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

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 particles (RTP) model 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.