Second-law like inequalities for transitions between non-stationary states D. Lacoste Laboratoire Physico-Chimie Théorique (PCT) Laboratoire Physico-Chimie Théorique (PCT) ESPCI, Paris
Outline of the talk I. Preliminaries on fluctuation theorems II. Modified Fluctuation-dissipation theorem off-equilibrium III. Second-law like inequalities for transitions between non-stationary states Acknowlegments: G. Verley, ESPCI, Paris R. Chétrite, Univ. Nice, France
Jarzynski relation t ∂ H ɺ ∫ Stochastic definition of work = τ W d h ( c h , ) τ τ τ t ∂ h 0 • Average over non-equilibrium trajectories leads to equilibrium behavior : − β − ∆ β W F C. Jarzynski, PRL 78, 2690 (1997) = e e t This leads to a formulation of the second-law for macroscopic systems : ≥ ∆ W F t • Derivation using Feyman-Kac relation : Hummer G and Szabo, PNAS 98, 3658 (2001) 1 − β − β W H c h ( , ) δ − = ( c c e ) e t t t Z A
Hatano-Sasa relation t ∂ φ ɺ ∫ Work like functional where = τ Y d h ( c h , ) φ = − ( , ) c h ln P c h ( , ) τ τ τ t st ∂ h 0 • Average over non-equilibrium trajectories leads to steady-state behavior − Y = e 1 T. Hatano and S. Sasa, PRL 86, 3463 (2001) t Now where the equality holds for a quasi-stationary process ≥ Y 0 t • Initial condition in a non-equilibrium steady state (NESS) • Expansion of the relation to first order in the perturbation leads to a modified FDT near a NESS
II. The three routes to modified Fluctuation-dissipation theorems (MFDT) • In terms of an additive correction (the asymmetry) which vanishes at equilibrium M. Baiesi et al. (2009); E. Lippiello et al. (2005) valid near any non-equilibrium state G. Diezemann (2005); L. Cugliandolo et al. (1994) • In terms of a local velocity/current valid near any non-equilibrium state R. Chétrite et al. (2008); U. Seifert et al. (2006) • In terms of a new observable constructed from the non-equilibrium stationary distribution J. Prost et al. (2009); valid near a NESS G. Verley, K. Mallick, D. L., EPL 93, 10002 (2011) Rk: in all 3 cases, markovian dynamics is assumed Is it possible to extend the third route for a general observable and a general non-equilibrium state ?
Three relevant probability distributions • Probability distribution ρ t (c) solution of unperturbed master equation : ∂ ρ ( ) c ∑ ∑ [ ] t = ρ − ρ = ρ w c c ( ', ) ( ') c w c c ( , ') ( ) c ( ') ( ', ) c L c c t t t t t t ∂ t c ' c ' • Probability P t (c,[h t ]) has a functional dependence on a perturbation [h t ] , [ ] [ ] ∂ ∂ P c h P c h ( , ( , ) ) ∑ ∑ ∑ ∑ [ ] [ ] [ ] [ ] [ ] [ ] t t t t h h h h h h = = − − = = w w ( ', ) ( ', ( ', ) ( ', c c P c c c P c h h ) ) w w ( , ') ( , ( , ') ( , c c P c h c c P c h ) ) P c P c ( ', ( ', h h ) ) L c c L c c ( ', ) ( ', ) t t t t t t t t t t t t ∂ t c ' c ' • Probability π t (c,h) defined for a constant time independent perturbation h, ∂ π ( , ) c h ∑ ∑ h h h t = π − π = π w c c ( ', ) ( ', ) c h w c c ( , ') ( , ) c h ( ', ) c h L c h ( ', ) t t t t t t ∂ t c ' c ' • Trajectory dependent quantity of interest constructed from π t (c,h) : ψ = − π ( , ) c h ln ( , ) c h t t t t t t
A particle obeying Langevin dynamics and submitted to a temperature quench k h 2 T ɺ • Model: with and t t t = − + + η η = ηη = δ − x x , 0, ( t t ') t t t t t t ' γ γ γ ∂ x t 1 k ∫ t [ ] h τ = = − τ • Response function is R t t ( , ') exp d , ∂ γ γ h t ' t ' → h 0 • Alternatively, one has 2 t t 1 1 h k ∫ ∫ [ ] τ τ = − − τ − τ ' P x h ( , ) exp x d exp d ' , t t 1/2 2 ( ) σ γ γ 2 2 πσ 2 t 0 τ t and thus for a constant protocol h 2 t t k 1 1 h ∫ ∫ τ ' π = − − τ − τ ( , ) x h exp x d exp d ' , t 1/2 2 ( ) σ γ γ 2 2 πσ 2 t 0 τ t • Using this together with the MFDT, the same response is recovered
t ɺ ∫ Our work like path functional = τ ∂ ψ Y d h ( , c h ) τ τ τ τ h 0 ∫ The Feyman-Kac approach : − Y = π = A c h e ( , ) dc ( , ) c h A c h ( , ) A c h ( , ) t t t t t t t t t [ ] [ ] π h t Generalized Hatano-Sasa relation − Y = e 1 [ ] h Through linear expansion, one obtains for t>t’>0, ∂ A c h ( , ) d [ ] t t t h = = = − = − ∂ ∂ ψ ψ R t t R t t ( , ') ( , ') ( , ) ( , ) c h c h A c h A c h ( , ) ( , ) h h t t ' ' t t ' ' t t t t t t → → ∂ h 0 h dt ' t ' h → 0 • This generalized Hatano-Sasa relation does not require any thermodynamic structure nor stationary reference process • It contains a very general modified Fluctuation-dissipation theorem which can be also obtained directly from linear response theory G. Verley, R. Chétrite, D. L., J.Stat. Mech., P10025 (2011)
Stochastic trajectory entropy [ ] • Stochastic trajectory entropy = − π = ψ s c ( , h ) ln ( , ) c h ( , ) c h t t t t t t o Distinct from Kolmogorov-Sinai entropy ɶ t o Distinct from [ ] U. Seifert PRL 95 ,040602 (2005) = − s c ( , h ) ln p c h ( , ) t t t o It can be decomposed into -Reservoir entropy + Total entropy production [ ] [ ] [ ] [ ] [ ] [ ] ∆ ∆ = −∆ = −∆ + + ∆ ∆ s c s c ( , ( , h h ) ) s s ( ( c c , , h h ) ) s s ( ( c c , , h h ) ) t t t t r r t t tot tot t t • Consequence of this decomposition for MFDT: d d r tot = ∂ ∆ = ∂ ∆ R ( , ') t t s c h ( , ) A c ( ) R ( , ') t t s ( , ) c h A c ( ) eq h t ' t ' t t neq h t ' t ' t t dt ' → dt ' → h 0 h 0 = = ν j c A c ( ) ( ) ( ) c A c ( ) t ' t ' t t t ' t ' t t • Additive structure of the MFDT involving local currents: = − ν R t t ( , ') ( j ( ) c ( c ) ) A ( c ) t ' t ' t ' t t
The 1D Ising model with Glauber dynamics • Classical model of coarsening : L Ising spins in 1D described by the hamiltonian L ∑ σ = − σ σ − σ H ({ }) J H , + i i 1 m m = i 1 • System intially at equilibrium at is quenched at time t=0 to a final temperature T. = ∞ T • At the time t’>0, a magnetic field H m is turned on: = θ − H ( ) t H ( t t '), m m • The dynamics is controlled by time-dependent (via H m ) Glauber rates α ( ) ( ) ( ) H i σ σ = − σ β σ + σ + β δ w ({ },{ } ) 1 tanh J H , m − + i i 1 i 1 m im 2
• Analytical verification : - MFDT can be verified although the distributions π t ({ σ },H m ) even for a zero magnetic field are not analytically calculable - Analytical form of the response is known C. Godrèche et al. (2000) • Numerical verification : the distributions π t ({ σ },H m ) can be obtained numerically for a small system size (L=14); and the MFDT verified: t = ∫ χ τ τ Integrated response function ( , ') t t d R ( , ) t − − n m n m t t ' '
III.Inequalities generalizing the second law of thermodynamics for transitions between non-stationary states Particular case:Transitions between periodically driven states o Vibrated granular medium o Electric circuits o Oscillations in biological systems o Manipulated colloids o Manipulated colloids Does a form of second law holds for such transitions ? G. Verley, R. Chétrite, D. L., Phys. Rev. Lett., 108, 120601 (2012)
The three faces of the second law • Two different mechanisms to put a system into non-equilibrium state : - from the breaking of detailed balance via non-equilibrium boundary conditions - from an external driving • This leads to a splitting of the total entropy production into M. Esposito et al., PRL 104 , 090601 (2010) ∆ = ∆ + ∆ S S S tot a na where each part satisfies, each separately, a detailed and an integral FT: where each part satisfies, each separately, a detailed and an integral FT: ∆ P ( S ) tot = ∆ exp( S ) tot −∆ P ( S ) tot ∆ P ( S ) ∆ P ( S ) na = ∆ exp( S ) a = ∆ exp( S ) na + + a −∆ P ( S ) −∆ P ( S ) a na leading to a splitting of the second law into ∆ ≥ ∆ ≥ ∆ ≥ S 0, S 0, S 0, tot a na Is it possible to generalize this decomposition using a non-stationary distribution as reference ?
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