Density large-deviations of
nonconserving driven models
Or Cohen and David Mukamel
STATPHYS 25 Conference, SNU, Seoul, Korea, July 2013
Grand canonical ensemble out of equilibrium ?
Equilibrium
H[ ]
T,µ
Grand canonical ensemble out of equilibrium ?
Equilibrium
H[ ]
T,µ
r   d d x  ( x)
P[  ]  exp   H [  ]  Vr

*
 exp log Pcons
. [  ]   F (  , r )  Vr
conserving
steady state
Helmholtz
free energy

Grand canonical ensemble out of equilibrium ?
Equilibrium
r   d d x  ( x)
H[ ]
P[  ]  exp   H [  ]  Vr

*
 exp log Pcons
.[  ]   F (  , r )  Vr
T,µ
conserving
steady state
Driven system
q
p
*
Pcons
.[  ]
conserving
steady state
Helmholtz
free energy

Grand canonical ensemble out of equilibrium ?
Equilibrium
r   d d x  ( x)
H[ ]
P[  ]  exp   H [  ]  Vr

*
 exp log Pcons
.[  ]   F (  , r )  Vr
T,µ
conserving
steady state
Driven system
q
p
*
Pcons
.[  ]
conserving
steady state
e-βμ 1
Helmholtz
free energy

Grand canonical ensemble out of equilibrium ?
Equilibrium
r   d d x  ( x)
H[ ]
P[  ]  exp   H [  ]  Vr

*
 exp log Pcons
.[  ]   F (  , r )  Vr
T,µ
conserving
steady state
Driven system
q
p
*
Pcons
.[  ]
conserving
steady state
Helmholtz
free energy
r


*
P[  ]  exp log Pcons.[  ]  V  dr ' s ( r ')  Vr 
0


s [ P
*
cons .
e-βμ 1

[  ]]
Dynamics-dependent
chemical potential
of conserving system
General particle-nonconserving driven model
wLC
w-NC
wRC
L sites
w+NC
General particle-nonconserving driven model
wLC
wRC
w-NC
t P( ) 
L sites
w+NC
w  P( ')  w P( )    w  P( ')  w P( )



' N
C
'
C
'
N ' N ' N '
NC
'
NC
'
conserving
nonconserving
(sum over η’ with same N)
(sum over η’ with N’≠N)
General particle-nonconserving driven model
wLC
wRC
w-NC
t P( ) 
L sites
w+NC
w  P( ')  w P( )    w  P( ')  w P( )



' N
C
'
C
'
N ' N ' N '
NC
'
NC
'
conserving
nonconserving
(sum over η’ with same N)
(sum over η’ with N’≠N)
Guess a steady
state of the form :
P ( )  P
*
cons .
 higher order 
( ; N ) f ( N )  O 

in
L


General particle-nonconserving driven model
wLC
wRC
w-NC
 t P ( ) 
L sites
w+NC
w  P ( ')  w P ( )   



' N
C
'
C
'
N ' N
' N '
NC
wNC' P ( ')  w
' P ( )
conserving
nonconserving
(sum over η’ with same N)
(sum over η’ with N’≠N)
Guess a steady
state of the form :
It is consistent if :
P ( )  P
*
cons .
wNC'
 higher order 
( ; N ) f ( N )  O 

in
L


1
 cons
.
For diffusive systems
 cons. ~ L2
Slow nonconserving dynamics
To leading order in L we obtain
0

N ' N
f ( N ')[ 
 N


' N '
*
wNC' Pcons
. ( '; N ')]  f ( N )[ 
0
 N

N ' N


' N '
f ( N ')WN ', N  f ( N )WN , N '
*
wNC' Pcons
. ( '; N )]
Slow nonconserving dynamics
0

N ' N
f ( N ')[ 
 N


' N '
*
wNC' Pcons
. ( '; N ')]  f ( N )[ 
0
 N

N ' N
N min
' N '
f ( N ')WN ', N  f ( N )WN , N '
WN , N 1
V (N )


*
wNC' Pcons
. ( '; N )]
WN , N 1
N*
= 1D - Random
walk in a potential
N max
Slow nonconserving dynamics
0

N ' N
f ( N ')[ 
 N


' N '
*
wNC' Pcons
. ( '; N ')]  f ( N )[ 
0
 N

N ' N


' N '
*
wNC' Pcons
. ( '; N )]
f ( N ')WN ', N  f ( N )WN , N '
V ( N )  LG ( N / L)
WN , N 1
= 1D - Random
walk in a potential
WN 1, N
N min
N
r
L
N*
 N
WN ', N '1 
f ( N )  exp   log(
)
WN '1, N ' 
 N ' N0
N max
Slow nonconserving dynamics
0

N ' N
f ( N ')[ 
 N


' N '
*
wNC' Pcons
. ( '; N ')]  f ( N )[ 
0
 N

N ' N


' N '
*
wNC' Pcons
. ( '; N )]
f ( N ')WN ', N  f ( N )WN , N '
V ( N )  LG ( N / L)
WN , N 1
w-NC
WN 1, N
N min
N*
N max
r
N
r
L
w+NC
wNC
 

e
wNC
 L  dr '  s ( r )  L  r
 N
WN ', N '1 
f ( N )  exp   log(
) ~ e 0
 e  LG ( r )
WN '1, N ' 
 N ' N0
Outline
1. Limit of slow nonconserving
w NC
L
 2
r



*
P[  ]  Pcons.[  ]exp   L  dr ' s (r ')   Lr 

0

2. Example of the ABC model
3. Corrections to the rate function G (r ) using MFT
4. Conclusions
ABC model
Ring of size L
Dynamics :
q
AB
BC
CA
A
B
C
1
q
1
q
1
M R Evans, Y Kafri , H M Koduvely, D Mukamel - Phys. Rev. Lett. 80 425 (1998 )
BA
CB
AC
ABC model
Ring of size L
Dynamics :
q
AB
BC
CA
A
q=1
L
q<1
B
C
1
q
1
q
1
ABBCACCBACABACB
AAAAABBBBBCCCCC
M R Evans, Y Kafri , H M Koduvely, D Mukamel - Phys. Rev. Lett. 80 425 (1998 )
BA
CB
AC
ABC model
site index
time
q 1
A
L  480
B
N A  N B  NC
C
Conserving ABC model
1
2
q
AB
BC
CA
1
q
1
q
1
BA
0X
CB
B
C
0
1
X0 X=A,B,C
N A  N B  NC
r
fixed
L
AC
1
A
1
+
2
A Lederhendler, D Mukamel - Phys. Rev. Lett. 105 105602 (2010)
A Lederhendler, OC, D Mukamel - J. Stat. Mech. 11 11016 (2010)
Conserving model
(canonical ensemble)
Conserving ABC model
Weakly asymmetric
thermodynamic limit1
1.
q  exp(

L
Density profile
)
M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003)
i
 A ( )  Ai
L
Conserving ABC model
Weakly asymmetric
thermodynamic limit1
For low β’s
q  exp(

L
Density profile
)
* ( x, r )  r  N / L
* ( x, r )
c
2nd order
1.
2.
i
 A ( )  Ai
L
M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003)
OC, D Mukamel - J. Phys. A 44 415004 (2011)
known2
Conserving ABC model
Weakly asymmetric
thermodynamic limit1
For low β’s
q  exp(

L
Density profile
)
i
 A ( )  Ai
L
* ( x, r )  r  N / L
* ( x, r )
known2
c
2nd order
Stationary
measure3:
1.
2.
3.
P[  A ,  B , C ]  e
 LF [  A ,  B , C ;  r ]
M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003)
OC, D Mukamel - J. Phys. A 44 415004 (2011)
T Bodineau, B Derrida - Comptes Rendus Physique 8 540 (2007)
Nonconserving ABC model
1
2
q
AB
BC
CA
A
B
C
0
1
q
1
q
1
BA
1
0X
CB
1
3
pe-3βμ
ABC
AC
1
X0 X=A,B,C
1
+
2
+
2
+
p
000
Conserving model
(canonical ensemble)
3
A Lederhendler, D Mukamel - Phys. Rev. Lett. 105 105602 (2010)
A Lederhendler, OC, D Mukamel - J. Stat. Mech. 11 11016 (2010)
Nonconserving model
(grand canonical ensemble)
Slow nonconserving ABC model
Slow nonconserving limit
WN , N 3
V (N )
N min

p~L
WN , N 3
N*
,  2
ABC
N max
pe-3βμ
p
000
Slow nonconserving ABC model

p~L
Slow nonconserving limit
WN , N 3
V (N )
WN , N 3
WN , N 3 




 N
ABC
N*
N min
' N 3
,  2
pe-3βμ
N max
1
NC *
 ' cons .
w P
p
N 3
( '; N )   dx(  ( x, ))
L
0
*
0
saddle point approx.
e  LF [  A ,  B , C ; r ]
000
Slow nonconserving ABC model
WN , N 3
V (N )
ABC
WN , N 3
pe-3βμ
p
r
N*
N min
000
N A  N B  NC
L
N max
1
WN , N 3
1
1
 S (r ) 
log(
)
log[ 1
3
WN 3, N
3
*
3
dx
(

(
x
,
r
))
 0
0
*
*
*
dx

(
x
,
r
)

(
x
,
r
)

B
C ( x, r )
 A
0
r
P(r )
exp[ L (  dr '  S (r ')   r )]  e  LG ( r )
]
Slow nonconserving ABC model
WN , N 3
V (N )
ABC
WN , N 3
pe-3βμ
p
r
N*
N min
000
N A  N B  NC
L
N max
1
WN , N 3
1
1
 S (r ) 
log(
)
log[ 1
3
WN 3, N
3
*
3
dx
(

(
x
,
r
))
 0
0
]
*
*
*
dx

(
x
,
r
)

(
x
,
r
)

B
C ( x, r )
 A
0
r
P(r )
exp[ L (  dr '  S (r ')   r )]  e  LG ( r )
This is similar to equilibrium :
P(r )  exp  L [ f (r ,  )   r ]
f = Helmholtz free energy density
Rate function of r, G(r)
rA  rB 
r
r
 , rC   2
3
3
  40
  0.025
G (r )
High µ
  0.05
G (r )
Low µ
  0.052
First order phase transition (only in the nonconserving model)
OC, D Mukamel - Phys. Rev. Let. 108, 060602 (2012)
Inequivalence of ensembles
For NA=NB≠NC :
rA  rB 
Conserving (Canonical)
disordered
r
r
 , rC   2
3
3
  0.01
Nonconserving (Grand canonical)
disordered
ordered
ordered
1st order transition
Stability line
2nd order transition
tricritical point
OC, D Mukamel - Phys. Rev. Let. 108, 060602 (2012)
Different nonconserving ABC model: J Barton, J L Lebowitz, E R Speer - J. Phys. A 44 065005 (2011)
Discussion about ensemble inequivalence: OC, D Mukamel - J. Stat. Mech. 12 12017 (2012)
Corrections to G(r) using MFT
pe-3βμ
ABC
p
p
000
T


2
L
1
1
d  dx Lc ( ρ ( x , ), j( x , ))  Lnc ( ρ ( x , ), k ( x , )) 

0
0
Pr[ j( x, ), k ( x, )] ~ e
L
  ( x, )   x j ( x, ) 
Lc
- conserving action1
L nc -
nonconserving action2,3

3
k ( x,  )
j
k
  A, B, C
- conserving current
- nonconserving current
1. T Bodineau, B Derrida - Comptes Rendus Physique 8 540 (2007)
2. G Jona-Lasinio, C Landim and M E Vares - Probability theory and related fields 97 339 (1993)
3. T Bodineau, M Lagouge - J. Stat. Phys. 139 201 (2010)
Corrections to G(r) using MFT
pe-3βμ
ABC
p
p
000
T


2
L
1
1
d  dxLc ( ρ ( x , ), j( x , ))  Lnc ( ρ( x , ), k ( x, )) 

0
0
Pr[ j( x, ), k ( x, )] ~ e
L
r
rμ
Instanton path:
τ
0
T
 ( x, )  * ( x, r ( ))  O( )
  A, B, C
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.
( µs(r) is dynamics dependent ! )
4. Corrections to rate function of r using MFT