Macroscopic fluctuations in non-equilibrium mean-field diffusions Krzysztof Gaw¸ edzki GGI, Florence, May 2014 • Mean field approximation has served us for over 100 years ( Curie 1895, Weiss 1907) giving hints about the behavior of in high dimensions systems with short range interactions ր ց systems with long range interactions in any dimension • Developed originally for equilibrium systems (ordered and disordered), it has been applied more recently to nonequilibrium dynamics • Here, I shall employ the Macroscopic Fluctuation Theory of the Rome group ( Bertini - De Sole - Gabrielli - Jona-Lasinio - Landim ) to describe fluctuations around mean field approximation
• Informally, the Roman theory may be viewed as a version of Freidlin - Wentzell large deviations theory applied to stochastic lattice gases (zero range, SSEP, WASEP, ABC, ...) • We shall keep a similar point of view in application to general non-equilibrium d -dimensional diffusions with mean-field coupling: N � � dx n 1 = X ( t, x n ) + Y ( t, x n , x m ) + X a ( t, x n ) ◦ η na ( t ) dt N ր m =1 a independent white noises with Y ( x, y ) = − Y ( y, x ) and ◦ for the Stratonovich convention • Based on joint work with F. Bouchet and C. Nardini
• Prototype model : N planar rotators with angles θ n and mean field coupling, undergoing Langevin dynamics N � � dθ n = F − H sin θ n − J sin( θ n − θ m ) + 2 k B T η n ( t ) dt N ր m =1 independent white noises Shinomoto - Kuramoto , Prog. Theor. Phys. 75 (1986), · · · · · · , Giacomin - Pakdaman - Pellegrin - Poquet , SIAM J. Math. Anal. 44 (2012) • Close cousin of the celebrated Kuramoto (1975) model for synchronization (with F → F n random and T = 0 ) whose versions were recently studied by the long-range community (papers by Gupta - Campa -( Dauxois )- Ruffo ) • Originally thought as a model of cooperative behavior of coupled nerve cells • Close to models of depinning transition in disordered elastic media
• The Shinomoto - Kuramoto system N � � dθ n = F − H sin θ n − J sin( θ n − θ m ) + 2 k B T η n ( t ) dt N m =1 may also be re-interpreted as a classical ferromagnetic XY model � with a mean-field coupling of planar spins S n • F = 0 case ( equilibrium ) : � � � H = H, 0 in constant external magnetic field � � � S n = S cos θ n , sin θ n • F � = 0 case ( non-equilibrium ) : � � � H = H cos( F t ) , − sin( F t ) in rotating external magnetic field � � � ( i . e . spins are viewed in S n = S cos( θ n − F t ) , sin( θ n − F t ) frame ) the co − moving S S n n θ n H θ −Ft H n F = 0 F � = 0
• Macroscopic quantities of interest in the general case N � � dx n 1 = X ( t, x n ) + Y ( t, x n , x m ) + X a ( t, x n ) ◦ η na ( t ) dt N m =1 a • empirical density N � 1 ρ N ( t, x ) = δ ( x − x n ( t )) N n =1 • empirical current N � 1 δ ( x − x n ( t )) ◦ dx n ( t ) j N ( t, x ) = N dt n =1 • They are related to each other by the continuity equation: ∂ t ρ N + ∇ · j N = 0 • Macroscopic Fluctuation Theory applies to their large deviations at N ≫ O (1) around N = ∞ mean field
• Effective diffusion in the density space dxn ( t ) • Substitution of the equation of motion for and the passage dt to the Itˆ o convention give: N � j N ( t, x ) = 1 δ ( x − x n ( t )) ◦ dx n ( t ) = j ρN ( t, x ) + ζ ρN ( t, x ) N dt n =1 where � � � j ρ = ρ X + Y ∗ ρ − D ∇ ρ ← − ρ quadratic in with � � � � � X = X − 1 1 ∇ · X a X a , D = X a ⊗ X a 2 2 a a � ( Y ∗ ρ )( t, x ) ≡ Y ( t, x, y ) ρ ( t, y ) dy and N � � 1 � ζ ρN ( t, x ) = X a ( t, x ) δ x − x n ( t )) η na ( t ) N n =1 a
• Conditioned w.r.t. ρ N , the noise ζ ρN ( t, x ) has the same law � 2 N − 1 D ( t, x ) ρ N ( t, x ) ξ ( t, x ) where as the white noise � � = δ ij δ ( t − s ) δ ( x − y ) ξ i ( t, x ) ξ j ( s, y ) • Follows from the fact that for functionals Φ[ ρ ] of (distributional) densities, the standard stochastic differential calculus gives � � � � � � d Φ[ ρ Nt ] = L Nt Φ [ ρ Nt ] dt where � � L Nt Φ � [ ρ ] = − δ Φ[ ρ ] δρ ( x ) ∇· j ρ ( t, x ) dx � � � δ 2 Φ[ ρ ] + 1 δρ ( x ) δρ ( y ) ∇ x ∇ y D ( t, x ) ρ ( t, x ) δ ( x − y ) dx dy N is the generator of the (formal) diffusion in the space of densities evolving according to the Itˆ o SDE � 2 N − 1 Dρ ξ � = 0 ∂ t ρ + ∇ · � j ρ +
• N = ∞ closure • When N → ∞ , the evolution equation for the empirical density reduces to Nonlinear Fokker - Planck Equation ( NFPE ) � � � ˆ � ∂ t ρ = −∇ · j ρ = −∇ · ρ X + Y ∗ ρ − D ∇ ρ a nonlinear dynamical system in the space of densities → (autonomous or not) • If Y = 0 then the N = ∞ empirical density coincides with instantaneous PDF of identically distributed processes x n ( t ) and NFPE reduces to the linear Fokker - Planck equation for the latter • The N = ∞ phase diagram of an autonomous system with mean-field coupling is obtained by looking for stable stationary and periodic solutions of NFPE and their bifurcations • In principle, more complicated dynamical behaviors may also arise
N = ∞ phases of the rotator model • • Stationary solutions of NFPE satisfy ∂ θ j ρ ( θ ) = 0 , i.e. � � � � 2 π � ∂ θ ρ ( θ ) F − H sin( θ ) − J sin( θ − ϑ ) ρ ( ϑ ) dϑ − k B T ∂ θ ρ ( θ ) 0 տ = sinθ cos ϑ − cos θ sin ϑ � � � � = ∂ θ ρ ( θ ) F − ( H + x 1 ) sin θ + x 2 cos θ − k B T ∂ θ ρ ( θ ) = 0 2 π � 2 π � with x 1 = J cos ϑ ρ ( ϑ ) dϑ , x 2 = J sin ϑ ρ ( ϑ ) 0 0 and the solution θ +2 π � F θ +( H + x 1) cos θ + x 2 sin θ − F ϑ +( H + x 1) cos ϑ + x 2 sin ϑ ρ ( θ ) = 1 kBT kB T Z e e dϑ θ • The coupled equations for 2 variables x 1 , x 2 may be easily analyzed
• N = ∞ phase diagram for the rotator model for F � = 0 ( Shinomoto - Kuuramoto 1984, Sakaguchi - Shin .- Kur . 1986, ... ) Bogdanov H −Takens ordered H o e op d n e f l d d sa F periodic � � disordered �� �� � � � � � � �� �� � � � � �� �� � � � � � � � � �� �� �� �� � � � � �� �� � � � � � � � � �� �� 0 J 0.5 T k B • For H = 0 the periodic phase coincides with the ordered low-temp. equilibrium phase viewed in the co-rotating phase • When F ց 0 the periodic phase reduces to the equilibrium disordered phase at H = +0 • Global properties of the NFPE dynamics for the rotator model have been recently studied by Giacomin and collaborators
• Fluctuations for N large but finite • Formally, domain of applications of the small-noise Freidlin - Wentzell large deviations theory • In Martin - Rose - Siggia formalism, the joint PDF of empirical density and current profiles is � �� � �� � � � � � � δ ρ − ρ N δ j − j N = δ ∂ t ρ + ∇ · j δ j − j ρ − ζ ρ � � � � � � a · ( j − jρ − ζρ ) D a e iN = δ ∂ t ρ + ∇ · j � � � � � a · ρD a D a e iN a · ( j − jρ ) − N = δ ∂ t ρ + ∇ · j � � � e − 1 ( j − jρ )( ρD ) − 1( j − jρ ) ∼ e − N I [ ρ,j ] 4 N ∼ δ ∂ t ρ + ∇ · j where the rate function(al) � � ( j − j ρ )( ρD ) − 1 ( j − j ρ ) dtdx 1 if ∂ t ρ + ∇ · j = 0 4 I [ ρ, j ] = ∞ otherwise
• Large-deviations rate function(al)s for empirical densities or empirical currents only � � � � N →∞ e − N I [ ρ ] N →∞ e − N I [ j ] δ [ ̺ − ρ N ] ∼ δ [ j − j N ] ∼ are obtained by the contraction principle � � � ( −∇ · ρD ∇ ) − 1 � � 1 I [ ρ ] = min I [ ρ, j ] = ∂ t ρ + ∇· j ρ ∂ t ρ + ∇· j ρ dtdx 4 j I [ j ] = min I [ ρ, j ] with appropriate boundary limiting conditions for ρ ρ • That empirical densities have dynamical large deviations with rate function given above was proven by Dawson - Gartner in 1987 • To our knowledge, the large deviations of currents for mean field models were not studied in math literature • The formulae above have similar form as for the macroscopic density and current rate functions in stochastic lattice gases studied by the Rome group and B. Derrida with collaborators
• Elements of the (Roman) Macroscopic Fluctuation Theory • Instantaneous fluctuations of empirical densities • Time t distribution of the empirical density � � ∼ e − N F t [ ̺ ] leading ← P t [ ̺ ] = δ [ ̺ − ρ Nt ] WKB term ∂ t P t = L † satisfies the functional equation Nt P t which reduces for the large-deviations rate function F t [ ̺ ] to the functional Hamilton - Jacobi Equation ( HJE ) � � � � � � j ρ · ∇ δ F t [ ̺ ] ∇ δ F t [ ̺ ] ∇ δ F t [ ̺ ] ∂ t F t [ ̺ ] + + · ρD = 0 δ̺ δ̺ δ̺ • In a stationary state the latter becomes the time-independent HJE for the rate function F [ ̺ ]
Recommend
More recommend
