Quasi-potential for Burgers equation

Transcription

1 Quasi-potential for Burgers equation Open Systems, Zurich, June 1, 29 L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim

2 Boundary Driven WASEP State Space: Fix N 1. {, 1} {1,...,N 1} η = (η(1),...,η(n 1)) Bulk dynamics: 1 x N 2 x x + 1 at rate x + 1 x at rate α + N 1 α Quasi-potential for Burgers equation p.1/21

3 Boundary Driven WASEP State Space: Fix N 1. {, 1} {1,...,N 1} η = (η(1),...,η(n 1)) Bulk dynamics: 1 x N 2 x x + 1 at rate α + N 1 x + 1 x at rate α Boundary dynamics: Fix ρ < ρ 1 1. Right boundary: creation at rate ρ 1, annihilation at rate (1 ρ 1 ) Left boundary: creation at rate ρ, annihilation at rate (1 ρ ) Quasi-potential for Burgers equation p.1/21

4 Stationary state Stationary state: µ N α (depends on ρ, ρ 1 ) ϕ = log(ρ/1 ρ) chemical potential. α c = [ϕ(1) ϕ()] 1. No current, process reversible, µ N α is product measure. α > α c. Current from right boundary to the left. α < α c. Current from left boundary to the right. Quasi-potential for Burgers equation p.2/21

5 Nonequilibrium free energy π N ( ) = 1 N N 1 x=1 η(x) δ( x/n) lim N µn α [ π N ρ α ] = 1 { = αρxx f(ρ) x ρ() = ρ, ρ(1) = ρ 1. f(ρ) = ρ (1 ρ) µ N α [ π N ρ ] e NS(ρ), S nonequilibrium free energy Quasi-potential for Burgers equation p.3/21

6 Equilibrium free energy α = α c. S(ρ) = { ρ(x) log ρ(x) ρ α (x) 1 ρ(x) } + [1 ρ(x)] log dx 1 ρ α (x) Quasi-potential for Burgers equation p.4/21

7 α > α c Enaud and Derrida (4), Derrida, Lebowitz and Speer (1) S(ρ) = {h(ρ) + (1 ρ)ϕ log ( 1 + e ϕ) } + αϕ x log αϕ x (αϕ x 1) log(αϕ x 1) dx A α h(a) = a log a + (1 a) log(1 a), A α a constant αϕ xx ϕ x (αϕ x 1) e ϕ = ρ ϕ() = ϕ, ϕ(1) = ϕ 1, αϕ x > 1 Quasi-potential for Burgers equation p.5/21

8 Dynamical approach Approach does not depend on the model π N t ( ) = 1 N N 1 x=1 η tn 2(x) δ( x/n) π N ρ P N ρ [ π N t u(t, ) ] 1 u t + f(u) x = αu xx u(, ) = ρ( ) u(t, ) = ρ, ρ(t, 1) = ρ 1. Quasi-potential for Burgers equation p.6/21

9 Dynamical large deviations Fix trajectory u(t, ), t. Let ρ( ) = u(, ). P ρ {π N u in [, T] } exp{ NI [,T] (u)} I [,T] (u) = α σ(u) [H x ] 2 dx σ(a) = a(1 a) mobility { ut + f(u) x αu xx = 2α [ ] σ(u)h x H(t, ) = H(t, 1) = x Quasi-potential for Burgers equation p.7/21

10 Quasi-potential I [,T] (u): cost of observing u. Cost of connecting ρ α to ρ in [ T, ]: Quasi-potential: inf I [ T,] (u) u( T, )=ρ α u(, )=ρ V (ρ) = inf T> inf I [ T,] (u) u( T, )=ρ α u(, )=ρ Quasi-potential for Burgers equation p.8/21

11 Nonequilibrium free energy Theorem 1 Bodineau-Giacomin (97), Farfan (9) If ρ α is a global attractor, S = V. S(ρ) = inf T> inf I [ T,] (u) u( T, )=ρ α u(, )=ρ Remark: Any dimension, model. Questions: Optimal trajectory. DLS formula for S, DLS equation α < α c Quasi-potential for Burgers equation p.9/21

12 Action functional I [,T] (u) = α σ(u) [H x ] 2 dx { ut + f(u) x αu xx = 2α [ σ(u)h x ] H(t, ) = H(t, 1) = x I [,T] (u) = 1 4α 1 { 1[ ] } 2 u t + f(u) x αu xx dx σ(u) I [,T] (u) = L(u, u t ) Quasi-potential for Burgers equation p.1/21

13 Hamiltonian formalism I [ T,] (u) = T L(u, u t ) V (ρ) = inf T> inf I [ T,] (u) u( T, )=ρ α u(, )=ρ H(ρ, H) = sup G { H, G L(ρ, G) } H(ρ, H) = α σ(ρ), H 2 x + αρ xx f(ρ) x, H Hamilton-Jacobi equation ( H ρ, δv δρ ) = Quasi-potential for Burgers equation p.11/21

14 Idea of the proof { ut = α u xx f(u) x 2α (σ(u)h x ) x H t = α H xx f (u)h x α σ (u) Hx 2 (ρ α, ) is fixed point Hyperbolic: (ρ, ) (ρ α, ) M s stable manifold, M u unstable manifold Quasi-potential for Burgers equation p.12/21

15 Statement M u is a graph M u is Lagrangian: Let r : [, 1] M u such that r() = r(1) Let r(t) = (ρ(t), H(t)). Then, ρ t, H =. Claim: V (ρ) = C + Γ ρ t, H Quasi-potential for Burgers equation p.13/21

16 Sketch of the proof I (,] (v) = inf I (,] (u) = inf u(, )=ρ α T> u(, )=ρ inf I [ T,] (u) u( T, )=ρ α u(, )=ρ I (,] (v) = v satisfies the Euler-Lagrange equation: d L(v, v t ) δl δv t (v, v t ) = δl δv (v, v t) Let H = H(v, v t ) = (δl/δv t )(v, v t ) (v, H) follows the Hamiltonian flow Quasi-potential for Burgers equation p.14/21

17 Sketch of the proof v(t) ρ α as t Assume that (v, H) M u : H(t) as t I (,] (v) = = = L(v, v t ) { v t, H H(v, H) } v t, H = V (ρ) V (ρ α ) Quasi-potential for Burgers equation p.15/21

18 DLS equation α > α c { ut = α u xx f(u) x 2α (σ(u)h x ) x Fix ρ. Let ϕ be given by H t = α H xx f (u)h x α σ (u) H 2 x αϕ xx ϕ x (αϕ x 1) e ϕ = ρ H = log ρ 1 ρ ϕ = (ρ, H) M u Γ u = {(ρ, H)} M u Γ u is a graph by definition Γ u is Lagrangian Quasi-potential for Burgers equation p.16/21

19 Proof that (ρ,h) M u Need (u(t), H(t)) (ρ α, ) as t Equation of motion: { ut = α u xx f(u) x 2α (σ(u)h x ) x H t = α H xx f (u)h x α σ (u) H 2 x Time reversed: { wt = α w xx + f(w) x + 2α (σ(w)j x ) x J t = α J xx + f (w)j x + α σ (w) J 2 x (1) Alternative formulation: w t = α w xx + f(w) x + 2α (σ(w)j x ) x J = log w 1 w ϕ Quasi-potential for Burgers equation p.17/21

20 Miracle Let F = eϕ 1 + e ϕ. Then F t = α F xx f(f) x Fix ρ. Solve DLS to get ϕ. Let F = e ϕ /(1 + e ϕ ) and let F evolve according to the Hyd. Eq. Let ϕ(t) = log[f(t)/1 F(t)]. Set w(t) = αϕ xx (t) ϕ x (t)(αϕ x (t) 1) + 1 J(t) = log 1 + e ϕ(t) w(t) 1 w(t) ϕ(t) Then, (w(t), J(t)) evolves according the backward equation (1) w(t) is the hydrodynamic path of the adjoint process Quasi-potential for Burgers equation p.18/21

21 Hamilton-Jacobi equation α σ(ρ), ( δv δρ ) 2 x + αρ xx f(ρ) x, δv δρ = No unique solution, not easy to solve Lemma: V is the maximal solution. Assume W solution Fix a trajectory u. u( T) = ρ α u() = ρ Then, I [ T,] (u) W(ρ) W(ρ α ) Quasi-potential for Burgers equation p.19/21

22 Lower bound I [ T,] (u) = α α σ(u) ( H x [ δw σ(u) δρ ] 2 x [ δw δρ ] x dx + 2α ) 2 dx σ(u) H x [ δw δρ ] x dx H-J: 2α α [ δw σ(u) δρ σ(u) H x [ δw δρ ] x ] 2 x dx = α dx = 2α { uxx + f(u) x }δw δρ dx [ σ(u) Hx ] x δw δρ dx I [ T,] (u) u t δw δρ dx = W(u()) W(u( T)) Quasi-potential for Burgers equation p.2/21

23 α < α c αϕ xx ϕ x (1 αϕ x ) = e ϕ ρ Exists a solution but not unique! Γ u = {(ρ, H)} is Lagrangian W(ρ, H) = C + Γ ρ t, H Γ u M u Similar proof, but H may be discontinuous. Phase transition } V (ρ) = inf {W(ρ, H) : (ρ, H) Γ u Quasi-potential for Burgers equation p.21/21