Entropy as a measure of diffusion

TL;DR

This paper proposes using the time variation of entropy as a novel measure of diffusion rate, providing theoretical analysis and comparisons with variance for different systems, including fractional-derivative diffusions.

Contribution

It introduces entropy as an alternative to variance for measuring diffusion, analyzing its behavior in various systems, especially those without stationary densities.

Findings

Entropy tends exponentially to a constant in systems with stationary densities.

In systems without stationary densities, entropy grows logarithmically with time.

Comparison shows entropy and variance behaviors differ across fractional-derivative diffusions.

Abstract

The time variation of entropy, as an alternative to the variance, is proposed as a measure of the diffusion rate. It is shown that for linear and time-translationally invariant systems having a large-time limit for the density, at large times the entropy tends exponentially to a constant. For systems with no stationary density, at large times the entropy is logarithmic with a coefficient specifying the speed of the diffusion. As an example, the large time behaviors of the entropy and the variance are compared for various types of fractional-derivative diffusions.