Exact position distribution of a harmonically-confined run-and-tumble particle in two dimensions

TL;DR

This paper derives the exact position distribution of a 2D run-and-tumble particle under harmonic confinement, revealing detailed probabilistic behavior and extending to multiple particles, higher dimensions, and stochastic resetting.

Contribution

It provides the first exact analytical solutions for the time-dependent and steady-state distributions of a confined 2D run-and-tumble particle, including extensions to interacting particles and higher dimensions.

Findings

Exact steady-state distribution derived

Decomposition into independent 1D problems achieved

Extended results to multiple particles and higher dimensions

Abstract

We consider an overdamped run-and-tumble particle in two dimensions, with self propulsion in an orientation that stochastically rotates by 90 degrees at a constant rate, clockwise or counter-clockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness $\mu$, and possibly diffuses. We find the exact time-dependent distribution $P\left(x,y,t\right)$ of the particle's position, and in particular, the steady-state distribution $P_{\text{st}}\left(x,y\right)$ that is reached in the long-time limit. We also find $P\left(x,y,t\right)$ for a "free" particle, $\mu=0$. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically-independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.