Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. II: construction of SDS with nonlinear force and multiplicative noise

Sto c hastic Dynamical Structure (SDS) of Nonequilibrium Pro cesses in the Ab sence of Detailed Balance. I I: construction of SDS with nonlinear force and m ultiplicativ e noise P . Ao Dep artments of Me chanic al Engine ering and Physics, University of Washington, Se attle, W A 98 195, USA (Dated: Ma rch 30, 200 8) Abstract There is a whole range of emergen t ph enomena in non-equilibriu m b eha viors can b e we ll d e- scrib ed b y a set of sto c hastic differentia l equ at ions. Inspired b y an insight gained du r ing our study of r obustness and stabilit y in ph ag e lam b da genetic switc h in mo dern biology , we found that there exists a classificatio n of generic nonequilibriu m pro cesses: In the contin uous description in terms of sto c hastic different ial equations, ther e exists four dynamical elemen ts: the p oten tial fu nctio n φ , the f ricti on matrix S , the anti -symm etric matrix T , and th e noise. T he generic feature of abs ence of detailed balance is then p r ec isely repr esen ted by T . F or dyn amic al near a fixed p oin t, whether or not it is stable or not, the s to c hastic dyn amics is linear. A rather complete analysis has b een car- ried out (Kw on, Ao, Th o u le ss, cond-mat/0506280 ; PNAS , 102 (2005) 13029) , referred to as SDS I. One imp ortant and p ersistent question is the existence of a p oten tial fu nctio n with nonlinear force and with m ultiplicativ e noise, with b oth nice lo cal dynamical and global steady s tate p rop ertie s. Here we demonstrate that a dyn amic al structure built into sto c hastic different ial equation allo ws us to constru ct suc h a global optimizatio n p oten tial function. First, we provide th e construction. One of most imp ortan t ingred ie nt is th e generalized Einstein relation. W e then present an ap- pro ximation scheme: T he gradien t expans ion w hic h tu rns ev ery order int o linear matrix equations. The consisten t of such metho dology with other kno wn sto c hastic treatmen ts will b e discus s ed in next pap er, S DS I I I; and th e explicitly connection to statistica l mec hanics and thermo dynamics will b e discu ssed in a forthcoming pap er, SDS IV. (Th e main results w ere pub lished. Please cite the pr esen t p aper as Potential in Sto chastic D iffer ential Equ at ions: Novel Construction , P . Ao, J. Ph ys. A37 L25-L30 (2004). h ttp://www.iop.org/EJ/abstract/ 0305-4470/37/3/L01/ ) 1 Let us consider an n comp onen t net w ork whose dynamics is describ ed b y a set o f stochastic differen tial equations 1 : ˙ q tj = f j ( q t ) + ζ j ( q t , t ) . (1) The question is whether or not we can find a p oten tia l function from Eq.(1) whic h giv es a global description of dynamics. Here ˙ q tj = dq tj /dt with j = 1 , 2 , ..., n and the subscript t for q indicates that the state v ariable q is a function of time. The v alue of j th comp onen t is denoted b y q j . The net w ork state v aria ble forms an n dimensional v ector q τ = ( q 1 , q 2 , ..., q n ) in the state space. Here t he sup erscript τ denotes the tra nspose. The state v ariable ma y b e the v alues of particle co ordinates in ph ysics or the prot ein n um b ers in a signal trans- duction pathw ay , or any other p ossible quantities sp ecifying the net work. Let f j ( q ) b e the deterministic nonlinear force on the j th comp onen t, whic h includes b oth the effects from other comp onen ts and itself, and ζ j ( q , t ) the r a ndom force. F or simplicit y we will assume that f j is a smo oth f unc tio n explicitly indep enden t of time. T o b e sp ecific, the noise will b e assumed to b e Gaussian and white with the v ariance, h ζ i ( q t , t ) ζ j ( q t ′ , t ′ ) i = 2 D ij ( q t ) δ ( t − t ′ ) , (2) and zero mean, h ζ j i = 0. Here δ ( t ) is the Dir a c delta function, and h ... i indicates the a v erage with respect to the dynamics of the sto c hastic force. By the ph ysics and che mistry con v en tion the semi-p o sitiv e definite symmetric matrix D = { D ij } with i, j = 1 , 2 , ..., n is the diffusion matrix. Eq.(2) also implies that, in situations whe re the temp erature T can b e defined, w e hav e set k B T = 1 with k B the Boltzmann constant. W e remark that if an a v erage ov er the sto c hastic force ζ , a Wiener noise, is p erformed, Eq.(1) is reduced to the follo wing equation in dynamical systems: h ˙ q t i = h f ( q t ) i = f ( h q t i ) . The last equalit y is due to the fact that at same time t , the noise and the state v aria bles are indep enden t of each other. It is equiv a len t to the fact that the noise can b e switc h out without affecting t he deterministic f o rce, a pro ces s demonstrated p ossib le in phys ics in dealing with en vironmen tal effects 2 . A broad range of phenomena in b oth natural a nd so cial sciences has b een describ ed b y such a deterministic equation 3 . Because o f its imp ortance and usefulness, rep eated attempts hav e b een made t o construct a p oten tial function 4,5,6,7 . The effort had, ho w ev er, only limited success 8 . The usefulness of a 2 p oten tial reemerges in the current study of dynamics of gene regulatory net w orks 9 ,10 , whic h w ould again require its construction in complex netw ork dynamics. It has b een observ ed that the nonlinear dynamics is in g eneral dissipativ e ( t r ( F ) 6 = 0 ) , asymmetric ( F j i 6 = F ij ), and sto c hastic ( ζ 6 = 0 ). Here the force matrix F is defined as F ij = ∂ f i /∂ q j i, j = 1 , ..., n . (3) and the tra ce is equal t o the div ergence of the force: tr( F ) = ∂ · f = P n j =1 ∂ f j /∂ q j . The com bination of those three features prev en ts any direct application of the insigh t fro m Hamil- tonian dynamics a nd has b een the main obstacle prev en ting the p oten tial construction. In fact, the a sy mmetry of dynamics has b een characterize d as the hallmark o f the net work in a state far from thermal equilibrium, and has b een pro claimed that it mak es the usual theoretical approach near thermal equilibrium un w ork able 4 . It is the goal of this pap er to rep ort that w e ha ve, nev ertheless, disco v ered a nov el construction that can tak e care of those dynamical features and can giv e us a p oten tial function. W e state, the explicit construction will b e given b elo w, that there exists a unique decom- p osition suc h that Eq.(1) can b e rewritten in the following form: [ S ( q t ) + A ( q t )] ˙ q t = − ∂ φ ( q t ) + ξ ( q t , t ) , (4) with the semi-p ositiv e definite symmetric matrix S ( q t ), the an ti-symmetric matrix A ( q t ), the single-v alued scalar function φ ( q t ), and t he sto c hastic force ξ ( q t , t ). Here ∂ is the gradient op erator in the state v a riable space. It is straightforw ard to v erify that the semi-p ositiv e definite symmetric matrix term is ‘dissipativ e’: ˙ q τ t S ( q t ) ˙ q t ≥ 0; the anti-symmetric part do es no ‘w ork’: ˙ q τ t A ( q t ) ˙ q t = 0, therefore non-dissipativ e. Hence, it is natural to identify that the dissipation is represen ted by the semi-p ositiv e definite symmetric matrix S ( q ), the friction matrix, and the tra ns ve rse force by the anti-sym metric matrix A ( q ), the t r ansv erse matrix. The scalar function φ ( q ) then acquires the meaning of p oten tial energy . The decomp osition from Eq.(1) to (4 ) may b e called the φ -decomp osition. Ho w ev er, without f ur t her constrain t, Eq.(4) w ould b e not unique. This ma y b e illustrated by a simple coun ting: There are four apparen t indep enden t quan tities in Eq.(4), while there are only t w o in Eq.(1). In order t o hav e a unique form for Eq.(4), w e may choose to imp ose the constrain t on the sto c hastic force a nd the semi-p ositiv e definite symmetric matrix in the follo wing manner: h ξ ( q t , t ) ξ τ ( q t ′ , t ′ ) i = 2 S ( q t ) δ ( t − t ′ ) , (5) 3 and h ξ ( q t , t ) i = 0. W e observ e that this constraint is consisten t with the Gaussian and white noise assumption for ζ in Eq.(1). It ma y b e called the sto c hasticity -dissipation relation. W e further o bse rve that that the forms of Eq.(4) and (5) strongly resem ble those of dissipativ e dynamics in quan tum mec hanics when b oth dissipativ e and Berry phase exists 2, 11 . The constrained φ - decomp osition will be called the gaug ed φ -decomp osition, whic h is indeed unique, a s w e will now demonstrate. W e prov e the existence and uniqueness of the gauged φ - decomp osition from Eq.(1) to (4) b y an explicit construction. Using Eq.(1) to eliminate the ve lo cit y ˙ q t in Eq.(4), w e ha ve [ S ( q t ) + A ( q t )][ f ( q t ) + ζ ( q t , t )] = − ∂ φ ( q t ) + ξ ( q t , t ) . Noticing that the dynamics of noise is indep enden t of that o f the state v ariables w e require that b oth the deterministic force and the noise satisfying follo wing t w o equations separately . F or t he deterministic force, this leads t o [ S ( q ) + A ( q )] f ( q ) = − ∂ φ ( q ) , (6) suggesting a ‘rotatio n’ from the force f to the g r a dien t o f the p oten tial φ at eac h p oin t in the state space. W e hav e dropp ed the subscript t . F or sto c hastic force, w e ha v e: [ S ( q ) + A ( q )] ζ ( q , t ) = ξ ( q , t ) , (7) whic h sho ws the same ‘rotation’ b et w een the sto c hastic forces. Here w e ha v e also dropp ed the subscript t for the state v ariable. Using Eq.(2) and (5), Eq.(7) implies [ S ( q ) + A ( q )] D ( q )[ S ( q ) − A ( q )] = S ( q ) , (8) whic h suggests a dualit y b et w een Eq.(1) and (4): a large friction matrix implies a small diffusion matrix. It is a generalization of the Einstein relation 12 to finite tra nsv erse matrix A . Next w e introduce an auxiliary mat r ix function G ( q ) = [ S ( q ) + A ( q )] − 1 . (9) Here the in v ersion ‘ − 1 ’ is with resp ect to the matrix. Using the prop ert y of the p oten tial function φ : ∂ × ∂ φ = 0, Eq.(6) leads to ∂ × [ G − 1 f ( q )] = 0 , (10) 4 whic h g iv es n ( n − 1) / 2 conditions to determine the n × n auxiliary matrix G . The generalized Einstein relatio n, Eq.(8), leads to the follo wing equation G + G τ = 2 D , (11) whic h readily determines the symmetric part of the auxiliary matr ix G , anot he r n ( n + 1) / 2 conditions fo r G . Eq. (10) and (1 1 ) give the needed n × n conditions to completely determine G . Here w e giv e a solution of G as a series in gradient expansion: G = D + ∞ X j =0 ∆ G j , (12) with ∆ G j = P ∞ l =1 ( − 1) l [( F τ ) l ˜ D j F − l + ( F τ ) − l ˜ D j F l ], ˜ D 0 = D F − F τ D , ˜ D j ≥ 1 = ( D + ∆ G j − 1 ) n [ ∂ × ( D − 1 + ∆ G − 1 j − 1 )] f o ( D − ∆ G j − 1 ). The zeroth or der a ppro ximatio n to Eq.(10) is GF τ − F G τ = 0. A fo r ma l solutio n to this approx imated equation together with Eq.(11) has b een constructed under a ra ther restrictiv e condition 13 , a nd the explicit solution under a generic condition has b een o btained in Ref. [14]. The gauged φ -decomp osition is therefore uniquely determined:            φ ( q ) = − R C d q ′ · [ G − 1 ( q ′ ) f ( q ′ )] S ( q ) = [ G − 1 ( q ) + ( G τ ) − 1 ( q )] / 2 A ( q ) = [ G − 1 ( q ) − ( G τ ) − 1 ( q )] / 2 . (13) The end a nd initia l p oin ts of t he in tegration contour C are q a nd q 0 resp ec tive ly . D uring the construction a sufficien t condition det ( F ) det( S + A ) 6 = 0 is a ss umed, with exception at some isolated p oin ts. W e remark o n the sp ecial role play ed by the force matrix: If F D = D F τ , i.e., ˜ D 0 = 0. If in addition ˜ D j ≥ 1 = 0 in this case, whic h can b e realized if D is a constan t, ∆ G j = 0 f o r a ll j . This means that G = D and the transv erse matrix A = 0. Suc h a condition has b een noticed in the linear case where b oth F and D are constan t matrices, and na me d the in tegrability condition 15 . In man y exp erimen tal studies of a complex net w ork, a question is often ask ed on the distribution o f the state v ariable after a transien t p erio d instead of fo cusing on the individual tra jectory of the net w ork. This implies that either there is an ensem ble of iden tical net w orks or r ep etitiv e ex p erimen ts are b een carried out. F rom statistical mec hanics, if viewing the p oten tial function φ a s an energy , a steady distribution function can b e exp ected from Eq.(4): ρ 0 ( q ) = 1 Z exp {− φ ( q ) } , (14) 5 with the partition function Z = R d n q exp {− φ ( q ) } . This is a Bo ltzmann-Gibbs distribution for the state v ariable, and giv e the strongest manifestation of the usefulness of the p oten tial function φ . W e remark that it is ho w ev er not obv ious that the steady state distribution, if exists, should b e giv en b y Eq.(14). In the fo llowing we giv e a heuristic demonstration that Eq.(14) is indeed a righ t steady distribution for the net work as the steady state solution of the corresp onding F okk er-Planc k equation. The connection b et w een the standard sto c hastic differen tial equation, Eq.(1), and F okk er- Planc k eq uatio n is necessarily am biguous for the generic nonlinear case as exemplified b y the Ito-Stratonovic h dilemma 1,16,17 . W e attribute this la c k o f definiteness to the asymptotic nature of the connection in whic h a pro cedure m ust b e explicitly defined: Different pro ce- dures will in general lead to different results. Here we presen t another pro cedure whic h may b e natural fro m the theoretical ph ysics p oin t of view. Our starting p oin t will b e Eq.(4), not Eq.(1) fr o m which most previous deriv ations started. The existence of b oth the deterministic for ce and the sto c hastic force in Eq.(4) suggests that there are tw o well separated time scales in the net w ork: microscopic time scale for the description of the sto c hastic force and macroscopic time scale for the net w ork motio n. The former time scale is m uch smaller than t he latter. This separatio n of t ime scales further suggests that the macroscopic motion of the netw ork has an inertial: It cannot resp onse instan taneously to the microscopic mot io n. T o capture this feature, we in tro duce a small inertial ”mass” m and a kinetic momen tum ve ctor p for the net w ork. The dynamical equation fo r the net work no w t ak es the form: ˙ q t = p t /m (15) defines the kinetic momen tum, and ˙ p t = − [ S ( q t ) + A ( q t )] p t /m − ∂ φ ( q t ) + ξ ( q t , t ) (16) is the extension of Eq.(4). W e note that there is no dep ende nt of friction matrix and the sto c ha stic force on the kinetic momen tum, therefore no Ito-Stratonovic h dilemma in the connection b et wee n the sto c hastic differen tial equation and the dynamical equation for the distribution function. The F okk er-Planck equation in this enlarged state space, the Klein- Kramers equation, can b e immediately obtained 1 :  ∂ t + p m · ∂ q + f · ∂ p − ∂ p S  p m + ∂ p  ρ ( q , p , t ) = 0 . (17) 6 Here f = p A/m − ∂ q φ , and t , q , and p are indep enden t v ariables. The steady distribution can b e found as 1 ρ ( q , p ) = exp {− [ p 2 / 2 m + φ ( q )] } / Z , (18) with Z = R d n q d n p exp {− [ p 2 / 2 m + φ ( q )] } the partitio n function. There is an explicit separation of state v ariable and it s kinetic momen tum in Eq.(18). The zero ”mass” limit can then b e tak en with no effect on the state v aria ble distribution. This confirms that Eq.(14), the Boltzmann-Gibbs distribution exp ected from Eq.(4), is t he righ t c hoice under this pro cedure. T o conclude, w e p oin t out a ma j or differenc e b et w een the presen t construction of the p oten tial and those in literature suc h as represen ted by the Gra ham-Hak en construction 5,6 : The presen t construction is based on a structure built in to sto c hastic differen tial equation. There is no explicit use of F okk er-Planc k equation. Therefore there is no need to mak e assumption on the distribution function in the limit time go es to infinite as assumed in the Graham-Hake n construction. In particular, the p oten tial in the presen t pap er can b e time dep ende nt. Critical discussions with N. Goldenfeld, A.J. Leggett, C. K won, J. Ross, D .J. Thouless, L. Yin, and X.-M. Zhu are highly appreciated. This work was supp orted in part by a USA NIH gra n t under the num b er HG0028 94-01. 1 N.G. v an Kamp en, Sto c h astic pr ocesses in ph ysics and chemistry , Elsevier, Amsterdam, 1992. 2 A.J. Leggett, in Quantum tunn eling in condensed media, edited by Y u. K a gan and A.J. Leggett, North-Holland, Ams terd am, 1992. 3 D. K aplan and L. Glass, Understand in g nonlinear d ynamics, Springer-V erlag, Berlin, 1995. 4 G. Nicolis and I. Prigogine, S elf-o rganization in nonequ ili b rium systems: from d issipativ e struc- ture to order through fluctuations, J ohn Wiley and sons, New Y ork, 1977; G. Nicolis, Physica A213 (1995) 1. 5 H. Hak en, Adv anced Synergetics, 3rd edition, S p ringer-V erlag, Berlin, 1987 6 R. Graham, in Noise in nonlinear dynamical systems, v.1, pp225-278, edited by F. Moss and P .V.E. McClin to c k, Cambridge Univ ersity P ress, Cam br idge, 1989. 7 J. Ross and M.O. Vlad, Ann u. Rev. Phys. Chem. 50 (1999) 51. 7 8 M.C. C r oss and P .C. Hohenberg, Rev. Mo d. Phys. 65 (1993) 851; J .S. Langer, in Critical Prob- lems in Physic s, pp11-27, edited b y V.L. Fitc h, D.R. Marlo w, and M.A.E. Demen ti, Princeton Univ ersit y Pr ess, Princeton, 1997. 9 M. Sasai and P .G. W olynes, Pro c. Nat. Acad. Sci. (USA) 100 (2003) 2374. 10 X.-M. Z h u, L. Yin, L. Ho o d , and P . Ao, prepr int, Calculating biological b eha viors of epigenetic states in phage λ life cycles (su bmitted to F unctional and Int egrativ e Genomics). 11 P . Ao and D.J. Thouless, P hys. Rev. Lett.70 (1993 ) 2158; P . Ao and X.-M. Z h u, Phys. Rev., B 60 (1999) 6850. 12 A. Einstein, Ann . Ph ysik 17 (1905 ) 549-56 . 13 P . Ao, prepr in t, S to c hastic force defined ev olution in dynamical systems (ph ysics/0302081) (submitted to Phys. Rev. Lett.). 14 C.-L. Kwon, P . Ao, and D.J. T houless, to b e pub lish ed. 15 M. Lax, Rev. Mo d. Phys. 38 (1966) 359. 16 C.W. Gardiner, Handb o ok of sto c hastic metho ds for p h ysics, chemistry and th e natural sciences, Springer-V erlag, Berlin, 1983. 17 H. Risken, The F okk er-Planc k equation, Sp ringer, Berlin, 1989. 18 The idea of sto c hastic dyn amical str ucture (S DS) w as p osted in 2003 s h ortly after its emergence from the biological study: a) Sto c h astic F or c e Define d Evolution in Dynamic al Systems , P . Ao, physic s/0302081, h − cac h e/ph ysics/p df/0302 /0302081 v1.p d f ; Ab o v e analysis w as sup erseded by Kwon, Ao, Thouless, referred to as SDS I: b) Structur e of Sto chastic D ynamics ne ar Fixe d Points , C. Kw on, P . Ao, and D.J. Thouless, Pro c. Natl Acad. S ci. (USA) 102 (2005) 13029 -13033. cond-mat/0 506280 . h − cac h e/c ond-mat/p df/0506/05 06280v1.p df ; The explicit and exact construction of SDS f o r the known dynamics without detailed balance, limit cycle, can b e foun d in c) Li mit Cycle and Conserve d Dynamics , X.-M. Z hu, L. Yin, P . Ao, Int . J. Mo d. Phy . B20 (2006 ) 817-8 27. nlin/0412059. ht − cac h e/nlin/pd f/0 412/04120 59v1.p df ; An int eresting realistic physical example with both singule matrix S and T , but nonsingule S + T , can b e f ound in d) None qu i librium Appr o ach to Blo ch-Peierls-Berry Dynamics , J.C. Olson and P . Ao, 8 Ph ys. Rev. B75 , 035114 (2007). h ttp://link.aps.org/abstract/ PRB/v75/e035114 (xxx.lanlg.go v : physic s/0605101) An app lica tion of SDS analysis to econophysics can b e found here: e) Boltzmann-Gibbs Distribution of F ortune and Br oken Time-R eversible Symmetry in E c ono dynamics , P . Ao, Comm un . Nonlin ear Sci. Num. Sim u lation 12 (2 007) 619-6 26. (DOI link: h ttp://dx.doi.org/10.10 16/j.cnsns .2005.07.004 ) physics/ 0506103; h − cac h e/ph ysics/p df/0506 /0506103 v1.p d f 9