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

Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar, Gregory Schehr

Published 2022-07-21Version 1

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 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 taking diffusion into account.