Joint probability density with radial, tangential, and perturbative forces

TL;DR

This paper derives the joint probability density for an active particle influenced by radial, tangential, and perturbative forces, revealing super-diffusive and diffusive behaviors in different velocity components over long times.

Contribution

It provides an analytical solution to the Fokker-Planck equation for active particles with combined forces, extending understanding of their velocity distributions.

Findings

Radial velocity exhibits super-diffusive behavior (~t^2)

Tangential velocity follows Gaussian diffusion (~t)

Analytical solution for joint probability density

Abstract

We study the Fokker-Planck equation for an active particle with both the radial and tangential forces and the perturbative force. We find the solution of the joint probability density. In the limit of the long-time domain and for the characteristic time=0 domain, the mean squared radial velocity for an active particle leads to a super-diffusive distribution, while the mean squared tangential velocity with both the radial and tangential forces and the perturbative force behaviors as the Gaussian diffusion. Compared with the self-propelled particle, the mean squared tangential velocity is matched with the same value to the time ~t^2, while the mean squared radial velocity is the same as the time ~t.