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 ?