Temporal Diffusion

TL;DR

This paper introduces the propagation-dispersion equation as a new model for describing the first passage distribution of particles with time delays, highlighting its distinction from traditional advection-diffusion models by reversing the roles of space and time.

Contribution

It derives the propagation-dispersion equation from microscopic first visit equations, providing a novel framework for temporal diffusion phenomena.

Findings

Derivation of the propagation-dispersion equation from microscopic principles

Identification of temporal diffusion as a generic behavior in certain systems

Contrast with advection-diffusion equation emphasizing reversed roles of space and time

Abstract

We consider the general problem of the first passage distribution of particles whose displacements are subject to time delays. We show that this problem gives rise to a \emph{propagation-dispersion equation} which is obtained as the continuous limit of the exact microscopic \emph{first visit equation}. The propagation-dispersion equation should be contrasted with the advection-diffusion equation as the roles of space and time are reversed, hence the name \emph{temporal diffusion}, which is a generic behavior encountered in an important class of systems.