Run-and-tumble oscillator: moment analysis of stationary distributions

TL;DR

This paper derives a recurrence relation for moments of the stationary distribution of a 3D run-and-tumble particle in a harmonic trap, enabling the reconstruction of the distribution despite the lack of an explicit form.

Contribution

It introduces a novel recurrence relation for moments of the 3D RTP stationary distribution, facilitating its analysis and reconstruction.

Findings

Derived recurrence relation for moments of the 3D RTP stationary distribution

Reconstructed the distribution using Fourier-Lagrange expansion

Indicated the distribution does not follow a beta function form

Abstract

When it comes to active particles, even an ideal-gas model in a harmonic potential poses a mathematical challenge. An exception is a run-and-tumble model (RTP) in one-dimension for which a stationary distribution is known exactly. The case of two-dimensions is more complex but the solution is possible. Incidentally, in both dimensions the stationary distributions correspond to a beta function. In three-dimensions, a stationary distribution is not known but simulations indicate that it does not have a beta function form. The current work focuses on the three-dimensional RTP model in a harmonic trap. The main result of this study is the derivation of the recurrence relation for generating moments of a stationary distribution. These moments are then used to recover a stationary distribution using the Fourier-Lagrange expansion.