From a microscopic solution to a continuum description of active particles with a recoil interaction in one dimension

TL;DR

This paper derives a continuum model for active particles with recoil interactions in one dimension, focusing on boundary conditions for the distribution functions and their discontinuities.

Contribution

It introduces a continuum limit for a microscopic model of active particles with recoil, highlighting the importance of boundary condition matching from discrete to continuum descriptions.

Findings

Stationary distribution functions satisfy an inhomogeneous fourth-order differential equation.

Boundary conditions are not naturally derived but require careful matching from discrete models.

Distribution functions or their derivatives are generally discontinuous at boundaries.

Abstract

We consider a model system of persistent random walkers that can jam, pass through each other or jump apart (recoil) on contact. In a continuum limit, where particle motion between stochastic changes in direction becomes deterministic, we find that the stationary inter-particle distribution functions are governed by an inhomogeneous fourth-order differential equation. Our main focus is on determining the boundary conditions that these distribution functions should satisfy. We find that these do not arise naturally from physical considerations, but need to be carefully matched to functional forms that arise from the analysis of an underlying discrete process. The inter-particle distribution functions, or their first derivatives, are generically found to be discontinuous at the boundaries.