Analytic physical model of anisotropic anomalous diffusion

TL;DR

This paper presents an analytical model for anisotropic anomalous diffusion using the temporal Fokker-Planck equation, providing insights into the growth of uncertainty in space over time.

Contribution

It introduces a physically reasonable model of anisotropic anomalous diffusion with drift based on a power-law ansatz for diffusion coefficients, with analytical solutions in multiple dimensions.

Findings

Analytical solutions for 2D and 3D diffusion cases.

Growth rate of uncertainty proportional to time raised to the half of the sum of exponents.

Interpretation of drift and diffusion coefficients in the model.

Abstract

The temporal Fokker-Planck equation is analytically integrated in an arbitrary number of spatial dimensions but with the 2D and 3D results highlighted. It is shown that a temporal power-law ansatz for the anisotropic diffusion coefficients leads naturally to a physically reasonable model of normal and anomalous diffusion with drift. An interpretation of the drift and diffusion coefficients is provided and the analytic growth rate of the area and volume of uncertainty is determined. It is shown that the asymptotic growth of the uncertainty in the volume of space about the mean position is proportional to the square-root of time raised to the power of the sum over all exponents of the anisotropic diffusion coefficients.