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

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 interparticle 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 they need to be carefully matched to functional forms that arise from the analysis of an underlying discrete process. The interparticle distribution functions, or their first derivatives, are generically found to be discontinuous at the boundaries.