density large-deviations of nonconserving driven models
DESCRIPTION
Density large-deviations of nonconserving driven models. Or Cohen and David Mukamel. 5 th KIAS Conference on Statistical Physics, Seoul, Korea, July 2012. Ensemble theory out of equilibrium ?. Equilibrium. T , µ. Ensemble theory out of equilibrium ?. Driven diffusive systems. Equilibrium. - PowerPoint PPT PresentationTRANSCRIPT
Density large-deviations of
nonconserving driven models
5th KIAS Conference on Statistical Physics, Seoul, Korea, July 2012
Or Cohen and David Mukamel
Ensemble theory out of equilibrium ?
T , µ
Equilibrium
. .( , ) [ , , ]G CanP N F H T
( )H
Ensemble theory out of equilibrium ?
T , µ
Equilibrium Driven diffusive systems
pq
w- w+
( )H conserving steady state
. .( , ) [ , , ]G CanP N F H T
( )CNP
Ensemble theory out of equilibrium ?
T , µ
Equilibrium Driven diffusive systems
pq
w- w+
Can we infer about the nonconserving system from
the steady state properties of the conserving system ?
( , ) ( ), ???NC CN
wP N F P
w
( )H ( )C
NP
conserving steady state
. .( , ) [ , , ]G CanP N F H T
Ensemble theory out of equilibrium ?
T , µ
Equilibrium Driven diffusive systems
pq
w- w+
( , ) ( ) ( )NC C NCNP N P P N
, ,w w q p
( , ) ( ), ???NC CN
wP N F P
w
. .
. . .
( , )
( ) ( )
G Can
Can G CanN
P N
P P N
( )H ( )C
NP
. .( , ) [ , , ]G CanP N F H T
Generic driven diffusive model
'
' ' ' '' ' 1 '
( ) ( ') ( ) ( ') ( )N N
NC C NC C NC NC NC NC NCt
N N
P w P w P w P w P
conserving
(sum over η’ with same N)nonconserving
(sum over η’ with N’≠N)
wRCwL
C
w-NC w+
NC
L sites
Generic driven diffusive model
conserving(sum over η’ with same N)
nonconserving(sum over η’ with N’≠N)
'
' ' ' '' ' 1 '
( ) ( ') ( ) ( ') ( )N N
NC C NC C NC NC NC NC NCt
N N
P w P w P w P w P
wRCwL
C
w-NC w+
NC
higher order( , ) ( ) ( )
in NC C
NP N P f N OL
Guess a steadystate of the form :
L sites
Generic driven diffusive model
conserving(sum over η’ with same N)
nonconserving(sum over η’ with N’≠N)
'
' ' ' '' ' 1 '
( ) ( ') ( ) ( ') ( )N N
NC C NC C NC NC NC NC NCt
N N
P w P w P w P w P
' 2NCw L It is consistent if :
wRCwL
C
w-NC w+
NC
In many cases :
higher order( , ) ( ) ( )
in NC C
NP N P f N OL
Guess a steadystate of the form :
L sites
Slow nonconserving dynamics
' '
' ' '' 1 ' '
0 ( ')[ ( ')] ( )[ ( ')]N N N N
NC C NC CN N
N N
f N w P f N w P
', , '' 1
0 ( ') ( )N N N NN N
f N W f N W
To leading order in ε we obtain
Slow nonconserving dynamics
= 1D - Random walk in a potential
', , '' 1
0 ( ') ( )N N N NN N
f N W f N W
, 1N NW
maxNminN *N
log[ ( )]V f N , 1N NW
To leading order in ε we obtain
' '
' ' '' 1 ' '
0 ( ')[ ( ')] ( )[ ( ')]N N N N
NC C NC CN N
N N
f N w P f N w P
Slow nonconserving dynamics
= 1D - Random walk in a potential
Steady state solution :
', , '' 1
0 ( ') ( )N N N NN N
f N W f N W
0
', ' 1
' ' 1, '
( ) ( )N
N NNC
N N N N
Wf N P N
W
, 1N NW , 1N NW
maxNminN *N
log[ ( )]V f N
To leading order in ε we obtain
' '
' ' '' 1 ' '
0 ( ')[ ( ')] ( )[ ( ')]N N N N
NC C NC CN N
N N
f N w P f N w P
Outline
1. Limit of slow nonconserving
2. Example of the ABC model
3. Nonequilibrium chemical potential
(dynamics dependent !)
4. Conclusions
( , ) ( ) ( )
( ) [ ( ), ]
NC
NC C NCN
NC C NCN
w L
P N P P N
P N F P w
( )N
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
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<1L
Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998
ABBCACCBACABACBAAAAABBBBBCCCCC
ABC model
t
x
A B C
Nonconserving ABC model
0X X0 X=A,B,C1
1
A B
C 0
Lederhendler & Mukamel - Phys. Rev. Lett. 2010
AB BA
BC CB
CA AC
q
1
q
1
q
1
1 2
1 2 Conserving model(canonical ensemble)
+
A B CN N Nr
L
fixed
Conserving model
Clincy, Derrida & Evans - Phys. Rev. E 2003
)exp(L
q
Weakly asymmetric thermodynamic limit
Conserving model
Clincy, Derrida & Evans - Phys. Rev. E 2003
)exp(L
q
Weakly asymmetric thermodynamic limit
knownFor low β’s
Density profile
2nd order
( )A i
iA
L
* ( , )x r
c
* ( , ) /x r r N L
Conserving model
Clincy, Derrida & Evans - Phys. Rev. E 2003
)exp(L
q
Weakly asymmetric thermodynamic limit
Density profile
known
2nd order
For low β’s
[ , ][ ] , ,L FP e A B C
( )A i
iA
L
* ( , )x r
c
* ( , ) /x r r N L
Nonconserving ABC model
0X X0 X=A,B,C1
1
A B
C 0
Lederhendler & Mukamel - Phys. Rev. Lett. 2010
AB BA
BC CB
CA AC
q
1
q
1
q
1
ABC 000pe-3βμ
p
1 2
3
1 2
1 2 3
Conserving model(canonical ensemble)
Nonconserving model(grand canonical ensemble)
+
++
Slow nonconserving model
Slow nonconserving limit 2,~ Lp
, 3N NW , 3N NW
maxNminN *N
ABC 000pe-3βμ
p
( )V N
Slow nonconserving model
Slow nonconserving limit
ABC 000pe-3βμ
p
saddle point approx.
1
1* 3
, 3 ' 0' 0
( ') ( ( , ))N N
NC CN N N
NW w P dx x
L
2,~ Lp
, 3N NW
maxNminN *N
( )V N , 3N NW
[ ]L Fe
Slow nonconserving model
ABC 000pe-3βμ
p
maxNminN *N
, 3N NW , 3N NW ( )V N
( )( ) exp[ ( ' ( ') )]r
L G rNCSP r L dr r r e
1* 30
, 3 01
* * *3,
0
( ( , ))1 1
( ) log( ) log[ ]3 3
( , ) ( , ) ( , )
N NS
N NA B C
dx x rW
rW
dx x r x r x r
A B CN N Nr
L
Slow nonconserving model
ABC 000pe-3βμ
p
This is similar to equilibrium :
( )( ) exp[ ( ' ( ') )]r
L G rNCSP r L dr r r e
. ( ) exp[ ( ( , ) )]G CanP r L F r r
maxNminN *N
, 3N NW , 3N NW ( )V N
1* 30
, 3 01
* * *3,
0
( ( , ))1 1
( ) log( ) log[ ]3 3
( , ) ( , ) ( , )
N NS
N NA B C
dx x rW
rW
dx x r x r x r
A B CN N Nr
L
Large deviation function of r
r maxrminr r
r maxrminr r
High µ
Low µ
First order phase transition (only in the nonconserving model)
( )G r
( )G r
Inequivalence of ensembles
Conserving = Canonical
Nonconserving = Grand canonical
2nd order transition
ordered
1st order transition tricritical point
disordered
ordered
disordered
23
,3
rr
rrr CBA 01.0For NA=NB≠NC :
Why is µS(N) the chemical potential ?
N1 N2
SLOW1 1 2 2
1 2
( / ) ( / )S SN L N L
N N N
Why is µS(N) the chemical potential ?
N1 N2
1 1 2 2
1 2
( / ) ( / )S SN L N L
N N N
( / )S N LGauge measures
SLOW
SLOW
Conclusions
1. Nonequlibrium ‘grand canonical ensemble’ - Slow
nonconserving dynamics
2. Example to ABC model
3. 1st order phase transition for nonmonotoneous µs(r) and
inequivalence of ensembles.
4. Nonequilibrium chemical potential
( dynamics dependent ! )
Thank you ! Any questions ?
Cohen & Mukamel - PRL 108, 060602 (2012)