phase diagram and density large deviation of a nonconserving a b c model
DESCRIPTION
Phase diagram and density large deviation of a nonconserving A B C model Or Cohen and David Mukamel. International Workshop on Applied Probability, Jerusalem, 2012. Driven diffusive systems. Bulk driven . Boundary driven . T 2. T 1. Studied via simplified. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
Phase diagram and density large
deviation of a nonconserving
ABC model
Or Cohen and David Mukamel
International Workshop on Applied Probability, Jerusalem, 2012
T2
Driven diffusive systems
T1
Boundary driven Bulk driven
Studied via simplified
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 ?
Outline
1. ABC model
2. Phase diagram under conserving dynamics
3. Slow nonconserving dynamic
4. Phase diagram and inequivalence of ensembles
5. Conclusions
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
Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998
t
x
A B C
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
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
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
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
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]
Nonequal densities ?AAAAABBAABBBCBBCCC• No detailed balance
(Kolmogorov criterion violated)• Steady state current• Stationary measure unknown
1 2min( , )N NJ q
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
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
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)+
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)
+
++
Nonequal densitiesHydrodynamics equations :
CBAA
CBAA ep
dxd
dxd
Ldtd 33
02
1
Drift Diffusion Deposition Evaporation
11 iiii BABA ( )A ii AL
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
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
Nonconserving steady-state
CBAA
CBAA ep
dxd
dxd
Ldtd 33
02
1
Drift Diffusion Deposition Evaporation
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
Dynamics of particle density
1rr
21 ~ L L~2 12
)(xA )(xB )(xC
Dynamics of particle density
1rr
21 ~ L L~2 12
After time τ1 :
),( 1* rx
Dynamics of particle density
2rr
21 ~ L L~2 12
After time τ2 :
Dynamics of particle density
2rr
21 ~ L L~2 12
After time τ1 :
),( 2* rx
Dynamics of particle density
3rr
21 ~ L L~2 12
After time τ2 :
Dynamics of particle density
3rr
21 ~ L L~2 12
After time τ1 :
),( 3* rx
Large deviation function of r
3rr
21 ~ L L~2 12
After time τ1 :
),( 3* rx )3(,)3( 3434 L
rrRL
rrR
),( 3* rx
Large deviation function of r
= 1D - Random walk in a potential
)(rV
r maxrminr r
)(rR
)(rR
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
Large deviation function of r
r maxrminr r
High µ
( )F r
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)
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 :
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 ?