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

Derek Frydel

Published 2022-07-18Version 1

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. Since the analogous equation for passive Brownian particles is first order, a second and third order term are features of active motion. This manner of formulating the problem allows us to make advancements in theoretical treatment and understanding of active particles, given that exact results even for basic models are elusive. Third order equations are less common in physics and solutions are more challenging to get. Often such solutions cannot be represented in a closed form. Such is the case for systems considered in this work. The formulation of the problem as third order equation can offer insights into the origin of this elusiveness and play a classifying role. For example, one of the results established in this work is that an equation and its solution for one- and two-dimensional system can be scaled onto each other.