Persistent random walkers with shocks

We study a generalisation of a model of bacterial dynamics where two persistent random walkers interact via hard-core exclusion on a one-dimensional ring lattice. In this work, we replace the hard-core exclusion interaction with a generalised ‘shock’ interaction inspired by the octoflagellate microalga Pyramimonas octopus, individuals of whom are observed to rapidly swim away from each other upon contact. The interaction comprises an instantaneous displacement of one of the walkers according to an arbitrary distribution, as well as a velocity reversal with probability r (hard-core exclusion is the special case where the walker shocks to the same site without a velocity reversal). We use a novel generating-function technique to solve this problem in the continuum, whereby a set of highly non-trivial boundary conditions are derived via the kernel method. In the case where shocks are smoothly distributed across the domain, a rich set of behaviours are observed – for example, a set of ‘re-entrant’ states in which an effective attraction exists between the walkers for only a finite range of persistence lengths. Finally, we show that a rich variety of inter-particle distribution functions may be accessed by tuning the model parameters.