‘Low temperature’ expansion of steady-state measures of nonequilibrium Markov chains

Or Cohen and David Mukamel, Weizmann Institute of Science, Rehovot, Israel

1. Motivation – driven lattice gases

2. Algorithm for computing εC

3. Tree representation3

4. Application: Random Energy Model6

5. Conclusions

…p q

α

γ

δ

β

ASEP1

(Asymmetric Simple Exclusion Process)ABC model2

A B C

AB q

1BA

BC q

1CB

CA q

1AC

1exp2

][W CC CCò‘Energy differences’ between configurations:

In the limit the steady-state is ( exp[) ]AP åCC

( , ) ( , ) ( , )tP t W P t W P t

CC CCC C

C C CMaster equation:

1. Draw maximal outgoing current

2. Follow the maximal current until loop is closed(Solving the leading order term in the Master equation)

3. Contract (renormalizing) the loop into a vertex(Looking at the next order in the Master equation)

4. Repeat 1-3 until one vertex remains, C0

5. Set P(C0)=1 and compute other P’s by backtracking

Exact for non-degenerate εCC’ , how to deal with degeneracy?

1

Loop Γ

3

2

5 4

A

C

B

Γ

AC

B

'min )( CCC

ò

1

1

max

min( )

m1 ax( )

i ii

i

j j

i

i i

i i

j

P P P

WP P

W

W P WP

C CCC

C CCC

CC

C C CC

Contract the sub-graph?

The coefficient A remains unknown

1 2

3

TC = All spanning trees rooted in C

{ , }

1( )t T t

P WZ

C

åCC

C CC

• In the limit we only need the

Minimal Directed Spanning Tree.

• Alternative algorithms derived earlier in Ref. [4,5]

In degenerate cases, A is the number of trees constructed from the graph

obtained in step 5.

2 2/2

1( ) E NJf E eNJ

2N micro-configurations

E

2 2ln ( ) ln 2 /n E N E NJ

0

2 2c

lnJ

c

Drive by adding infinite temp. bath to a fully-connected graph

( )1( ) EP eZ

CCwith energies drawn from

1 [ ( ) ( )]2

E EW e e

C CCC

ln /E P

ln ( )n E

0minE

2 2minmin

min

( )/2 2E EN E NJ E N

En d S AE Pe e

min ln 2NJE

c

min| |E

2nd order

Low-T bound

ln 0n

The same phase diagram obtained for other random graphs

• General algorithm for computing the steady-state in ‘low T limit’• Useful for studying driven systems numerically• Application to the REM coupled to an infinite temp. bath.• Higher order terms?• Application to other systems?

1. F. Spitzer, Adv. Math. 5 (1970) 246.2. M. R. Evans, Y. Kafri, H. M. Koduvely and D. Mukamel , Phys. Rev. Lett. 80 (1998) : 4253. R. K. P. Zia and B. Schmittmann, J. Stat. Mech. (2007) : P07012

4. J Edmonds, J. Res. Nat. Bur. Standards 71B (1967) 233-2405. Gabow, Harold N., et al. , Combinatorica 6.2 (1986): 109-1226. Derrida, Bernard, Phys. Rev. B, 24.5 (1981): 2613

Colour scale :1 1( ) ln ( ) lns P P nN N

C

C C

Phase diagram: Numerics:

Developed in collaboration with Bernard Derrida, ENS, Paris