Точные решения в неравновесной статистической механике В.Б. Приезжев ЛТФ ОИЯИ

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) Substitution “ Fourier ansatz ” gives

We put

From the initial conditions 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

As in the one-particle case, we have Bethe Ansatz

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

Integrating, we obtain From initial conditions

ASEP as a combinatorial problem

Free fermions TASEP Discrete formulation

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

Cancellation for the TASEP (step 1) Reference coordinates for A,B,C,D

Shift operators

Cancellation for the TASEP (step 2)

Solution for two particles

General solution for infinite lattice