Driven ABC model under particle-nonconserving dynamics Or Cohen and - - PowerPoint PPT Presentation

Driven ABC model under particle-nonconserving dynamics

Or Cohen and David Mukamel

International Seminar on Large Fluctuations in Non-Equilibrium Systems, Dresden, July 2011

T2

Motivation

• Much remains unknown
• Exhibit long-range correlations
• Exhibit similar phenomena ?

T1

Equilibrium systems with Long-range interactions System driven out of equilibrium

• Much is known
• Exhibit long-range correlations
• Exhibit unique phenomena :

inequivalence of ensembles, negative specific heat in MC ensemble, slow relaxation, quasi-stationary states r GMm r v  ) (

Outline

• 1. Long-range interactions
• 2. Inequivalence of ensembles

Driven Models Equilibrium Long-range

• 3. ABC model
• 4. Inequivalence of ensembles
• 5. Conclusions

Long-range interactions

r

 



d

r r v 1 ) (

Short range Long range σ>0

Canonical

CV≥0 CV≥0

Energy scaling Additive

YES NO

Micro- canonical

S

CV≥0 CV<0

S

CV<0

1

E

2

E

E

1

E

2

E

E

d

V E

/ 1  



V E 

Canonical Microcanonical

1st order transition 2nd order transition

T K

• rdered

disordered

CV<0

Inequivalence of ensembles

T K

• rdered

disordered

T K

• rdered

disordered inequivalence

K

= interaction strength

Outline

• 1. Long-range interactions
• 2. Inequivalence of ensembles

Driven Models Equilibrium Long-range

• 3. ABC model
• 4. 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 q=1 q<1

  L

Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998

ABBCACCBACABACB AAAAABBBBBCCCCC

Currents & Detailed Balance

• 1. Equal densities NA=NB=NC
• 2. Nonequal densities, e.g. NB≠NC

Although q ≠ 1 detailed balance obeyed with respect with to

}) ({

}) ({

i

X H i

q X P 

 



     

  

L i L k k i i k i i k i i i

L A C C B B A L k X H

1 1 1 2

~ }) ({

No effective Hamiltonian

Weak asymmetry

) exp( L q   

L S L E ~ ~

2 / ( )

( ) ( )

E S E L L f

P E E q e e

    

   

Clincy, Derrida & Evans - Phys. Rev. E 2003

Weak asymmetry

) exp( L q   

L S L E ~ ~

2

2nd order phase transiton at the critical temp. Clincy, Derrida & Evans - Phys. Rev. E 2003

9 . 10 3 2    c

/ ( )

( ) ( )

E S E L L f

P E E q e e

    

   

Outline

• 1. Long-range interactions
• 2. Inequivalence of ensembles

Driven Models Equilibrium Long-range

• 3. ABC model
• 4. Inequivalence of ensembles
• 5. Conclusions

ABC model + vacancies 0X X0 X=A,B,C

1 1

A B C

‘Canonical’ ensemble :

Lederhendler & Mukamel - Phys. Rev. Lett. 2010

Nonconserving dynamics ABC 000

pe-3βμ p

A B C

N N N r L   

Fluctuating parameter : Conjugate field :



A B C

0X X0 X=A,B,C

1 1

‘Grand Canonical’ ensemble :

Lederhendler & Mukamel - Phys. Rev. Lett. 2010

2nd order transition

• rdered

1st order transition tricritical point disordered

• rdered

disordered T= T=

Inequivalence of ensembles

Conserving = Canonical Nonconserving = Grand canonical For NA=NB=NC :

Lederhendler, Cohen & Mukamel - J. Stat. Mech: Theory Exp. 2010

Nonequal densities

Hydrodynamics equations :

 

 

C B A A C B A A

e p dx d dx d L dt d         

 3 3 2

1



          

Drift Diffusion Deposition Evaporation

1 1  



i i i i

B A B A

( )

A i

i A L  

Nonequal densities

ABC 000

pe-3βμ p

0X X0

1 1

AB BA

e-β/L 1

BC CB

e-β/L 1

CA AC

e-β/L 1

X= A,B,C

Hydrodynamics equations :

 

 

C B A A C B A A

e p dx d dx d L dt d         

 3 3 2

1



          

Drift Diffusion Deposition Evaporation

1 1  



i i i i

B A B A

( )

A i

i A L  

Conserving steady-state

Conserving model

 p

Steady-state profile

L N N N r

C B A

  

   

        

   d x c sn b a d x c sn r r x , , 1 ) , (

*

    

 

 

C B A A C B A A

e p dx d dx d L dt d         

 3 3 2

1



          

Drift Diffusion

Nonequal densities : Cohen & Mukamel - Preprint Equal densities : Ayyer et al. - J. Stat. Phys. 2009

Nonconserving steady-state

 

 

C B A A C B A A

e p dx d dx d L dt d         

 3 3 2

1



          

Drift Diffusion Deposition Evaporation

Nonconserving steady-state

2 , ~ 







L p

? 



r

Steady-state profile Steady-state density

) , (

*  

 r x

Nonconserving model

with slow nonconserving dynamics

 

 

C B A A C B A A

e p dx d dx d L dt d         

 3 3 2

1



          

Drift + Diffusion Deposition + Evaporation

Dynamics of particle density

1

r r 

2 1 ~ L





 L ~

2 1 2

  

) (x

A

 ) (x

B

 ) (x

C



Dynamics of particle density

1

r r 

2 1 ~ L





 L ~

2 1 2

  

After time τ1 :

) , (

1 *

r x





Dynamics of particle density

2

r r 

2 1 ~ L





 L ~

2 1 2

  

After time τ2 :

Dynamics of particle density

2

r r 

2 1 ~ L





 L ~

2 1 2

  

After time τ1 :

) , (

2 *

r x





Dynamics of particle density

3

r r 

2 1 ~ L





 L ~

2 1 2

  

After time τ2 :

Dynamics of particle density

3

r r 

2 1 ~ L





 L ~

2 1 2

  

After time τ1 :

) , (

3 *

r x





Large deviation function of r

3

r r 

2 1 ~ L





 L ~

2 1 2

  

After time τ1 :

) , (

3 *

r x





) 3 ( , ) 3 (

3 4 3 4

L r r R L r r R    

) , (

3 *

r x





Large deviation function of r

= 1D - Random walk in a potential

) (r V



r

max

r

min

r

r

) (r R ) (r R

Large deviation function of r

= 1D - Random walk in a potential



r

max

r

min

r

r )] ( exp[ ) ' ( ) ' ( ) (

'

r LF r R r R r P

r r r 

  

  

  

            

 r r C B A

dx dx e dr r F

1 * * * 1 3 * 3

) ( log ' ) (    

 

ABC 000

pe-3βμ p

Large deviation function

) (r V

) (r R ) (r R

Inequivalence of ensembles

Conserving = Canonical Nonconserving = Grand canonical

2nd order transition

• rdered

1st order transition tricritical point disordered

• rdered

disordered

       2 3 , 3 r r r r r

C B A

01 .  

For NA=NB≠NC :

Locating 1st order transition

Conserving Nonconserving

μ

r

Maxwell’s construction

μ r

1

r

2

r

) ( ) (

2 1

r F r F

 



Critical point Ordered phase Homogenous phase

1st order trans. 2nd order trans.

         

 

1 3 * * * *

) ) ( ( log 3 1 ) ' ( )) ' ( ( ' ) (         

 C B A r r

dx r r dr r F

Large deviation function ‘Chemical potential’ in conserving system

Fast evaporation & deposition

Conserving Nonconserving

r x  1 ) (  const x  ) (  ) (  x d d



 

) ( ) , (    

 

v x x  

2 2

36 3 2    r

c

 

2 1 2 2 1 2

) 2 1 1 ( 36 ) 1 ( 3 2 k k r r

c

       

NESS is sensitive to the dynamics

2 ~ 







L p

Flat vacancies profile No moving solutions Oscillatory vacancies profile Moving solutions

Results & Conclusions

• 1. Inequivalence of ensembles in the ABC model

Open questions : Other similarities to system with LRI ? (dynamical features etc.) In other driven models ?

• 2. Dynamical definition of ensembles in driven models ?

Conserving ABC model + slow nonconserving dynamics Obtain LDF of particle density Applies to other driven models