The delayed uncoupled continuous-time random walks do not provide a model for the telegraph equation

TL;DR

This paper demonstrates that delayed continuous-time random walks (DCTRWs) do not accurately model the telegraph equation at the microscopic level, as they do not capture finite propagation speed and are better approximated by diffusion equations.

Contribution

The paper clarifies that DCTRWs are not suitable microscopic models for the telegraph equation and shows diffusion equations provide better approximations to DCTRWs.

Findings

Diffusion equations approximate DCTRWs better than the telegraph equation in $L_2$ norm.

DCTRWs have infinite particle velocity, unlike the finite speed in the telegraph equation.

DCTRWs do not offer a correct microscopic interpretation of the telegraph equation.

Abstract

It has been alleged in several papers that the so called delayed continuous-time random walks (DCTRWs) provide a model for the one-dimensional telegraph equation at microscopic level. This conclusion, being widespread now, is strange, since the telegraph equation describes phenomena with finite propagation speed, while the velocity of the motion of particles in the DCTRWs is infinite. In this paper we investigate how accurate are the approximations to the DCTRWs provided by the telegraph equation. We show that the diffusion equation, being the correct limit of the DCTRWs, gives better approximations in $L_2$ norm to the DCTRWs than the telegraph equation. We conclude therefore that, first, the DCTRWs do not provide any correct microscopic interpretation of the one-dimensional telegraph equation, and second, the kinetic (exact) model of the telegraph equation is different from the model based on the DCTRWs.