line ˜ x = ˜ t/v , where a source term appears. This line coincides with the b oundary of the region where G v anishes identically , and therefore marks the precise lo cus at whic h analyticit y , and hence kinetic admissibility , breaks down. 2 One can show (see Appendix C) that the p ortion of G with ˜ t < 0 is a superp osition of unstable modes (red curve in Fig. 1), whereas for ˜ t > 0 it inv olv es only stable mo des (blue curve in Fig. 1). 3 Indeed, it is sho wn in [16] that no perturbation δf belonging to the diffusiv e sector can be compactly supported, even when the associated density p erturbation δ n is. 6 B. Bounds on localizability The imp ossibilit y for δ n (0 , ˜ x ) to b e compactly supported is only one manifestation of the strong lo calization con- strain ts ob eyed b y Paley-Wiener functions. Additional limitations follo w from standard uncertain t y-type arguments. In particular, the uncertain ty relation ∆ ˜ x ∆ ˜ k ≥ 1 / 2 implies a low er b ound on spatial lo calization. Since functions in P W Λ ha ve F ourier supp ort of length 2Λ, one has ∆ ˜ k ≤ 2Λ, and therefore ∆ ˜ x > (4Λ) − 1 . (16) The inequalit y is strict, as saturation of the uncertaint y b ound occurs only for Gaussian w a v epack ets, whic h do not b elong to the P aley-Wiener class. A related constrain t follows from a point wise bound on the density profile. Indeed, from | δ n ( ˜ x ) | ≤ 1 2 π ∥ φ ∥ L 1 = 1 2 π ( e i arg φ , φ ) and the Cauch y-Sc hw arz inequalit y , one finds [31] | δ n ( ˜ x ) | ≤ r Λ π ∥ δ n ∥ L 2 . (17) Both b ounds indicate that there is a fundamen tal limit to ho w strongly the densit y can be concen trated at a single lo cation. If either bound is violated at a giv en time, a bo osted obser ver can immediately infer that the subsequent densit y evolution cannot b e gov erned solely by the diffusiv e sector of F okker-Planc k kinetic theory , but must inv olve additional, non-h ydro dynamic contributions. C. Shannon-Whittak er representation The final structural property of the space P W Λ that will b e relev ant for our purposes is the Shannon-Whittaker sampling theorem [32], whose con tent w e briefly recall b elow. The starting p oint is the observ ation that the set { e − iπ a ˜ k/ Λ } a ∈ Z forms an orthogonal basis of L 2 ([ − Λ , Λ]). As a consequence, the F ourier represen tation (11) admits the decomp osition δ n (0 , ˜ x ) = Z Λ − Λ d ˜ k 2 π + ∞ X a = −∞ c a e i ˜ k ( ˜ x − πa Λ ) = + ∞ X a = −∞ c a Z Λ − Λ d ˜ k 2 π e i ˜ k ( ˜ x − πa Λ ) , (18) whic h yields δ n (0 , ˜ x ) = + ∞ X a = −∞ c a sin[Λ( ˜ x − π a Λ )] π ( ˜ x − π a Λ ) . (19) Ev aluating this expression at the sampling p oin ts ˜ x a = π a/ Λ immediately gives c a = π δ n (0 , ˜ x a ) / Λ, and therefore δ n (0 , ˜ x ) = + ∞ X a = −∞ δ n (0 , ˜ x a ) sin[Λ( ˜ x − ˜ x a )] Λ( ˜ x − ˜ x a ) . (20) This iden tit y sho ws that any initial profile δ n (0 , ˜ x ) ∈ P W Λ can be uniquely reconstructed from its v alues on the discrete set of p oints ˜ x a = π a/ Λ, which motiv ates the terminology “sampling theorem” . An imm ediate consequence is that , once t he time evoluti on of the initial profile sinc(Λ ˜ x ) is known, the corresponding solution K ( ˜ t, ˜ x ) can b e used to generate all other solutions via the expansion δ n ( ˜ t, ˜ x ) = + ∞ X a = −∞ δ n (0 , ˜ x a ) K ( ˜ t, ˜ x − ˜ x a ) . (21) In this sense, K plays the role of a “discrete Green” function, or fundamental solution, for b o osted diffusion. It is worth stressing, how ev er, that K is not a Green function in the usual sense (and it is not retarded). In fact, since sinc(Λ ˜ x ) ∈ P W Λ , the function K ( ˜ t, ˜ x ) is necessarily smooth and real-analytic, and therefore cannot v anish on an y dense set of spacetime p oints. Accordingly , K solves the bo osted diffusion equation (3) throughou t R 2 , without the app earance of Dirac-delta source terms at ( ˜ t, ˜ x ) = (0 , 0), and it describes b oth forward and backw ard time ev olution. W e also emphasize that the discrete represen tation (21) is not an appro ximation scheme, nor a n umerical truncation. Rather, it is an exact identit y within the space P W Λ . The emergence of a discrete sampling structure is the direct coun terpart of the strong limitations on spatial lo calization imp osed by band limitation, which ultimately originates from the b ound (7). 7 IV. THE FUNDAMENT AL SOLUTION In this section, we compute the function K ( ˜ t, ˜ x ) explicitly , and analyze its properties. This function pro vides the basic building block from whic h any other kinetically admissible soluti on of (3) can b e reconstructed, through the sampling formula (21). A. Expressing the F ourier in tegral in the rest frame Since the initial condition is K (0 , ˜ x ) = sinc(Λ ˜ x ), the fundamental solution admits the integral represen tation K ( ˜ t, ˜ x ) = Z Λ − Λ d ˜ k 2Λ e i ˜ k ˜ x − i ˜ ω ˜ t , (22) where ˜ ω ( ˜ k ) is the stable disp ersion relation, giv en in (12). In this form, there is no hop e of ev aluating the in tegral explicitly , since ˜ ω appears in the exp onen t as a complicated function of ˜ k . How ev er, w e can transform bac k to the rest frame, where the dynamics are muc h simpler. By construction, e i ˜ k ˜ x − i ˜ ω ˜ t = e ikx − k 2 t . Accordingly , the in tegral o ver ˜ k ∈ [ − Λ , Λ] may b e rewritten as an in tegral o ver the complex k -plane along the image of this inte rv al under the mapping k ( ˜ k ) = γ ( ˜ k − v ˜ ω ( ˜ k )). Since k ( ˜ k ) is analytic and one-to-one on [ − Λ , Λ], this c hange of v ariables defines a smooth contour in the complex k -plane with endpoints k ( − Λ) and k (Λ). This con tour av oids the branch singularities associated with the square root in ˜ ω . F or the cutoff v alue Λ used here, the endp oin ts take the simple form (see App endix B) k ( ± Λ) = i ± r 1 + 1 v . (23) Moreo ver, the relation k ( ˜ k ) can b e inv erted to yield ˜ k = γ [ k + v ω ( k )] = γ ( k − iv k 2 ), so that the measure transforms as d ˜ k = γ (1 − 2 iv k ) dk . W e therefore obtain K ( t, x ) = Z k (Λ) k ( − Λ) dk 2Λ γ (1 − 2 iv k ) e ikx − k 2 t . (24) Since the integrand is an entire function of k , the v alue of the integral depends only on the endp oints of the contour (and not on the particular choice of path connecting them). Before ev aluating the in tegral explicitly for generic v alues of t and x , it is instructiv e to verify that (24) indeed p ossesses the prop erties required of K . First, by construction, it is an exact solution of the diffusion equation, b eing a sup erp osition of modes ∝ e ikx − k 2 t . Moreov er, the in tegral conv erges for all ( t, x ) ∈ R 2 and defines a real-analytic function, by the same argumen ts discussed in Section I I I A. The integration con tour may be chosen as k = Re k + i ( Re k ) 2 1 + 1 /v , Re k ∈ − r 1 + 1 v , r 1 + 1 v ! , (25) whic h lies entirely within the strip (7) and is therefore compatible with kinetic admissibilit y . The final property to b e chec k ed is the initial condition. When ˜ t = 0, the expression (24) must reduce to sinc(Λ ˜ x ). When ˜ t = γ ( t + v x ) = 0, the prefactor app earing in the integrand b ecomes proportional to the deriv ativ e of the exponential, allo wing the integral to b e ev aluated explicitly . One finds K ( − v x, x ) = Z k (Λ) k ( − Λ) dk 2Λ γ (1 − 2 iv k ) e ikx + v k 2 x = γ 2 i Λ x Z k (Λ) k ( − Λ) dk ( ix + 2 v k x ) e ikx + v k 2 x = γ 2 i Λ x  e ik (Λ) x + v k (Λ) 2 x − e ik ( − Λ) x + v k ( − Λ) 2 x  = γ Λ x sin x γ 1 + 2 v p v (1 − v ) ! = sinc(Λ ˜ x ) , (26) as required. 8 B. Ev aluating the F ourier integral Since (24) is a contour integral of an entire function, its v alue is given b y the difference of the primitive ev aluated at the endp oints of the contour. Recalling that deriv atives ma y b e taken outside the integral sign, one finds K ( t, x ) = γ √ π 4Λ √ t (1 − 2 v ∂ x )  e − x 2 4 t erf  √ t k (Λ) − ix 2 √ t  − e − x 2 4 t erf  √ t k ( − Λ) − ix 2 √ t  . (27) This expression is w ell defined also for negative v alues of t , in whic h case, the square root con tributes a factor of i . Equiv alently , for t < 0 one may replace √ t with p | t | and the error function erf with the imaginary error function erfi. Moreo ver, ev aluating (24) directly at t = 0 yields (see figure 3) K (0 , x ) = 1 1+2 v (1 − 2 v ∂ x ) " e − x sinc x r 1 + 1 v !# . (28) The appearance of the exp onen tial factor e − x deserv es some explanation. In constructing K , we are effectiv ely ask ed to determine an initial density profile K (0 , x ) suc h that, up on time evolution, it repro duces the profile sinc(Λ ˜ x ) on the sim ultaneit y line ˜ t = 0 (i.e. t = − v x ) of an observer O mo ving with v elocity − v (with v > 0). F or large p ositiv e x , this simultaneit y line lies in the far past relativ e to t = 0, so one exp ects that diffusion has exp onentially damped the profile by the time it reac hes the line t = 0. Accordingly , K (0 , x ) must decay exponentially for x → + ∞ . By con trast, for large negativ e x , the sim ultaneit y line ˜ t = 0 lies far in the future of t = 0, so the initial profile m ust con tain oscillations of exponentially large ampli tude in order for them to surviv e up to ˜ t = 0. The fact that the resulting asymmetry is precisely enco ded by the factor e − x reflects the circumstance that K is constructed as a superp osition of all kinematically admissible mo des in the bo osted frame, and therefor e lies at the boundary of the kinetic admissibilit y condition (7). F rom a mathematical standp oint, this b ehavior illustrates that the natural class of spatially lo calized functions for whic h the b o osted initial-v alue problem is well p osed cannot be obtained b y simp ly bo osting solutions that are spatially localized in the rest frame. In this sense, spatial lo calizability is a frame-dependent notion for diffusive dynamics. This sharply con trasts with the situation in causal theories suc h as Cattaneo’s, where compact supp ort in one reference frame implies compact supp ort in all reference frames [6]. Finally , it is instructiv e to consider the limits of small and large b o ost v elo cities. As v → 0, one finds that K (0 , x ) ≈ sinc( x/ √ v ), whic h con verges to π √ v δ ( x ). This behavior is exp ected: in the limit v → 0, the b o osted initial-v alue problem reduces to the rest-frame one, the cutoff Λ div erges as 1 / √ v , and the sampling formula (21) b ecomes indistinguishable from the usual Green-function representation, with the replacement P a → R dx/ ( π √ v ). In the opp osite limit v → 1, no singular b ehavior occurs, and K conv erges to a finite and smo oth function. This reflects the fact that b oth the sim ultaneity h yp ersurface of O and the cutoff wa v en umbers (23) approac h finit e limits. The fact that Λ diverges as v → 1 (see figure 2) is entirely due to the cont raction of the b o osted co ordinates. There is no in trinsic singularity in K itself. - 15 - 10 - 5 0 5 10 15 - 10 - 5 0 5 10 x  FIG. 3. Snapshot at t = 0, in the rest frame of the medium, of the fundamental solution K associated with the bo osted initial-v alue problem for an observer moving with velocity − 1 / 2. The profile is mo dulated by an exp onential factor e − x , which arises due to relativity of simultaneit y , and reflects the saturation of the kinetic admissibility b ound (7). The corresp onding analytic formula is giv en in equation (28). 9 C. Building exact solutions By b o osting the expression (27) bac k to the moving frame, one obtains the Green function K ( ˜ t, ˜ x ) (see figure 4), whic h enters the sampling formula (21). As suc h, K pro vides the basic building blo c k from which all solutions of the initial-v alue problem introduced in Section I I C can b e reconstructed. It is worth emphasizing that the sampling formula is most naturally interpreted not only as an ev olution rule, but also as a constructive prescription for admissi ble initial data. Indeed, one ma y start from an arbitrary sequence of co efficien ts { c a } a ∈ Z ∈ ℓ 2 (whic h is needed for conv ergence), and define δ n ( ˜ t, ˜ x ) = + ∞ X a = −∞ c a K ( ˜ t, ˜ x − ˜ x a ) , (29) where ˜ x a = π a/ Λ. By construction, the result ing profile b elongs to P W Λ and is th us kinetically admissible. Moreov er, it solv es the b o osted diffusion equation (3) exactly , and its initial profile satisfies δ n (0 , ˜ x a ) = c a at the sampling p oin ts. Since K (0 , ˜ x ) = sinc(Λ ˜ x ), the qualitativ e features of th e reconstructed initial profil e are directl y controlled by the b eha vior of the coefficients c a . If these v ary rapidly from one integer to the next, the resulting profile δ n (0 , ˜ x ) will oscillate on the shortest admissible spatial scale, of order 1 / Λ, whic h reflects the presence of modes near the cutoff. Con versely , if the coefficients are obtained by sampling a reference function g ( ˜ x ) that v aries smoothly on scales m uc h larger than 1 / Λ, namely c a = g ( ˜ x a ), then the reconstructed profile δ n (0 , ˜ x ) pro vides an accurate P W Λ -appro ximation of g ( ˜ x ). In this regime, the effect of the cutoff becomes negligible, and the sampling represen tation is effectiv ely indistinguishable from a contin uum description. As a simple illustration, one finds + ∞ X a = −∞ e − ( a 2 ) 2 K (0 , ˜ x − ˜ x a ) ≈ e − ( Λ ˜ x 2 π ) 2 (v ery accurate) , + ∞ X a = −∞ e − ( a 2 ) 4 K (0 , ˜ x − ˜ x a ) ≈ e − ( Λ ˜ x 2 π ) 4 (sligh tly worse, because more sharply lo calized) . (30) As can be seen in figure 5, the oscillatory features asso ciated with the cutoff scale are strongly suppressed 4 , and they b ecome progressiv ely smaller at later times. T ak en together, these examples indicate that, in the long-wa v elength regime (relative to the scale 1 / Λ), and upon restricting atten tion to forw ard time evolution, the exact, kinetically admissible solutions (29) are effectiv ely indistinguishable from those obtained by solving the b o osted diffusion equa- tion (3) for generic smo oth initial densit y profiles n (0 , ˜ x ), where the initial time deriv ativ e ∂ ˜ t n (0 , ˜ x ) is fixed so as to pro ject out the unstable branch and retain only the stable modes. In this regime, the kinetic admissibilit y constrain t (7) plays no dynamical role b eyond enforcing stabilit y forw ard in time. t ~ = 0 t ~ = 0.5 t ~ = 4 - 4 - 2 0 2 4 - 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x ~ - v t ~  t ~ = 0 t ~ = - 0.5 t ~ = - 1 - 4 - 2 0 2 4 - 2 - 1 0 1 2 x ~ - v t ~  FIG. 4. Snapshots of the forward (left panel) and backw ard (right panel) time evolution of the Green function K ( ˜ t, ˜ x ), obtained b y b o osting (27). T o compensate for adv ection by the moving medium, each snapshot cov ers a region shifted by v ˜ t . F or ˜ t > 0, the evolution exhibits asymmetric diffusion and rapid suppression of oscillations associated with the cutoff scale. Under backw ard time evolution, those same oscillatory features are progressiv ely amplified. Again, we chose v = 1 / 2. 4 Note that, ev en if oscillations are not visible to the naked ey e (at least in the case with e − a 2 / 4 ), oscillatory tails are generically presen t for band-limited functions (although highly exceptional non-oscillating P aley-Wiener functions do exist [33]). Such oscillatory components are progressively amplified under backw ard time evolution, as sho wn in figure 5. 10 t ~ = 0 t ~ = 1 t ~ = 10 - 10 - 5 0 5 10 0.0 0.2 0.4 0.6 0.8 1.0 x ~ - v t ~ δ n c a   - a 2 4 t ~ = 0 t ~ = - 0.5 t ~ = - 1.5 - 10 - 5 0 5 10 0.0 0.5 1.0 1.5 x ~ - v t ~ δ n c a   - a 2 4 t ~ = 0 t ~ = 1 t ~ = 10 - 10 - 5 0 5 10 0.0 0.2 0.4 0.6 0.8 1.0 x ~ - v t ~ δ n c a   - a 4 16 t ~ = 0 t ~ = - 0.5 t ~ = - 1.5 - 10 - 5 0 5 10 - 1.0 - 0.5 0.0 0.5 1.0 1.5 2.0 x ~ - v t ~ δ n c a   - a 4 16 FIG. 5. Exact solutions (29) of the b o osted diffusion equation (3) for the initial data (30) at b o ost velocity v = 1 / 2. Left panels: forw ard ev olution. W e see a progressive loss of sensitivit y to the detailed structure of the initial profile, and con v ergence to ward a universal (asymmetric) diffusiv e shap e. Right panels: backw ard evolut ion. Here, an tidiffusion amplifies mo des near the cutoff, leading to the emergence or enhancemen t of spatial oscillations on the scale 1 / Λ. D. Kinetic embedding Up to this p oint, we ha v e implicitly assumed that an y solution of (1) constructed as a linear sup erp osition of kinetically admissible mo des is itself kinetically admissible. This assumption, ho wev er, hides a subtle mathematical issue. Even if a sup erp osition of density mo des of the form (6) conv erges to a regular density profile n ( t, x ), this do es not automatically imply that the corresp onding sup erp osition of kinetic mo des (5) also con v erges to a regular distribution function f ( t, x, p ) [16]. Therefore, in order to rigorously establish that the solutions (29) admit a realization within F okker-Planc k kinetic theory , one m ust expl icitly verify that the kernel (27) arises as the particl e density asso ciated with a physically admissible solution of the kinetic equation (4). In the case of massless particles, it is known that, if suc h a kinetic realization exists, it must tak e the form [16] δ f K ( t, x, p ) = π β e − β | p | (1 − ∂ 2 x ) K ( t, x − β p ) . (31) It is straightforw ard to verify that (31) indeed solves (4). The associated density reads δ n ( t, x ) = Z R dp 2 π π β e − β | p | (1 − ∂ 2 x ) K ( t, x − β p ) ξ ≡− β p = Z R dξ 2 e −| ξ | (1 − ∂ 2 ξ ) K ( t, x + ξ ) = Z 0 −∞ dξ 2 e ξ (1 − ∂ 2 ξ ) K ( t, x + ξ ) + Z + ∞ 0 dξ 2 e − ξ (1 − ∂ 2 ξ ) K ( t, x + ξ ) = K ( t, x ) −  e ξ 2 ( K − ∂ ξ K )  ξ = −∞ −  e − ξ 2 ( K + ∂ ξ K )  ξ =+ ∞ , (32) where the final line follo ws from in tegration by parts (recall that K is smooth). The b oundary term at + ∞ deca ys exponentially , as e − 2 ξ , while the con tribution at −∞ deca ys 1 /ξ . Hence, both these terms v anish at infinit y , meaning that the in tegral con v erges (and this is precisely our admissibility criterion), so (31) defines a kinetically admissible solution of the Vlasov-F okker-Planc k equation with asso ciated density δ n ( t, x ) = K ( t, x ), as required. 11 V. CONCLUSIONS In this work, w e hav e reconsidered ho w an initial-v alue problem should b e formulated for Loren tz-b oosted diffusion, starting from its microscopic em b edding within relativistic F okker-Planc k kinetic theory [16, 17]. By exploiting the fact that Fick-t yp e diffusion arises as the exact h ydro dynamic sector of the kinetic theory , we identified a ph ysically motiv ated space of admissible initial data for b o osted profiles, based on the requirement that the resulting evolution should admit a kinetic realization. This criterion singles out a space of band-limited functions, and allo ws one to define an initial-v alue problem for the b o osted diffusion equation that is well posed and microscopically meaningful. Within this restricted setting, w e show ed that the resulting dynamics are wel l defined not only forw ard in time, but also bac kward. While the forward evolution undergo es ordinary diffusiv e smo othing, the backw ard ev olution is correspondingly “an ti-diffusiv e” , with spatial struct ures on the shortest admissible scales b eing progressiv ely amplified. Crucially , ho w ever, this growth is not arbitrarily fast. The kinetic admissibility condition enforces a sharp wa velength cutoff, whic h in turn bounds the growth rate to b e con trolled b y a maximal exponent 1 / ( γ v ). As a result, the evoluti on remains well-posed in the sense of Hadamard [29, § 3.10]: small changes in the initial data lead to proportionally small c hanges in the solution at any finite time. In ph ysical terms, the cutoff prev ents the catastrophic short-wa v elength instabilities that would otherwise render the backw ard problem ill-p osed, and ensures a controlled, contin uous time- rev ersed dynamics within the kinetically admissible sector. Finally , we ha v e v erified that the mathematical tec hnicalities associated with band-limited function spaces become largely irrelev ant in the regime that is most relev ant for practical applications. In the long-wa v elength limit, and up on restricting attent ion to forw ard time evolution, the exact solutions constru cted here are effectively indistinguishable from those obtained by solving the b o osted diffusion equation after simply discarding the unstable branch of mo des. F rom this p ersp ectiv e, the presen t analysis provides a microscopic justification for what is otherwise the most ob vious fix of b o osted diffusion. It is important to emphasize that the construction presen ted here is not intended as a “practical” framew ork for relativistic transp ort theory . In fact, our analysis is restricted to one spatial dimension, to the linear regime, and to a homogeneous bac kground medium. Moreo v er, it relies hea vily on the special structure of the F okker-Planc k kinetic equation. F or these reasons, w e do not exp ect this approac h to replace more con ven tional formulations of relativistic dissipation [7, 8, 34], nor to b e directly applicable in fully realistic set tings. Its primary in terest lies instead in its conceptual implications. Our results show that, in some cases, it is p ossible to mak e precise mathematical sense of ev olution equations that are acausal and linearly unstable when treated as stand-alone partial differential equations, pro vided one is willing to imp ose non-lo cal constraints on the admissible initial data. A CKNOWLEDGEMENTS This w ork is supported by a MERAC F oundation prize gran t, an Isaac Newton T rust Grant , and funding from the Cam bridge Centre for Theoretical Cosmology . Appendix A: Cattaneo theory as an exact kinetic equation F ollowing [4], w e decomp ose the ensem ble of massless particles in to righ t mov ers, whic h propagate with v elocity +1, and left mov ers, which propagate with v elocity − 1, disregarding the precise v alue of their momentum. In a kinetic- theory framew ork, where the fundamen tal degree of freedom is the single-particle distribution function f ( t, x, p ), this amoun ts to introducing the partial densities n + ( t, x ) = Z + ∞ 0 dp 2 π f ( t, x, p ) , n − ( t, x ) = Z 0 −∞ dp 2 π f ( t, x, p ) . (A1) In tegrating the Boltzmann equation ( ∂ t + V ∂ x ) f = “scattering integral” (with V = sgn( p )) ov er p ositiv e or negative momen ta, and inv oking particle-num b er conserv ation, one obtains ( ∂ t + ∂ x ) n + = R and ( ∂ t − ∂ x ) n − = −R . Here, R denotes the net rate of conv ersion b etw een right mov ers and left mo v ers induced by random scattering with the en vironment . The central assumption of the mo del is that all particles, independently of their momentum, hav e the same probability p er unit time of rev ersing their direction of motion. Under this assumption, the conv ersion rate tak es the form R = ( n − − n + ) / 2 (in appropriate units). Introducing the total particle density n = n + + n − and the asso ciated flux J = n + − n − , one finds ∂ t n + ∂ x J = 0 , ∂ t J + J = − ∂ x n , (A2) whic h is the Cattaneo theory of diffusion. Com bining these tw o equations yields the second-order evolution equation ∂ t n = ( ∂ 2 x − ∂ 2 t ) n , as exp ected. 12 A natural question is therefore which form of the scattering integral giv es rise to the conv ersion rate R = ( n − − n + ) / 2. The simplest example is provided by the relaxation-time appro ximation, ( ∂ t + V ∂ x ) f = f eq − f [35], wher e f eq denotes the lo cal equilibrium distribution with the same density as f (and with the temp erature and velocity of the environmen t). Indeed, integrating this equation ov er p ositive momen ta yields R = n/ 2 − n + = ( n − − n + ) / 2, as required. By con trast, F okk er-Planc k kinetic theory do es not share this prop ert y . Integrating ( ∂ t + V ∂ x ) f = β − 2 ∂ p [ f eq ∂ p ( f /f eq )] o ver positive momen ta gives R = −  f eq 2 π β 2 ∂ p ( f /f eq )  p =0 , (A3) whic h is not determined solely b y n ± , but dep ends on the detailed momentum-space structure of the distribution function. Appendix B: Determination of the cutoff w av enum b er Plugging (8) i nto iω = k 2 , w e obtain iγ ( ˜ ω − v ˜ k ) = γ 2 ( ˜ k − v ˜ ω ) 2 . W e seek solutions with ˜ k = Λ ∈ R and ˜ ω = Ω − i/ ( γ v ), where the real part Ω is to b e determined, and the imaginary part − ( γ v ) − 1 saturates (9). This leav es us with iγ (Ω − v Λ) + 1 v = γ 2 (Λ − v Ω) 2 + 2 iγ (Λ − v Ω) − 1 . (B1) Since Λ and Ω are assumed real, we can just set the real and the imaginary parts of (B1) to zero separately . This immediately giv es Λ − v Ω = ± 1 γ r 1 + 1 v , Ω = 2 + v 1 + 2 v Λ . (B2) Plugging the second equality in to the first, and taking the p ositive solution, w e arrive at (10). Appendix C: F ourier decomp osition of the rest-frame retarded Green function in b o osted frames W e ha ve the follo wing (note the change of integration v ariable y = ˜ t/v − ˜ x ): G ( ˜ t, ˜ k ) = Z R Θ( ˜ t − v ˜ x ) p 4 π γ ( ˜ t − v ˜ x ) exp  − γ ( ˜ x − v ˜ t ) 2 4( ˜ t − v ˜ x ) − i ˜ k ˜ x  d ˜ x = Z R Θ( y ) √ 4 π γ v y exp " − γ ( y − ˜ t γ 2 v ) 2 4 v y − i ˜ k ( ˜ t/v − y ) # dy = e − i ˜ k v ˜ t + ˜ t 2 γ v 2 √ 4 π γ v Z ∞ 0 1 √ y exp  −  γ 4 v − i ˜ k  y − ˜ t 2 4 γ 3 v 3 y  dy = e − i ˜ k v ˜ t + ˜ t 2 γ v 2 √ 4 π γ v s π γ 4 v − i ˜ k exp   − 2 s  γ 4 v − i ˜ k  ˜ t 2 4 γ 3 v 3   = 1 q γ ( γ − 4 iv ˜ k ) exp   − i ˜ t   ˜ k v + i 2 γ v 2 − | ˜ t | ˜ t i 2 γ v 2 s 1 − 4 iv ˜ k γ     , (C1) from which w e immediately obtain G ( ˜ t, ˜ k ) = 1 q γ ( γ − 4 iv ˜ k ) ( e − i ˜ ω − ( ˜ k ) ˜ t , t > 0 , e − i ˜ ω + ( ˜ k ) ˜ t , t < 0 , (C2) where ω ± are the tw o disp ersion relations of the bo osted diffusion equation. 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