Analytic Results for Hopping Models with Excluded Volume Constraint

Abstract

The Theory of Brownian Vacancy Driven Walk

(ABSTRACT)

We analyze the lattice walk performed by a tagged member of

an infinite 'sea' of particles filling a d-dimensional

lattice, in the presence of a single vacancy. The vacancy is

allowed to be occupied with probability 1/2d by any of its

2d nearest neighbors, so that it executes a Brownian walk.

Particle-particle exchange is forbidden; the only

interaction between them being hard core exclusion. Thus,

the tagged particle, differing from the others only by its

tag, moves only when it exchanges places with the hole.

In this sense, it is a random walk "driven" by the

Brownian vacancy. The probability distributions for its

displacement and for the number of steps taken, after

n-steps of the vacancy, are derived. Neither is a Gaussian!

We also show that the only nontrivial dimension where the

walk is recurrent is d=2. As an application, we compute the

expected energy shift caused by a Brownian vacancy in a

model for an extreme anisotropic binary alloy. In the last

chapter we present a Monte-Carlo study and a mean-field

analysis for interface erosion caused by mobile vacancies.

Part II.

One-Dimensional Periodic Hopping Models with

Broken Translational Invariance.Case of a Mobile

Directional Impurity

(ABSTRACT)

We study a random walk on a one-dimensional periodic lattice

with arbitrary hopping rates. Further, the lattice contains

a single mobile, directional impurity (defect bond), across

which the rate is fixed at another arbitrary value. Due to

the defect, translational invariance is broken, even if all

other rates are identical. The structure of Master equations

lead naturally to the introduction of a new entity,

associated with the walker-impurity pair which we call the

quasi-walker. Analytic solution for the distributions in

the steady state limit is obtained. The velocities and

diffusion constants for both the random walker and impurity

are given, being simply related to that of the

quasi-particle through physically meaningful equations.

As an application, we extend the Duke-Rubinstein reputation

model of gel electrophoresis to include polymers with

impurities and give the exact distribution of the steady

state.