5 th KIAS Conference on Statistical Physics Seoul Korea July 2012 Or Cohen and David Mukamel Ensemble theory out of equilibrium T µ Equilibrium Ensemble theory out of equilibrium ID: 264866
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Slide1
Density large-deviations of nonconserving driven models
5th KIAS Conference on Statistical Physics, Seoul, Korea, July 2012
Or Cohen and David
MukamelSlide2
Ensemble theory out of equilibrium ?
T , µ
EquilibriumSlide3
Ensemble theory out of equilibrium ?
T , µ
Equilibrium
Driven diffusive systems
p
q
w
-
w
+
conserving steady stateSlide4
Ensemble theory out of equilibrium ?
T , µ
Equilibrium
Driven diffusive systems
p
q
w
-
w
+
Can we infer about the
nonconserving
system from the steady state properties of the conserving system ?
conserving steady stateSlide5
Ensemble theory out of equilibrium ?
T , µ
Equilibrium
Driven diffusive systems
p
q
w
-
w
+Slide6
Generic driven diffusive model
conserving
(sum over
η’ with same N)
nonconserving(sum over η
’ with N’≠N)
w
R
C
w
L
C
w
-
NC
w
+
NC
L sitesSlide7
Generic driven diffusive model
conserving(sum over η
’ with same N)
nonconserving(sum over
η’ with N’≠N)
w
R
C
w
L
C
w
-
NC
w
+
NC
Guess a steady
state of the form :
L sitesSlide8
Generic driven diffusive model
conserving(sum over η
’ with same N)
nonconserving(sum over
η’ with N’≠N)
It is consistent if :
w
R
C
w
L
C
w
-
NC
w
+
NC
In many cases :
Guess a steady
state of the form :
L sitesSlide9
Slow nonconserving dynamics
To leading order in
ε
we obtainSlide10
Slow nonconserving dynamics
= 1D - Random walk in a potential
To leading order in
ε
we obtainSlide11
Slow nonconserving dynamics
= 1D - Random walk in a potential
Steady state solution :
To leading order in
ε
we obtainSlide12
Outline
Limit of slow nonconserving
Example of the ABC model
Nonequilibrium chemical potential (dynamics dependent !)
ConclusionsSlide13
ABC model
A
B
C
AB BA
BC CB
CA AC
Dynamics :
q
1
q
1
q
1
Ring of size L
Evans,
Kafri
,
Koduvely
&
Mukamel
- Phys. Rev.
Lett
. 1998 Slide14
ABC model
A
B
C
AB BA
BC CB
CA AC
Dynamics :
q
1
q
1
q
1
Ring of size L
q=1
q<1
Evans,
Kafri
,
Koduvely
&
Mukamel
- Phys. Rev.
Lett
. 1998
A
BB
C
A
CC
B
A
C
A
B
A
C
B
AAAAA
BBBBB
CCCCCSlide15
ABC model
t
x
A
B
CSlide16
Nonconserving ABC model
0X X0 X=A
,B,
C
1
1
A
B
C
0
Lederhendler
&
Mukamel
-
Phys. Rev.
Lett
.
2010
A
B
B
A
B
C
C
B
C
A
A
C
q
1
q
1
q
1
1
2
1
2
Conserving model
(canonical ensemble)
+
fixedSlide17
Conserving model
Clincy
, Derrida & Evans -
Phys. Rev. E 2003
Weakly asymmetric
thermodynamic limitSlide18
Conserving model
Clincy
, Derrida & Evans -
Phys. Rev. E 2003
Weakly asymmetric
thermodynamic limit
known
For low
β
’s
Density profile
2
nd orderSlide19
Conserving model
Clincy
, Derrida & Evans -
Phys. Rev. E 2003
Weakly asymmetric
thermodynamic limit
Density profile
known
2
nd
order
For low
β’sSlide20
Nonconserving ABC model
0X X0 X=A
,B,
C
1
1
A
B
C
0
Lederhendler
&
Mukamel
-
Phys. Rev.
Lett
.
2010
A
B
B
A
B
C
C
B
C
A
A
C
q
1
q
1
q
1
A
B
C
000
pe
-3
βμ
p
1
2
3
1
2
1
2
3
Conserving model
(canonical ensemble)
Nonconserving
model
(grand canonical ensemble)
+
+
+Slide21
Slow nonconserving model
Slow nonconserving
limit
A
B
C
000
pe
-3
βμ
pSlide22
Slow nonconserving model
Slow nonconserving
limit
A
B
C
000
pe
-3
βμ
p
saddle point approx.Slide23
Slow nonconserving model
A
BC 000
pe
-3
βμ
pSlide24
Slow
nonconserving model
A
BC 000
pe
-3
βμ
p
This is similar to equilibrium : Slide25
Large deviation function of r
High µ
Low µ
First order phase transition (only in the
nonconserving
model)Slide26
Inequivalence of ensembles
Conserving = Canonical
Nonconserving =
Grand canonical
2
nd order transition
ordered
1
st order transition
tricritical point
disordered
ordered
disordered
For
N
A
=
N
B
≠
N
C
:Slide27
Why is µS(N) the chemical potential ?
N
1
N
2
SLOWSlide28
Why is µ
S
(N) the chemical potential ?
N
1
N
2
Gauge measures
SLOW
SLOWSlide29
Conclusions
Nonequlibrium ‘grand canonical ensemble’ - Slow
nonconserving
dynamicsExample to ABC model
1st order phase transition for
nonmonotoneous µs(r) and inequivalence
of ensembles.Nonequilibrium chemical potential
( dynamics dependent ! )
Thank you !
Any
questions ?
Cohen & Mukamel - PRL
108, 060602 (2012)