fluctuation relations in ising models
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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 - PowerPoint PPT PresentationTRANSCRIPT
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Fluctuation relations in Ising models
G.G. & Antonio Piscitelli (Bari)
Federico Corberi (Salerno)
Alessandro Pelizzola (Torino)
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• 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
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EQUILIBRIUM
External driving or thermal gradients
Maxwell-Boltzmann
?
Fluctuations in non-equilibrium systems.
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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
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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….
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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
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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
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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 (¿)
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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
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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]
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• 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
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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
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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.
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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
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Transient between different steady states
_°1 ! _°2
No symmetry under time-reversalForward and reverse pdfs do not coincide
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Fluctuation relation for work
The transient FR does not depend on the nature of the initial state.
G.G, Pelizzola, Saracco, Rondoni
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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.
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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¿
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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
¶
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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
´
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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
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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)
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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.