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

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.

Tuning attraction and repulsion between active particles through persistence

“…A further simplification that occurs when ρ(x) is differentiable at the boundaries is that the boundary conditions q(0) = 1 and q(1) = 0 carry over directly from the master equation ( 16). The fact that this does not occur in the general case is addressed in depth in [31]. Integrating (17) with these boundary conditions gives…”

“…In a forthcoming work [31], we solve a system equivalent to ( 15) and ( 16) for a general recoil distribution ρ(x), including those that have delta functions at the boundary points x = 0 and x = 1, which correspond to the particles having some finite probability of jamming into, or passing through, each other on contact. This solution has complex behavior in the region of size 1 √ L at each boundary, with discontinuities in the interparticle distribution functions and/or their derivatives at the boundary points.…”

“…The upshot of this is that the boundary conditions that naturally arise by considering probability fluxes at the boundaries are not satisfied by the limiting forms of p(x) and q(x). This has implications for constructing continuum descriptions of interacting active matter systems which are explored more fully in a forthcoming paper [31].…”

“…These two expressions (3) and ( 4) are the main result of this work, and apply if the recoil distribution ρ(x) is differentiable at the boundaries. The more general case, and the technical subtleties that arise at the boundaries, are discussed in detail in a forthcoming paper [31].…”