Positing the problem of stationary distributions of active particles as third-order differential equation

TL;DR

This paper derives a third-order differential equation for the stationary distribution of active particles in a harmonic trap, revealing the influence of active motion and thermal fluctuations.

Contribution

It formulates the stationary distribution problem as a third-order differential equation and provides solutions as convolutions of Gaussian and beta distributions, highlighting active motion effects.

Findings

Derived third-order differential equation for active particles

Solutions expressed as convolutions of Gaussian and beta distributions

Indicates independence of thermal and active processes

Abstract

In this work, we obtain third order linear differential equation for stationary distributions of run-and-tumble particles in two-dimensions in a harmonic trap. The equation represents the condition $j = 0$ where $j$ is a flux and is obtained from inference, using different known results in the limiting conditions. Since the analogous equation for passive Brownian particles is first order, a second and third order term must be a feature of active motion. In addition to formulating the problem as third order equation, we obtain solutions in the form of convolution of two distributions, the Gaussian distribution due to thermal fluctuations, and the beta distribution due to active motion at zero temperature. The convolution form of the solution indicates that the two random processes are independent and the total distribution is the sum of those two processes.