Density large-deviations of nonconserving driven models
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Density large-deviations of
nonconserving driven models
STATPHYS 25 Conference, SNU, Seoul, Korea, July 2013
Or Cohen and David Mukamel
T , µ
Equilibrium[ ]H
Grand canonical ensemble out of equilibrium ?
T , µ
Equilibrium[ ]H
*.
[ ] exp [ ]
exp log [ ] ( , )cons
P H Vr
P F r Vr
Grand canonical ensemble out of equilibrium ?
( )dr d x x
conserving steady state
Helmholtzfree energy
T , µ
Equilibrium[ ]H
Grand canonical ensemble out of equilibrium ?
( )dr d x x
conserving steady state
Helmholtzfree energy
Driven system
pqconserving
steady state*
.[ ]consP
*
.
[ ] exp [ ]
exp log [ ] ( , )cons
P H Vr
P F r Vr
T , µ
Equilibrium[ ]H
Grand canonical ensemble out of equilibrium ?
( )dr d x x
conserving steady state
Helmholtzfree energy
Driven system
pq
e-βμ 1
conserving steady state
*.[ ]consP
*
.
[ ] exp [ ]
exp log [ ] ( , )cons
P H Vr
P F r Vr
T , µ
Equilibrium[ ]H
Grand canonical ensemble out of equilibrium ?
( )dr d x x
conserving steady state
Helmholtzfree energy
Driven system
pq
e-βμ 1
*.
0
[ ] exp log [ ] ' ( ')r
cons sP P V dr r Vr
conserving
steady state*
.[ ]consP
*.[ [ ]]s consP
Dynamics-dependentchemical potential
of conserving system
*
.
[ ] exp [ ]
exp log [ ] ( , )cons
P H Vr
P F r Vr
General particle-nonconserving driven modelwR
CwLC
w-NC w+
NC
L sites
conserving(sum over η’ with same N)
nonconserving(sum over η’ with N’≠N)
wRCwL
C
w-NC w+
NC
L sites
'
' ' ' '' ' '
( ) ( ') ( ) ( ') ( )N N
C C NC NCt
N N
P w P w P w P w P
General particle-nonconserving driven model
wRCwL
C
w-NC w+
NC
*.
higher order( ) ( ; ) ( )
in consP P N f N OL
Guess a steadystate of the form :
L sites
'
' ' ' '' ' '
( ) ( ') ( ) ( ') ( )N N
C C 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)
General particle-nonconserving driven model
1' .NC
consw It is consistent if :
wRCwL
C
w-NC w+
NC
For diffusive systems
L sites
'
' ' ' '' ' '
( ) ( ') ( ) ( ') ( )N N
C C NC NCt
N N
P w P w P w P w P
2. ~cons L
*.
higher order( ) ( ; ) ( )
in consP P N f N OL
Guess a steadystate of the form :
conserving(sum over η’ with same N)
nonconserving(sum over η’ with N’≠N)
General particle-nonconserving driven model
Slow nonconserving dynamics
' '
* *' . ' .
' ' '
0 ( ')[ ( '; ')] ( )[ ( '; )]N N N N
NC NCcons cons
N N
f N w P N f N w P N
', , ''
0 ( ') ( )N N N NN N
f N W f N W
To leading order in L we obtain
Slow nonconserving dynamics
' '
* *' . ' .
' ' '
0 ( ')[ ( '; ')] ( )[ ( '; )]N N N N
NC NCcons cons
N N
f N w P N f N w P N
', , ''
0 ( ') ( )N N N NN N
f N W f N W
= 1D - Random walk in a potential
, 1N NW , 1N NW
maxNminN *N
( )V N
Slow nonconserving dynamics
0
', ' 1
' ' 1, '
( ) exp log( )N
N N
N N N N
Wf N
W
, 1N NW
maxNminN *N
( ) ( / )V N LG N L
' '
* *' . ' .
' ' '
0 ( ')[ ( '; ')] ( )[ ( '; )]N N N N
NC NCcons cons
N N
f N w P N f N w P N
', , ''
0 ( ') ( )N N N NN N
f N W f N W
1,N NW
NrL
= 1D - Random walk in a potential
Slow nonconserving dynamics
0
0
' ( )', ' 1 ( )
' ' 1, '
( ) exp log( ) ~
r
sL dr r L rNN N LG r
N N N N
Wf N e e
W
, 1N NW
maxNminN *N
( ) ( / )V N LG N L
' '
* *' . ' .
' ' '
0 ( ')[ ( '; ')] ( )[ ( '; )]N N N N
NC NCcons cons
N N
f N w P N f N w P N
', , ''
0 ( ') ( )N N N NN N
f N W f N W
1,N NW
NrL
w-NC w+
NC
NC
NC
w ew
Outline
1. Limit of slow nonconserving
2. Example of the ABC model
3. Corrections to the rate function using MFT
4. Conclusions
2NCw L
( )G r
*.
0
[ ] [ ]exp ' ( ')r
cons sP P L dr r Lr
ABC model
A B C
AB BA
BC CB
CA AC
Dynamics : q
1
q
1
q
1
Ring of size L
M R Evans, Y Kafri , H M Koduvely, D Mukamel - Phys. Rev. Lett. 80 425 (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
M R Evans, Y Kafri , H M Koduvely, D Mukamel - Phys. Rev. Lett. 80 425 (1998 )
ABBCACCBACABACBAAAAABBBBBCCCCC
ABC model
time
site index
A B C
1 480 A B Cq L N N N
Conserving ABC model
0X X0 X=A,B,C1
1
A B
C 0
A Lederhendler, D Mukamel - Phys. Rev. Lett. 105 105602 (2010) A Lederhendler, OC, D Mukamel - J. Stat. Mech. 11 11016 (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 ABC model
1. M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003)
)exp(L
q Weakly asymmetric
thermodynamic limit1
Density profile
( )A ii AL
Conserving ABC model
1. M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003)2. OC, D Mukamel - J. Phys. A 44 415004 (2011)
)exp(L
q Weakly asymmetric
thermodynamic limit1
Density profile
known2
2nd order
For low β’s
( )A ii AL
* ( , )x r
c
* ( , ) /x r r N L
Conserving ABC model
1. M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003)2. OC, D Mukamel - J. Phys. A 44 415004 (2011)3. T Bodineau, B Derrida - Comptes Rendus Physique 8 540 (2007)
)exp(L
q Weakly asymmetric
thermodynamic limit1
Density profile
known2
2nd order
For low β’s
[ , , ; ][ , , ] A B CLF rA B CP e
( )A ii AL
* ( , )x r
c
* ( , ) /x r r N L
Stationarymeasure3:
Nonconserving ABC model
0X X0 X=A,B,C1
1
A B
C 0
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)
+
++
A Lederhendler, D Mukamel - Phys. Rev. Lett. 105 105602 (2010) A Lederhendler, OC, D Mukamel - J. Stat. Mech. 11 11016 (2010)
Slow nonconserving ABC model
Slow nonconserving limit 2,~ Lp
, 3N NW , 3N NW
maxNminN *N
ABC 000pe-3βμ
p
( )V N
Slow nonconserving ABC model
Slow nonconserving limit
ABC 000pe-3βμ
p
saddle point approx.
3
1* * 3
, 3 ' . 0' 0
( '; ) ( ( , ))N N
NCN N cons
NW w P N dx xL
~ , 2p L
, 3N NW
maxNminN *N
( )V N , 3N NW
[ , , ; ]A B CLF re
Slow nonconserving ABC model
ABC 000pe-3βμ
p
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 NrL
( )( ) exp[ ( ' ( ') )]r
LG rSP r L dr r r e
Slow nonconserving ABC model
ABC 000pe-3βμ
p
This is similar to equilibrium :
( )( ) exp[ ( ' ( ') )]r
LG rSP r L dr r r e
( ) exp [ ( , ) ]P r L f r r
maxNminN *N
, 3N NW , 3N NW ( )V N
f = Helmholtz free energy density
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 NrL
Rate function of r, G(r)
High µ
Low µ
First order phase transition (only in the nonconserving model)
( )G r
( )G r
OC, D Mukamel - Phys. Rev. Let. 108, 060602 (2012)
0.05
0.052
23
,3
rrrrr CBA
0.025 40
Inequivalence of ensembles
Conserving (Canonical) Nonconserving (Grand canonical)
2nd order transition
ordered
1st order transition tricritical point
disordered
ordered
disordered
23
,3
rrrrr CBA 01.0For NA=NB≠NC :
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)
Stability line
Corrections to G(r) using MFT
2 1pL ABC 000
pe-3βμ
p
1
0 0( ( , ), ( , )) ( ( , ), ( , ))
[ ( , ), ( , )] ~T
c ncL d dx x x x k xPr x k x e
ρ j ρj
L L
cL
( , ) ( , ) ( ,3
,) ,x j B Cx x Ax k
- conserving action1
ncL - nonconserving action2,3
j - conserving current
k - 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
2 1pL ABC 000
pe-3βμ
p
τ
r
rμ0 T
*( , ) ( , ( )) ( ) , ,x x r O A B C
1
0 0( ( , ), ( , )) ( ( , ), ( , ))
[ ( , ), ( , )] ~T
c ncL d dx x x x k xPr x k x e
ρ j ρj
L L
Instanton path:
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
Thank you for listening ! Any questions ?
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