From Quenched Disorder to Continuous Time Random Walk

TL;DR

This paper presents an explicit mapping of the quenched trap model to continuous time random walk, providing analytical expressions for key transport properties in disordered systems, especially in the sub-diffusive regime.

Contribution

It introduces a linear temporal transformation that maps the quenched trap model to CTRW, with an exact form for the transformation constant and derived disorder-averaged probability densities.

Findings

Derived the asymptotic mapping for the quenched trap model to CTRW.

Obtained explicit expressions for diffusion coefficient and drift.

Provided analytic forms for the disorder-averaged position probability density.

Abstract

This work focuses on quantitative representation of transport in systems with quenched disorder. Explicit mapping of the quenched trap model to continuous time random walk is presented. Linear temporal transformation: $t\to t/\Lambda^{1/\alpha}$ for transient process on translationally invariant lattice, in the sub-diffusive regime, is sufficient for asymptotic mapping. Exact form of the constant $\Lambda^{1/\alpha}$ is established. Disorder averaged position probability density function for quenched trap model is obtained and analytic expressions for the diffusion coefficient and drift are provided.