Periodic One-Dimensional Hopping Model with one Mobile Directional Impurity

TL;DR

This paper provides an analytic steady-state solution for a one-dimensional periodic lattice model with a mobile, directional impurity, revealing how the impurity affects transport properties and connecting to models in driven diffusive systems.

Contribution

It introduces a novel analytical approach to solve for the steady state of a 1D hopping model with a mobile impurity, including new entities called quasi-walker.

Findings

Derived velocities and diffusion constants for walker and impurity

Established relations between quasi-walker and system transport properties

Connected the model to gel electrophoresis and reptation models

Abstract

Analytic solution is given in the steady state limit for the system of Master equations describing a random walk on one-dimensional periodic lattices with arbitrary hopping rates containing one mobile, directional impurity (defect bond). 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. 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. Applications in driven diffusive systems are shown, and connections with the Duke-Rubinstein reptation models for gel electrophoresis are discussed.