  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.

Statistical Mechanics