Fluctuation relations in Ising models

G.G. & Antonio Piscitelli (Bari)

Federico Corberi (Salerno)

Alessandro Pelizzola (Torino)

• Introduction• Fluctuation relations for stochastic systems:

- transient from equilibrium

- NESS• Heat and work fluctuations in a driven Ising

model• Systems in contact with two different heat

baths• Effects of broken ergodicity and phase

transitions

Outline

EQUILIBRIUM

External driving or thermal gradients

Maxwell-Boltzmann

?

Fluctuations in non-equilibrium systems.

Gallavotti-Cohen symmetry

q = entropy produced until time .

P(q) probability distribution for entropy production

Theorem:log P(q)/P(-q) = -q

Evans, Cohen&Morriss, PRL 1993Gallavotti&Cohen, J. Stat. Phys. 1995

¿ ! 1

From steam engines to cellular motors:thermodynamic systems at different scales

Ciliberto & Laroche, J. de Phys.IV 1994Wang, Evans & et al, PRL 2002Garnier & Ciliberto, PRE 2005….

• How general are fluctuation relations?

• Are they realized in popular statistical (e.g. Ising) models?

• Which are the typical time scales for their occuring? Are there general corrections to asymptotic behavior?

• How much relevant are different choices for kinetic rules or interactions with heat reservoir?

Questions

Discrete time Markov chains

• N states with probabilities

evolving at the discrete times with the law

and the transition matrix ( )

• Suppose an energy can be attributed to each state i. For a system in thermal equilibrium:

P j (s +1) =NX

1=1

P i (s)Qi j (s)

Qi j ¸ 08i; j ;NX

j =1

Qi j = 18i

s = 0;::::¿

P j (s) j = 1;:::;N

P eqi =

e¡ ¯ E i

P Nj =1 e¡ ¯ E j

= e¯ F ¡ ¯ E i ¯ =1

kB T

F (¯;E ) = ¡1¯

lnNX

i=1

e¡ ¯ E i

Qi j

E i

Microscopic work and heat

Heat = total energy exchanged with the reservoir due to transitions with probabilities . Work = energy variations due to external work

¢ E = E i (¿)(¿) ¡ E i (0)(0) = Q[¾]+W[¾]

¾´ (i(0); :::::::; i(¿))

¢ F = F (¯;fE (¿)g) ¡ F (¯ ;fE (0)g)

WD [¾]= W[¾]¡ ¢ F

Qi j

Trajectory in phase space withE i (0); :::E i (s); :::E i (¿)

Microscopic reversibility

Probability of a trajectory with fixed initial state::

i(s) = i(¿ ¡ s) E (s) = E (¿ ¡ s) s = 0;:::¿

P eqi Qi j = P eq

j Qj i 8i; j

P [¾ji(0);Qi j ]=¿¡ 1Y

s=0

Qi (s)i (s+1)(s)

P [¾ji(0);Qi j ]

P [¾ji(0);Qi j ]= e¡ ¯ Q [¾]

¾! ¾ Time-reversed trajectory:

Time-reversed transition matrix:Qi j

Averages over trajectories

function defined over trajectories

< f >F ´X

f i (0)g;f i (¿)g

P eqi (0)P [¾(i(0); ::::i(¿))ji(0);Qi j ]f [¾]

f [¾]

Microscopic reversibility

1-1 correspondence between forward and reverse trajectories

+ < f e¡ ¯ W d >F =< f >R

f [¾]= f [¾]

• Jarzynski relation (f=1):

< e¡ ¯ W >= e¡ ¯ ¢ F

• Transient fluctuation theorem starting from equilibrium ( ):

PF (¯Wd)PR (¡ ¯Wd)

= e¯ W d

Fluctuation relations

f [¾]= ±(¯Wd ¡ ¯Wd[¾])

Equilibrium state 1

Equilibrium state 2work

Jarzynski, PRL 1997

Crooks, PRE 1999

Fluctuation relations for NESS

½(i(0))P [¾ji(0);Qi j ]

½(i(0))P [¾ji(0);Qi j ]= e! [¾]

½(i(0)) initial phase¡ spacedistribution

! = ln½(i(¿)) ¡ ln½(i(0)) ¡ ¯Q[¾]

! » ¡ ¯Q if t ! 1

lim¿! 1

PF (¯Q)PR (¡ ¯Q)

= e¯ Q

Lebowitz&Spohn, J. Stat. Phys. 1999 Kurchan, J. Phys. A, 1998

Ising models with NESS

• Does the FR hold in the NESS?

• Does the work transient theorem hold when the initial state is a NESS?

• Systems in contact wth two heat baths.

Work and heat fluctuations in a driven Ising model ( )

Single spin-flip Metropolis or Kawasaki dynamics

Shear events: horizontal line with coordinate y is moved by y lattice steps to the right

Q[¾] =tFX

t=1

[H (¾(t)) ¡ H S (¾(t ¡ 1))]; W[¾]= ¡tFX

t=1

[H S (¾(t ¡ 1)) ¡ H (¾(t ¡ 1))]

H = ¯X

hi j i

¾i¾j

+

¾(t) collection of spin varables at elementary MC-time t

obtained applying shear at the configuration at MC-time t¾S (t)H S (¾(t)) = H (¾S (t)) if a shear event has occurred just after t

H S (¾(t)) = H(¾(t)) otherwise

G.G, Pelizzola, Saracco, Rondoni

Transient between different steady states

_°1 ! _°2

No symmetry under time-reversalForward and reverse pdfs do not coincide

Fluctuation relation for work

The transient FR does not depend on the nature of the initial state.

G.G, Pelizzola, Saracco, Rondoni

Work and heat fluctuations in steady state

• Start from a random configuration, apply shear and wait for the stationary state

• Collect values of work and heat measured over segments of length in a long trajectory.

Work (thick lines) and heat (thin lines) pdfs for L = 50, M = 2, = 1, r = 20 and = 0.2. = 1,8,16,24,32,38, 42from left to right. Statistics collected over 10^8 MC sweeps.

Fluctuation relation for heat and work

Slopes for as function of corresponding to the distributions of previous figure at = 4,16,28.

Slopes for at varying . Parameters are the same as in previous figures.

lnP (O¿ )=P (¡ O¿ ) (O = W;Q)P (O¿ )=P (¡ O¿ ) (O = W;Q)O¿

Fluctuation relation for systems coupled to two heat baths

reservoir reservoir

T1 T2

system

Q¿1 Heat exchanged with the hot heat-bath in the time

Heat exchanged with the cold heat-bath in the time Q¿2

lnP (Q¿

i )P (¡ Q¿

i )» ¡ Q¿

i

µ1Ti

¡1Tj

¶

Two-temperature Ising models (above Tc )

FR holds, independently on the dynamic rules and heat-exchange mechanisms

slope=ln P (Q (n ) (¿))

P (¡ Q (n ) (¿))

Q(n)(¿)³

1Tn 0

¡ 1Tn

´

Scaling behavior of the slope

²(¿;L) = f (¿=L); f (x) » 1=x

L x L square lattices

A. Piscitelli&G.G

²(¿;L) = 1¡ slope

Phase transition and heat fluctuations

Above Tc (T1=2.9, T2=3)

- Heat pdfs below Tc are narrow.- Slope 1 is reached before the ergodic time - Non gaussian behaviour is observable. - Scaling f(x) = 1/x holds

Below Tc =2.27 (T1=1, T=1.3)

2 typical time scales: - relaxation time of autocorrelation - ergodic time (related to magnetization jumps)

Conclusions

• Transient FR for work holds for any initial state (NESS or equilibrium).

• Corrections to the asympotic result are shown to follow a general scaling behavior.

• Fluctuation relations appear as a general symmetry for nonequilibrium systems.