Phase diagram and density large deviation of a nonconserving A B C model
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Phase diagram and density large
deviation of a nonconserving
ABC model
Or Cohen and David Mukamel
International Workshop on Applied Probability, Jerusalem, 2012
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T2
Driven diffusive systems
T1
Boundary driven Bulk driven
Studied via simplified
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Motivation
What is the effect of bulk nonconserving dynamics on bulk driven system ?
pq
w- w+
Can it be inferred from the conserving steady state properties ?
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Outline
1. ABC model
2. Phase diagram under conserving dynamics
3. Slow nonconserving dynamic
4. Phase diagram and inequivalence of ensembles
5. Conclusions
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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
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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
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ABC model
Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998
t
x
A B C
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Equal densitiesFor equal densities NA=NB=NC
})({})({ iXHi qXP
L
i
L
kkiikiikiii LACCBBA
LkXH
1
1
1
2~})({
AAAAABBAABBBCBBCCCCCC
BBBBB BB
Potential induced by other species
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Weak asymmetry
Clincy, Derrida & Evans - Phys. Rev. E 2003
})({})({ iXHi qXP { } { , , , , , , , }iX A C A B B C B
( ) , , [0,1)x A B C x Coarse graining
2[ ] [ ] log( [ ])H L S L
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Weak asymmetry
)exp(L
q
[ ]/[ ] L FS H LP e e
Clincy, Derrida & Evans - Phys. Rev. E 2003
Weakly asymmetric thermodynamic limit
})({})({ iXHi qXP { } { , , , , , , , }iX A C A B B C B
( ) , , [0,1)x A B C x
2[ ] [ ] log( [ ])H L S L
Coarse graining
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Phase transition
Clincy, Derrida & Evans - Phys. Rev. E 2003
1 1 1
10 0 0
[ ] ( ) ( ) ( ) log ( )F dx zdz x x z dx x x
For low β
is minimum of F[ρα]
( ) /x r N L
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Phase transition
2nd order phase transition at
Clincy, Derrida & Evans - Phys. Rev. E 2003
9.1032 c
For low β
1 1 1
10 0 0
[ ] ( ) ( ) ( ) log ( )F dx zdz x x z dx x x
( ) ( )x r x ( ) /x r N L
is minimum of F[ρα]
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Nonequal densities ?AAAAABBAABBBCBBCCC• No detailed balance
(Kolmogorov criterion violated)• Steady state current• Stationary measure unknown
1 2min( , )N NJ q
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Nonequal densities ?
Hydrodynamics equations :
2
1A AA B C
d dddt L dx dx
Drift Diffusion
11 iiii BABA ( )A ii AL
AAAAABBAABBBCBBCCC• No detailed balance (Kolmogorov criterion violated)• Steady state current• Stationary measure unknown
1 2min( , )N NJ q
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Nonequal densities ?
Hydrodynamics equations :
2
1A AA B C
d dddt L dx dx
Drift Diffusion
11 iiii BABA ( )A ii AL
2132 3 / 1 ( )c r
Full steady-state solution orExpansion around homogenous
AAAAABBAABBBCBBCCC• No detailed balance (Kolmogorov criterion violated)• Steady state current• Stationary measure unknown
1 2min( , )N NJ q
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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)+
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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)
+
++
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Nonequal densitiesHydrodynamics equations :
CBAA
CBAA ep
dxd
dxd
Ldtd 33
02
1
Drift Diffusion Deposition Evaporation
11 iiii BABA ( )A ii AL
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Nonequal densities
ABC 000pe-3βμ
p0X X0
1
1
AB BAe-β/L
1
BC CBe-β/L
1
CA ACe-β/L
1
X= A,B,C
Hydrodynamics equations :
CBAA
CBAA ep
dxd
dxd
Ldtd 33
02
1
Drift Diffusion Deposition Evaporation
11 iiii BABA ( )A ii AL
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Conserving steady-state
Conserving model 0pSteady-state profile
LNNNr CBA
dxcsnba
dxcsnrrx,
,1),(*
CBAA
CBAA ep
dxd
dxd
Ldtd 33
02
1
Drift Diffusion
Nonequal densities : Cohen & Mukamel - Phys. Rev. Lett. 2012 Equal densities : Ayyer et al. - J. Stat. Phys. 2009
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Nonconserving steady-state
CBAA
CBAA ep
dxd
dxd
Ldtd 33
02
1
Drift Diffusion Deposition Evaporation
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Nonconserving steady-state
2,~ LpNonconserving model with slow nonconserving dynamics
CBAA
CBAA ep
dxd
dxd
Ldtd 33
02
1
Drift + Diffusion Deposition + Evaporation
[ ] [ ; ] ( )nc c ncP P r P r
*argmax( [ ; ]) ( , )cP r x r ( ) ?ncP r
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Dynamics of particle density
1rr
21 ~ L L~2 12
)(xA )(xB )(xC
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Dynamics of particle density
1rr
21 ~ L L~2 12
After time τ1 :
),( 1* rx
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Dynamics of particle density
2rr
21 ~ L L~2 12
After time τ2 :
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Dynamics of particle density
2rr
21 ~ L L~2 12
After time τ1 :
),( 2* rx
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Dynamics of particle density
3rr
21 ~ L L~2 12
After time τ2 :
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Dynamics of particle density
3rr
21 ~ L L~2 12
After time τ1 :
),( 3* rx
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Large deviation function of r
3rr
21 ~ L L~2 12
After time τ1 :
),( 3* rx )3(,)3( 3434 L
rrRL
rrR
),( 3* rx
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Large deviation function of r
= 1D - Random walk in a potential
)(rV
r maxrminr r
)(rR
)(rR
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Large deviation function of r
= 1D - Random walk in a potentialr maxrminr r
r
rCBAdx
dxedrrF
0
1
0
***
1
0
3*0
3 )(log')(
ABC 000pe-3βμ
p
Large deviation function
)(rV )(rR
)(rR
0
( / )( ) exp[ ( )]( / )
rL
n r L
R n LP r LF rR n L
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Large deviation function of r
r maxrminr r
High µ
( )F r
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Large deviation function of r
r maxrminr r
r maxrminr r
High µ
Low µ
( )F r
( )F r
First order phase transition (only in the nonconserving model)
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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 :
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Conclusions
1. ABC model
2. Slow nonconserving dynamics
3. Inequivalence of ensemble, and links to long range
interacting systems.
4. Relevance to other driven diffusive systems.
Thank you ! Any questions ?