Transcript Fluctuation relations in Ising models

Fluctuation relations in Ising
models
G.G. & Antonio Piscitelli (Bari)
Federico Corberi (Salerno)
Alessandro Pelizzola (Torino)
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
Fluctuations in non-equilibrium systems.
EQUILIBRIUM
Maxwell-Boltzmann
?
External driving or
thermal gradients
Gallavotti-Cohen symmetry
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: ¿ ! 1
log P(q)/P(-q) = -q
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
….
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?
Discrete time Markov chains
•
N states with probabilities
Pj (s)
j = 1; :::; N
evolving at the discrete times s = 0; ::::¿ with the law
XN
Pj (s + 1) =
Pi (s)Q i j (s)
1= 1
and Qi j the transition matrix (Qi j ¸ 0 8i ; j ;
XN
Q i j = 1 8i )
j=1
• Suppose an energy E i can be attributed to each state i.
For a system in thermal equilibrium:
e¡ ¯ E i
1
eq
¯F ¡ ¯Ei
Pi = P N
=e
¯=
¡
¯
E
j
kB T
j=1e
N
1 X ¡
F (¯; E ) = ¡ ln
e
¯ i= 1
¯Ei
Microscopic work and heat
¾´ (i (0); :::::::; i (¿))
Trajectory in phase space with
E i (0) ; :::E i (s) ; :::Ei (¿)
Heat = total energy exchanged with the reservoir
due to transitions with probabilities Qi j .
Work = energy variations due to external work
¢ E = Ei (¿) (¿) ¡ Ei (0) (0) = Q[¾] + W[¾]
¢ F = F (¯; f E(¿)g) ¡ F (¯; f E(0)g)
WD [¾] = W[¾] ¡ ¢ F
Microscopic reversibility
¾!
¾
^
^i(s) = i (¿ ¡ s)
^i j
Q
Time-reversed trajectory:
^
E(s)
= E(¿ ¡ s)
s = 0; :::¿
Time-reversed transition matrix:
^ i j = P eq Qj i
Pieq Q
j
8i ; j
Probability of a trajectory with fixed initial state:
¿¡
Y1
P [¾ji (0); Q i j ] =
Qi ( s) i ( s+ 1) (s)
s= 0
P[¾ji (0); Qi j ]
= e¡
^i j ]
P[^
¾j^i(0); Q
¯ Q [¾]
:
Averages over trajectories
f [¾]
function defined over trajectories
X
< f >F´
Pieq
( 0) P [¾(i (0); ::::i (¿))ji (0); Q i j ]f [¾]
f i ( 0) g;f i ( ¿) g
Microscopic reversibility
+
1-1 correspondence between
forward and reverse trajectories
¡ ¯W d
< fe
> F = < f^ > R
f^[^
¾] = f [¾]
Fluctuation relations
• Jarzynski relation (f=1):
¡ ¯W
<e
Jarzynski, PRL 1997
¡ ¯¢ F
>= e
• Transient fluctuation theorem starting from equilibrium
( f [¾] = ±(¯Wd ¡ ¯Wd [¾]) ):
Crooks, PRE 1999
PF (¯Wd )
= e¯ W d
PR (¡ ¯Wd )
Equilibrium
state 1
work
Equilibrium
state 2
Fluctuation relations for NESS
½(i (0))
i ni ti al phase ¡ space di str i buti on
! = ln ½(i (¿)) ¡ ln ½(i (0)) ¡ ¯Q[¾]
½(i (0))P[¾ji (0); Qi j ]
= e!
^i j ]
½(^i(0))P[^
¾j^i(0); Q
[¾]
! » ¡ ¯Q if t ! 1
PF (¯Q)
lim
= e¯ Q
¿! 1 PR (¡ ¯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
X
Ising model ( H = ¯ ¾i ¾j )
hi j i
Single spin-flip Metropolis
or Kawasaki dynamics
¾(t)
¾S (t)
+
G.G, Pelizzola, Saracco, Rondoni
Shear events: horizontal line
with coordinate y is moved by
yl lattice steps to the right
collection of spin varables at elementary MC-time t
obtained applying shear at the configuration at MC-time t
H S (¾(t)) = H (¾S (t)) if a shear event has occurred just after t
H S (¾(t)) = H (¾(t))
Xt F
Q[¾] =
S
otherwise
[H (¾(t)) ¡ H (¾(t ¡ 1))];
t= 1
Xt F
W [¾] = ¡
t= 1
[H S (¾(t ¡ 1)) ¡ H (¾(t ¡ 1))]
Transient between different steady states
°_1 !
No symmetry under time-reversal
Forward and reverse pdfs do not coincide
°_2
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 P(O¿)=P(¡ O¿ ) (O = W; Q)
as function of O¿ corresponding to
the distributions of previous figure at t
= 4,16,28.
Slopes for ln P(O¿ )=P(¡ O¿ ) (O = W; Q)
at varying t. Parameters are the same
as in previous figures.
Fluctuation relation for systems coupled to
two heat baths
system
Q ¿1
Q ¿2
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
ln
P(Q¿i )
P(¡ Q¿i )
µ
» ¡
Q¿i
1
1
¡
Ti
Tj
¶
Two-temperature Ising models (above Tc )
(n )
)
ln PP( (¡ QQ ( n )( ¿)
( ¿) )
³
´
slope =
1
1
(
n
)
FR holds, independently on the dynamic
rulesT and
Q (¿)
¡ T n heat-exchange mechanisms
0
n
Scaling behavior of the slope
L x L square lattices
²(¿; L) = 1 ¡ slope
²(¿; L) = f (¿=L);
f (x) » 1=x
A. Piscitelli&G.G
Phase transition and heat fluctuations
2 typical time scales: - relaxation time of autocorrelation
- ergodic time (related to magnetization jumps)
Above Tc (T1=2.9, T2=3)
Below Tc =2.27 (T1=1, T=1.3)
- 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
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.