Analytical Study of Anomalous Diffusion by Randomly Modulated Dipole

TL;DR

This paper develops a Fokker-Planck equation for particles influenced by a randomly modulated dipole, explaining anomalous and anisotropic diffusion phenomena observed in turbulent systems, and introduces a mechanism to recover probability at singularities.

Contribution

It derives a new FP equation for potential-driven particles affected by a random dipole, linking singularities to fractional diffusion and anisotropic behavior.

Findings

Derived the FP equation for the system

Showed the model explains anomalous diffusion

Matched the fractal dimension with previous results

Abstract

This paper derives the Fokker-Planck (FP) equation for a particle moving in potential by a randomly modulated dipole. The FP equation describes the anomalous diffusion observed in the companion paper [1] and breaks the conservation of the total probability at the singularity by the dipole. It also shows anisotropic diffusion, which is typical in fluid turbulence. We need to modify the probability density by introducing a mechanism to recover the particle. After the modification, the latent fractal dimension matches with the results of the companion paper. We hope that our model gives a new example of fractional diffusion, where the random singularity triggers fractionality.