STATIONARY NON EQULIBRIUM STATES F STATES FROM A MICROSCOPIC AND A MACROSCOPIC POINT OF VIEW

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1 STATIONARY NON EQULIBRIUM STATES FROM A MICROSCOPIC AND A MACROSCOPIC POINT OF VIEW University of L Aquila 1 July 2014 GGI Firenze

2 References L. Bertini; A. De Sole; D. G. ; G. Jona-Lasinio; C. Landim Stochastic interacting particle systems out of equilibrium J. Stat. Mech. (2007) D. G. From combinatorics to large deviations for the invariant measures of some multiclass particle systems Markov Processes Relat. (2008) L. Bertini; D. G.; G. Jona-Lasinio; C. Landim Thermodynamic transformations of nonequilibrium states J. Stat. Phys. (2012)

3 SNS: Microscopic description Lattice: Λ N Configuration of particles: η {0, 1} Λ N or η N Λ N η t (x) = number of particles at x Λ N at time t

4 SNS: Microscopic description Stochastic Markovian dynamics r(η, η ) = rate of jump from configuration η to configuration η η = local perturbation of η µ N (η) = invariant measure of the process, probability measure on the state space µ N (η) η r(η, η ) = η µ N(η )r(η, η) µ N = MICROSCOPIC description of the SNS

5 SNS: Macroscopic description η = π N (η) Empirical measure (positive measure on [0, 1]) π N (η) = 1 N x Λ N η(x)δ x δ x = delta measure (Dirac) at x [0, 1]; since x Λ N we have x = i N, i N. Given f : [0, 1] R fdπ N = 1 η(x)f(x) [0,1] N x Λ N

6 SNS: Macroscopic description When η is distributed according to µ N and N is large LAW OF LARGE NUMBERS π N ρ(x)dx This means fdπ N [0,1] [0,1] f(x) ρ(x) dx ρ(x) = typical density profile of the SNS

7 SNS: Macroscopic description When η is distributed according to µ N and N is large, a refinement of the law of large numbers LARGE DEVIATIONS ( ) P π N (η) ρ(x)dx e NV (ρ) V = Large deviations rate function V = MACROSCOPIC DESCRIPTION OF THE SNS V contains less information than µ N but is easier to compute and is independent from microscopic details of the dynamics

8 Example: Equilibrium SEP Equilibrium: C L = C R = C; A L = A R = A Microscopic state: product of Bernoulli measures of parameter p = C A+C µ N (η) = x Λ N p η(x) (1 p) 1 η(x)

9 Example: Equilibrium SEP MACROSCOPIC DESCRIPTION ( ) P π N (η) ρ(x)dx = = {η,: π N (η) ρ(x)dx} {η,: π N (η) ρ(x)dx} µ N (η) e N ( [0,1] dπ N (η) log 1 p p log(1 p) ) Using the combinatorial estimate {η, : π N (η) ρ(x)dx} e N 1 0 ρ(x) log ρ(x)+(1 ρ(x)) log(1 ρ(x)) dx V (ρ) = 1 ρ(x) 0 ρ(x) log p + (1 ρ(x)) log (1 ρ(x)) (1 p) dx

10 Contraction P Average number of particles 1 η(i) = dπ N (η) N ( 1 N i [0.1] satisfies LDP ) η(i) y e NJ(y) i BY CONTRACTION J(y) = inf {ρ : 1 0 ρ(x) d x=y} V (ρ)

11 Relative entropy Relative entropy of the probability measure µ 2 N µ 1 N ( H µ 2 N µ 1 N ) = η µ2 N (η) log µ2 N (η) µ 1 N (η) H 0, not symmetric!! Density of relative entropy 1 h = lim N + N (µ H 2 N µ 1 N ) with respect to

12 An example µ 1 N (η) = x Λ N p η(x) (1 p) 1 η(x), product of Bernoulli measures of parameter p µ 2 N (η) = x Λ N ρ(x) η(x) (1 ρ(x)) 1 η(x), slowly varying product of Bernoulli measures associated to the density profile ρ(x) = η = 1 N µ 2 N(η) 1 N 1 ( N H ρ(x) log ρ(x) p x Λ N µ 2 N µ 1 N ) η(x) log ρ(x) p x Λ N + (1 ρ(x)) log + (1 η(x)) log (1 ρ(x)) (1 p) (1 ρ(x)) (1 p) Riemann sums, convergence when N + to V (ρ)

13 From microscopic to MACROSCOPIC Driving parameters (λ, E) λ = rates of injection and annihilation at the boundary E = external field driving the particles on the bulk µ λ,e N = corresponding invariant measure ρ λ,e = corresponding typical density profile V λ,e (ρ) = corresponding LD rate function V λ1,e 1 ( ρ λ2,e 2 ) = lim N + 1 N H ( µ λ 2,E 2 N µ λ 1,E 1 N )

14 From microscopic to MACROSCOPIC This relation between relative entropy and LD rate function can be easily verified for the boundary driven Zero Range Process It is true also for boundary driven SEP; proof based on matrix representation of µ N In general the computation of V through relative entropy is difficult An alternative powerful approach to compute V is the dynamic variational one of the Macroscopic Fluctuation Theory

15 Boundary driven TASEP: a microscopic view

16 Boundary driven TASEP: a microscopic view Duchi E., Schaeffer G A combinatorial approach to jumping particles, J. Comb. Theory A (2005)

17 Boundary driven TASEP: a microscopic view η = configuration of particles above ξ = configuration of particles below (η, ξ) = full configuration of particles Stochastic Markov dynamics for (η, ξ) Observing just η = still Markov and boundary driven TASEP ν N (η, ξ) = invariant measure for the joint dynamics, it has a combinatorial representation µ N (η) = ξ ν N(η, ξ)

18 Boundary driven TASEP: a microscopic view Complete configurations E(x) = y x (η(y) + ξ(y)) Nx 1 (η, ξ) is a complete configuration if { E(x) 0 E(1) = 0 ν N is concentrated on complete configurations (η, ξ) complete = N 1 (η, ξ), N 2 (η, ξ) ν N (η, ξ) = 1 Z N A N 1(η,ξ) C N 2(η,ξ) Special case A = C = 1 = ν N uniform measure on complete configurations

19 Boundary driven TASEP: a macroscopic view Joint Large deviations ( ) P (π N (η), π N (ξ)) (ρ(x), f(x)) e NG(ρ,f) Contraction principle ( ) NV (ρ) P π N (η) ρ(x) e V (ρ) = inf f G(ρ, f)

20 Boundary driven TASEP: a macroscopic view Complete density profiles E(x) = x 0 (ρ(y) + f(y)) dy x The pair (ρ, f) is a complete density profile if { E(x) 0 E(1) = 0 When C = A = 1 since ν N is uniform on complete configurations a classic simple computation gives G(ρ, f) = 1 if (ρ, f) is complete; here 0 [ h 1 2 h p (α) = α log α p ] (ρ(x)) + h 1 (f(x)) dx 2 + (1 α) log (1 α) 1 p

21 Boundary driven TASEP: a macroscopic view V (ρ) = inf f : (ρ,f) C 1 0 [ h 1 2 ] (ρ(x)) + h 1 (f(x)) dx 2 To be compared with B. Derrida, J.L. Lebowitz, E.R. Speer Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process J. Stat. Phys. (2003) V (ρ) = sup f 1 0 { ρ(x) log [ρ(x)(1 f(x))] + (1 ρ(x)) log [(1 ρ(x))f(x)] where f(0) = 1, f(1) = 0 and f is monotone } dx + log 4

22 Boundary driven TASEP: a macroscopic view Both variational problems have the same minimizer ( x ) f ρ (x) = CE (1 ρ(y)) dy 0 V (ρ) = G(ρ, f ρ ) See Bahadoran C. A quasi-potential for conservation laws with boundary conditions arxiv: for a dynamic variational approach, using MFT

23 2-class TASEP

24 The invariant measure

25 Collapsing particles ( η 1, η T ) : x η 1 (x) x η T (x) = (η 1, η T ) = C [ ( η 1, η T )) ] Flux across bond (x, x + 1) [ ] J(x) = sup η 1 (z) η T (z) y z [y,x] +

26 Collapsing measures ( ρ 1, ρ T )) : where d ρ 1 d ρ T = (ρ 1, ρ T ) = C [ ( ρ 1, ρ T )) ] S 1 S 1 Definition (a,b] dρ 1 = (a,b] d ρ 1 + J(a) J(b) J(x) := sup y [ d ρ 1 (y,x] (y,x] d ρ 2 ] +

27 Collapsing measures

28 Large deviations LD for the ( η 1, η T ) variables Ṽ ( ρ 1, ρ T ) = [h m1 ( ρ 1 ) + h m2 ( ρ T ))] d x S 1 LD for the SNS (not convex!) V (ρ 1, ρ T ) = inf Ṽ ( ρ 1, ρ T ) {( ρ 1, ρ T ) : C[( ρ 1, ρ T )]=(ρ 1,ρ T )} = [h m1 (ˆρ 1 ) + h m2 (ρ T ))] d x S 1 On any (a, b) where ρ 1 = ρ T x [ x ˆρ 1 (y)dy = CE a a ] ρ 1 (y)dy