The Fokker-Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

TL;DR

This paper derives a fractional Fokker-Planck equation for passive tracer diffusion in cytoplasm, linking experimental anomalous diffusion data to a novel mathematical model involving the Kr"atzel function.

Contribution

It introduces a new fractional diffusion model based on superstatistical fractional Brownian motion, specifically tailored for cytoplasmic tracer diffusion.

Findings

Probability density function related to Kr"atzel function

Derived Fokker-Planck equation as a fractional diffusion equation in Erdélyi-Kober sense

Discussion on the model's uniqueness compared to existing literature

Abstract

By collecting from literature data the experimental evidences of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live E. coli cells, we get the probability density function of molecules' displacement and we derive the corresponding Fokker-Planck equation. Molecules' distribution emerges to be related to the Kr\"atzel function and its Fokker-Planck equation be a fractional diffusion equation in the Erd\'elyi-Kober sense. The irreducibility of the derived Fokker-Planck equation to those of other literature models is also discussed.