Brownian motion of fractal particles: Levy flights from white noise

TL;DR

This paper extends the Langevin equation to include fractional derivatives, modeling Levy flights in fractal particle Brownian motion, with implications for analyzing complex data series.

Contribution

It introduces a local fractional Langevin equation to describe Levy flights in fractal particles, generalizing classical Brownian motion models.

Findings

Levy processes arise from the fractional Langevin equation.

Fractal particles exhibit Levy flight behavior.

Implications for complex data analysis are discussed.

Abstract

We generalise the Langevin equation with Gaussian white noise by replacing the velocity term by a local fractional derivative. The solution of this equation is a Levy process. We further consider the Brownian motion of a fractal particle, for example, a colloidal aggregate or a biological molecule and argue that it leads to a Levy flight. This effect can also be described using the local fractional Langevin equation. The implications of this development to other complex data series are discussed.