The velocity of 1D Mott variable range hopping with external field

TL;DR

This paper studies how an external bias affects the movement speed of a one-dimensional Mott variable range hopping model, providing conditions for when the walk is ballistic or sub-ballistic, and analyzing the environment's invariant distribution.

Contribution

It introduces a detailed analysis of the biased 1D Mott random walk, establishing criteria for ballisticity and sub-ballisticity, and explores the environment's invariant distribution in the ballistic regime.

Findings

Bias induces transience in the walk.

Conditions for positive and zero linear speed are established.

Invariant environment distribution exists in the ballistic regime.

Abstract

Mott variable range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps. We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias.