Long time behavior of run-and-tumble particles in two dimensions

TL;DR

This paper analyzes the long-time behavior of two-dimensional run-and-tumble particles, deriving a perturbative series for their position distribution and developing a recursive method to compute moments, revealing detailed asymptotic properties.

Contribution

It introduces a perturbative series approach for RTP position distribution in 2D and a recursive formalism to compute moments, advancing understanding of their asymptotic behavior.

Findings

Distribution approximates a Gaussian at long times

Higher order corrections satisfy inhomogeneous diffusion equations

Recursive method for computing position moments

Abstract

We study the long-time asymptotic behavior of the position distribution of a run-and-tumble particle (RTP) in two dimensions and show that the distribution at a time $t$ can be expressed as a perturbative series in $(\gamma t)^{-1}$, where $\gamma^{-1}$ is the persistence time of the RTP. We show that the higher order corrections to the leading order Gaussian distribution generically satisfy an inhomogeneous diffusion equation where the source term depends on the previous order solutions. The explicit solution of the inhomogeneous equation requires the position moments, and we develop a recursive formalism to compute the same.