Density large-deviations of nonconserving driven models

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


T , µ

Equilibrium Driven diffusive systems

pq

w- w+

( )H conserving steady state

. .( , ) [ , , ]G CanP N F H T

( )CNP


T , µ


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


T , µ


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


conserving(sum over η’ with same N)

nonconserving(sum over η’ with N’≠N)

'

' ' ' '' ' 1 '

( ) ( ') ( ) ( ') ( )N N


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


conserving(sum over η’ with same N)

nonconserving(sum over η’ with N’≠N)

'

' ' ' '' ' 1 '

( ) ( ') ( ) ( ') ( )N N


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


= 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


' '

' ' '' 1 ' '

0 ( ')[ ( ')] ( )[ ( ')]N N N N

NC C NC CN N

N N

f N w P f N w P


= 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


' '

' ' '' 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


)exp(L

q


knownFor low β’s

Density profile

2nd order

( )A i

iA

L

* ( , )x r

c

* ( , ) /x r r N L

Conserving model


)exp(L

q


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 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


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


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)

Density large-deviations of nonconserving driven models

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