Точные решения в неравновесной статистической...

Точные решения в неравновесной статистической механике

В.Б. ПриезжевЛТФ ОИЯИ

Totally Asymmetric Exclusion Process

Totally Asymmetric Exclusion Process

Applications to:

1. Hopping conductivity

2. Queuing problems

3. Directed polymers in random medium

4. Traffic problems

Master Equation

One-particle master equation (Poisson process)

).,(),1(),(

txPtxPt

txP

.1||,)( zzxP x

Substitution

).()1()( xPxPxP

“ Fourier ansatz ”

gives )()exp(),( xPttxP

We put

zzzz xxx 1

1;1

]2,0[, pez pi

)(),(2

0

pfedptxP xipt

From the initial conditions

tyx

yxiptipe

ypiyx

eyx

t

eedpP

epfxP

)!(

2

1t)(x,

Then,

)2()(havewe)0,(

)(2

0

)1(

1,

Poisson distribution

then (2) has the form (1).

Therefore, Eq.(1) + condition P(x,x)=P(x,x+1)

gives the Asymmetric Exclusion Process

Two-particle exclusion process

;t)(x,xP(x,x;t)-P

xxxx

txxPtxxPt

txxP

txxPtxxPtxxPt

txxP

1termstwoaddweif

)2(

havewe,1,If

)1(

);1,();1,1();1,(

21

);2,1(2);12,1();2,11();2,1(

As in the one-particle case, we have

),(2)1,(),1(),(

),();,(

21212121

2121

xxPxxPxxPxxP

xxPetxxP t

Bethe Ansatz

1221

2121211221 ),( xxxx zzAzzAxxP

2

1

21

1212

121

221

21

1

1S:matrix

;1

1;2

11

z

z

A

AS

Az

zA

zz

From condition P(x,x)=P(x,x+1), we have

)(

)();,(

2112221121

2,12

2

01

2

021

xipxpixipxpi

t

eSe

ppfedpdptxxP

Integrating, we obtain

2211221 )2(),( yipyipeppf

From initial conditions

),1(),(),(

);(),(;!

),(

2,1;2,1,|);(|det)0;,|;,(

001

0010

2121

tkFtkFtkF

tjkFtkFk

tetkF

jityxFyytxxP

j

kt

jiji

ASEP as a combinatorial problem

)0;,...,|;,...,( 11 pp yytxxP

Free fermions TASEP

Discrete formulation

of all free paths for time t. M.E. Fisher (1984):

"functionpartition");,( 00

000

xxtxx yzxx

ttxxF

|);(|det)0,|,(

);();();();(

,0

1,202,102,201,10

tyxFytxP

tyxFtyxFtyxFtyxF

ji

Cancellation for the TASEP (step 1)

Reference coordinates for A,B,C,D

)()|)(|1()()|1)(|(

)()|1)(|1()()|1)(|1(

2121

2121

DxxxxBxxxx

CxxxxAxxxx

Shift operators

1

2

1

1:matrix

a

aSS

Cancellation for the TASEP (step 2)

|);(|det)0,|,(

);();();();(

,

1,212,112,201,10

tyxFytxP

tyxFtyxFtyxFtyxF

jiji

Solution for two particles

),1(),(),(

);(),(;!

),(

2,1;2,1,|);(|det)0;,|;,(

001

0

010

2121

tkFtkFtkF

tjkFtkFk

tetkF

jityxFyytxxP

j

kt

jiji

General solution for infinite lattice

);,();,(

);,(1

1);,(

)!();,(

|);,(|det)0;,...,|;,...,(

00

00

0

11

tkyxFk

mtyxF

tkyxFm

mktyxF

yx

tetyxF

tyxFyytxxP

ji

m

kjim

jik

jim

ij

yxt

ji

jijipp

ij