Second-law like inequalities for transitions between non-stationary - - PowerPoint PPT Presentation

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

Stochastic definition of work

• Average over non-equilibrium trajectories leads to equilibrium behavior :

t

W F

e e

β β − − ∆

=

( , )

τ τ τ

τ ∂ = ∂

∫

ɺ

t t

H W d h c h h

• C. Jarzynski, PRL 78, 2690 (1997)

Jarzynski relation

This leads to a formulation of the second-law for macroscopic systems :

• Derivation using Feyman-Kac relation :

≥ ∆

t

W F

( , )

1 ( )

β β

δ

− −

− =

t t

W H c h t A

c c e e Z

Hummer G and Szabo, PNAS 98, 3658 (2001)

1

t

Y

e

−

=

( , )

τ τ τ

φ τ ∂ = ∂

∫

ɺ

t t

Y d h c h h

Work like functional where

• Average over non-equilibrium trajectories leads to steady-state behavior

Hatano-Sasa relation

• T. Hatano and S. Sasa, PRL 86, 3463 (2001)

( , ) ln ( , ) φ = −

st

c h P c h

Now where the equality holds for a quasi-stationary process

• 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

≥

t

Y

• In terms of an additive correction (the asymmetry) which vanishes at equilibrium

valid near any non-equilibrium state

• In terms of a local velocity/current

valid near any non-equilibrium state

• M. Baiesi et al. (2009); E. Lippiello et al. (2005)
• G. Diezemann (2005); L. Cugliandolo et al. (1994)
• R. Chétrite et al. (2008); U. Seifert et al. (2006)
• II. The three routes to modified Fluctuation-dissipation theorems

(MFDT)

• In terms of a new observable constructed from the non-equilibrium stationary distribution

valid near a NESS Rk: in all 3 cases, markovian dynamics is assumed

• J. Prost et al. (2009);
• G. Verley, K. Mallick, D. L., EPL 93, 10002 (2011)

Is it possible to extend the third route for a general

• bservable and a general non-equilibrium state ?

• Probability distribution ρt(c) solution of unperturbed master equation :
• Probability Pt(c,[ht]) has a functional dependence on a perturbation [ht],

[ ]

' '

( ) ( ', ) ( ') ( , ') ( ) ( ') ( ', )

t t t t t t t c c

c w c c c w c c c c L c c t ρ ρ ρ ρ ∂ = − = ∂

∑ ∑

Three relevant probability distributions

[ ] [ ] [ ] [ ]

( , ) ( ', ) ( ', ) ( , ') ( , ) ( ', ) ( ', )

t t h h h

P c h w c c P c h w c c P c h P c h L c c ∂   = − =

∑ ∑

• Probability πt(c,h) defined for a constant time independent perturbation h,
• Trajectory dependent quantity of interest constructed from πt(c,h) :

[ ] [ ] [ ] [ ]

' '

( , ) ( ', ) ( ', ) ( , ') ( , ) ( ', ) ( ', )

t t t

t t h h h t t t t t t t t t c c

P c h w c c P c h w c c P c h P c h L c c t ∂   = − =   ∂

∑ ∑

' '

( , ) ( ', ) ( ', ) ( , ') ( , ) ( ', ) ( ', )

h h h t t t t t t t c c

c h w c c c h w c c c h c h L c h t π π π π ∂   = − =   ∂

∑ ∑

( , ) ln ( , ) ψ π = −

t t t t t t

c h c h

• Model: with and
• Response function is

A particle obeying Langevin dynamics and submitted to a temperature quench

'

2 , 0, ( ') η η ηη δ γ γ γ = − + + = = − ɺ

t t t t t t t t t

k h T x x t t

[ ]

' '

1 ( , ') exp ,

τ

τ γ γ

→

∂   = = −   ∂  

∫

t t h t t h

x k R t t d h

• Alternatively, one has

and thus for a constant protocol h

• Using this together with the MFDT, the same response is recovered

[ ]

( )

2 ' 1/2 2 2

1 1 ( , ) exp exp ' , 2 2

τ τ τ

τ τ σ γ γ πσ         = − − −              

∫ ∫

t t t t t t

h k P x h x d d

( )

2 ' 1/2 2 2

1 1 ( , ) exp exp ' , 2 2

τ τ

π τ τ σ γ γ πσ         = − − −              

∫ ∫

t t t t t

k h x h x d d

Our work like path functional The Feyman-Kac approach : Generalized Hatano-Sasa relation Through linear expansion, one obtains for t>t’>0,

( , )

τ τ τ τ

τ ψ = ∂

∫

ɺ

t h

Y d h c h

[ ] [ ]

( , ) ( , ) ( , ) ( , )

π

π

−

= =

∫

t

Y t t t t t t t t t h

A c h e dc c h A c h A c h

[ ]

1

−

=

Y h

e

[ ]

' '

( , ) ( , ') ( , ) ( , ) ψ

→

∂ = = − ∂

t t t h h t t t t t

A c h d R t t c h A c h

• 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

' ' '

( , ') ( , ) ( , ) ' ψ

→ →

= = − ∂ ∂

h t t t t t h t h

R t t c h A c h h dt

• G. Verley, R. Chétrite, D. L., J.Stat. Mech., P10025 (2011)

• Stochastic trajectory entropy
• Distinct from Kolmogorov-Sinai entropy
• Distinct from
• It can be decomposed into

[ ] [ ] [ ]

( , ) ) ( , ) ( , ∆ = −∆ ∆ +

r t tot t t t

s c s h c c h s h

Stochastic trajectory entropy

• Reservoir entropy + Total entropy production

[ ]

( , ) ln ( , ) ( , ) π ψ = − =

t t t t t t

s c h c h c h

• U. Seifert PRL 95,040602 (2005)

[ ]

( , ) ln ( , ) = − ɶt

t t t

s c h p c h

• Consequence of this decomposition for MFDT:
• Additive structure of the MFDT involving local currents:

[ ] [ ] [ ]

( , ) ) ( , ) ( , ∆ = −∆ ∆ +

r t tot t t t

s c s h c c h s h

' ' ' '

( , ') ( , ) ( ) ' ( ) ( )

→

= ∂ ∆ =

r eq h t t t t h t t t t

d R t t s c h A c dt j c A c

' ' ' '

( , ') ( , ) ( ) ' ( ) ( ) ν

→

= ∂ ∆ =

tot neq h t t t t h t t t t

d R t t s c h A c dt c A c

' ' '

( ) ( , ') ( ) ( ) ) ( ν = −

t t t t t

R t t A j c c c

The 1D Ising model with Glauber dynamics

• Classical model of coarsening : L Ising spins in 1D described by the hamiltonian
• System intially at equilibrium at is quenched at time t=0 to a final temperature T.

1 1

({ }) , σ σ σ σ

+ =

= − −

∑

L i i m m i

H J H = ∞ T

• At the time t’>0, a magnetic field Hm is turned on:
• The dynamics is controlled by time-dependent (via Hm) Glauber rates

( )

( )

( )

1 1

({ },{ } ) 1 tanh , 2 α σ σ σ β σ σ β δ

− +

= − + +

m

H i i i i m im

w J H ( ) ( '), θ = −

m m

H t H t t

• Analytical verification :
• MFDT can be verified although the distributions πt({σ},Hm)

even for a zero magnetic field are not analytically calculable

• Analytical form of the response is known
• Numerical verification : the distributions πt({σ},Hm) can be obtained numerically for a small

system size (L=14); and the MFDT verified: Integrated response function

'

( , ') ( , ) χ τ τ

− −

=∫

t n m n m t

t t d R t

• C. Godrèche et al. (2000)

' t

III.Inequalities generalizing the second law of thermodynamics for transitions between non-stationary states

Particular case:Transitions between periodically driven states

• Vibrated granular medium
• Electric circuits
• Oscillations in biological systems
• Manipulated colloids
• 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

where each part satisfies, each separately, a detailed and an integral FT:

∆ = ∆ + ∆

tot a na

S S S

• M. Esposito et al., PRL 104, 090601 (2010)

where each part satisfies, each separately, a detailed and an integral FT: leading to a splitting of the second law into

( ) exp( ) ( ) ∆ = ∆ −∆

tot tot tot

P S S P S ( ) exp( ) ( )

+

∆ = ∆ −∆

na na na

P S S P S ( ) exp( ) ( )

+

∆ = ∆ −∆

a a a

P S S P S 0, 0, 0, ∆ ≥ ∆ ≥ ∆ ≥

tot a na

S S S

Is it possible to generalize this decomposition using a non-stationary distribution as reference ?

• Now Duality transformation (^) with respect to a non-stationary distribution :
• Second term is a difference of traffic between the direct and dual dynamics,

Traffic is the time-integrated escape rate

• 1

( , ') ( , ) ( ', ) ( ', ) π π

−

=

h t t t t

w c c c h w c c c h

[ ] ɵ ( )

( ) ( ) ln ( , ), λ λ π   ∆Τ = − = − ∂    

∫ ∫

t t

T T h h t t t t t t t t

c dt c c dt c h

• C. Maes et al., PRL 96, 240601 (2006)

'

( ') ( ', ), λ

≠

=∑

t t

h h t t c c

c w c c

It is symmetric with respect to time-reversal: unlike the entropy

• When

and , the action A is called non-adiabatic :

• When and , the action A is called adiabatic :

similar but different from the 3FTs of

*

( ) (^) ( ) ( ) = = ɶ

[ ]

[

]

ln ,

na na na

P A A P A ∆ ∆ = −∆

*

( ) (^) ( ) Id = = ɶ

[ ]

[

]

ln ,

a a a

P B B P B ∆ ∆ = −∆

[ ]

,   ∆ = ∆   T c T c

• Fast relaxation of the accompagnying distribution towards a stationary distribution,

then and one recovers the 3 FTs.

• For transitions between non-stationary states, the generalized Hatano-Sasa relation follows

0, T ∆ = exp( ) 1, − = Y

• Modified second law (Clausius type inequality)

Modified second law (Clausius type inequality)

• Equality corresponds to the adiabatic limit (slow driving) :

where and

ex

S S T ∆ ≥ − ∆ + ∆ ( || ) π ≥ − ∆ = ≥

b T T

Y S D p

b

S ∆ =

na

A T ∆ = ∆ =

• r

• S. Vaikunanathan and C. Jarzynski (2009)

1 (

|| ) β − ≥

eq diss T T

W D p p

For an initial equilibrium state, « Dissipated work dictates the maximum extend to which equilibrium can be broken – equivalently the maximum amount of lag – at a given instant during the process. » For an arbitrary non-stationary initial state, the lag between PT and πΤ distributions provides a bound for

( || ) π ≥ ≥

T T T

Y D p

Non-stationarity due to relaxation

• A reference dynamics is created by some initial conditions different from steady state values
• The model (with two states dynamics) is further driven
• Direct simulation of trajectories from which distributions and of Y are obtained

/2 /2

( , ) ( , ) ; ( , ) ( , )

t t t t

h h h h

w a b w a b e w b a w b a e

−

= =

ln

t

π

• DFT holds independently of

the relaxation time of

t

π

Non-stationarity from periodic driving

• A sinusoidally driven two states model is further perturbed using

As expected and

sin( ) sin( )

( , ) ( , ) ; ( , ) ( , )

t

h t h h t h

w a b w a b e w b a w b a e

ω ω − − +

= =

• As expected

and in the quasi-static limit

T

Y ≥

T na

T Y S ∆ = = ∆ =

na T

S T Y ∆ − ∆ =

Conclusions

• A formalism based on fluctuation relations leads to a modified

fluctuation-dissipation theorem and modified second law of thermodynamics off equilibrium.

• Such a formalism could be useful for studying transitions between

periodically driven states, or between states which are undergoing relaxation due to coarseing or aging.