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Fluctuation relations in Ising models

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.: A A A A A A A A A A. Outline. Introduction

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Fluctuation relations in Ising models

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  1. 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.: AAAAAAAAAA

  2. Outline • 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

  3. Fluctuations in non-equilibrium systems. EQUILIBRIUM Maxwell-Boltzmann ? External driving or thermal gradients

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

  5. 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 ….

  6. Questions • 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?

  7. 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:

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

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

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

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

  12. Fluctuation relations for NESS Lebowitz&Spohn, J. Stat. Phys. 1999 Kurchan, J. Phys. A, 1998

  13. 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.

  14. Workand heat fluctuations in a driven Ising model ( ) G.G, Pelizzola, Saracco, Rondoni Shear events: horizontal line with coordinate y is moved by yl lattice steps to the right + Single spin-flip Metropolis or Kawasaki dynamics 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

  15. Transient between different steady states No symmetry under time-reversal Forward and reverse pdfs do not coincide

  16. Fluctuation relation for work The transient FR does not depend on the nature of the initial state. G.G, Pelizzola, Saracco, Rondoni

  17. 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.

  18. 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.

  19. Fluctuation relation for systems coupled to two heat baths system T1 T2 reservoir reservoir Heat exchanged with the hot heat-bath in the time t Heat exchanged with the cold heat-bath in the time t

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

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

  22. Phase transition and heat fluctuations 2 typical time scales:- relaxation time of autocorrelation - ergodic time (related to magnetization jumps) Below Tc =2.27 (T1=1, T=1.3) 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

  23. 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.

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