Fluctuation relations in Ising models G.G. & Antonio Piscitelli (Bari) Federico Corberi (Salerno) Alessandro Pelizzola (Torino) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A A A AA A

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 t. P(q) probability distribution for entropy production Theorem: log P(q)/P(-q) = -q Evans, Cohen&Morriss, PRL 1993 Gallavotti&Cohen, J. Stat. Phys. 1995

The second law of thermodynamics Example : T1T1 T2T2 Q1 heat adsorbed from thermostat 1 d i S = -Q1(1/T1 – 1/T2)

From steam engines to cellular motors: thermodynamic systems at different scales Ciliberto & Laroche, J. de Phys.IV 1994 Wang, Evans & et al, PRL 2002 Garnier & Ciliberto, PRE 2005 ….

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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:

Microscopic work and heat (2) Total energy variation Dissipated work Note: work can be also simply produced by changes of the state of the system not due to interactions with the thermal bath.

Microscopic work and heat Heat = total energy exchanged with the reservoir due to transitions with probabilities. Work = energy variations due to external work Trajectory in phase space with

Microscopic reversibility Probability of a trajectory with fixed initial state: : Time-reversed trajectory: Time-reversed transition matrix:

Averages over trajectories function defined over trajectories Microscopic reversibility 1-1 correspondence between forward and reverse trajectories +

Jarzynski relation (f=1): Transient fluctuation theorem starting from equilibrium ( ): Fluctuation relations Equilibrium state 1 Equilibrium state 2 work Jarzynski, PRL 1997 Crooks, PRE 1999

Fluctuation relations for NESS Lebowitz&Spohn, J. Stat. Phys 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 l lattice steps to the right + collection of spin varables at elementary MC-time t obtained applying shear at the configuration at MC-time t if a shear event has occurred just after t otherwise G.G, Pelizzola, Saracco, Rondoni

Phase diagram of the driven Ising model shear rate measuring the strength of the driving number of shear events per each MC sweep The phase diagram is similar to that of the Ising model, but with a different critical temperature (depending on the dynamics) and different exponents horizontal and vertical size G.G&Saracco, in preparation Corberi, G.G, Lippiello&Zannetti, Europh. Lett.2003

Stationary sheared systems are symmetric under time-reversal ( reversing all velocities gives a simple reflection of the original system) Time-reversal invariance Time-reversal symmetry

Transient between different steady states No symmetry under time-reversal Forward 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 t in a long trajectory. Work (thick lines) and heat (thin lines) pdfs for L = 50, M = 2, l = 1, r = 20 and b = 0.2. t = 1,8,16,24,32,38, 42 from 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 t = 4,16,28. Slopes for at varying t. Parameters are the same as in previous figures.

Fluctuation relation for systems coupled to two heat baths reservoir T1T1 T2T2 system Heat exchanged with the hot heat-bath in the time t Heat exchanged with the cold heat-bath in the time t

Sketch of the proof of the fluctuation relation Detailed balance is generalized to Transition rate for the jump i -> j with heats q a, q b,.. transferred from heat baths at T a, T b,.. to the system Ratio of probabilities of a trajectory and of its time-reversed Bounded internal energy and probabilities imply Large t limit Derrida&Bodineau, 2007

Two-temperature Ising models (above T c ) FR holds, independently on the dynamic rules and heat-exchange mechanisms

Scaling behavior of the slope L x L square lattices A. Piscitelli&G.G

Scaling behavior of the slope for dynamical coupling L x L square lattices A. Piscitelli&G.G

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 e = 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)

Magnetization jums

TEST OF GALLAVOTTI-COHEN SYMMETRY IN A LANGEVIN SYSTEM FOR BINARY MIXTURES The model: = order parameter Evolution equation: with: (Noise verifies the fluctuation-dissipation relation) and NOTE: This model is used in practise in the study of phase separation and of mixtures dynamics with a convective term when the fluctuation of the velocity field are negligible.

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.