Fokker Planck Equation on Fractal Curves

TL;DR

This paper derives a Fokker-Planck equation on fractal curves using fractal calculus, revealing subdiffusive behavior and enabling estimation of fractal dimension from the distribution function.

Contribution

It introduces a novel Fokker-Planck framework on fractal curves utilizing fractal calculus, extending classical diffusion models to fractal geometries.

Findings

The equation exhibits subdiffusive behavior.

Fractal dimension can be estimated from the distribution.

Diffusion deviates from classical behavior on fractal paths.

Abstract

A Fokker Planck equation on fractal curves is obtained, starting from Chapmann-Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order $\alpha$, $\alpha$ being the dimension of the curve. The solution of this equation with localized initial condition shows deviation from ordinary diffusion behaviour due to underlying fractal space in which diffusion is taking place. An exact solution of this equation manifests a subdiffusive behaviour. The dimension of the fractal path can be estimated from the distrubution function.