Reversal Collision Dynamics

TL;DR

This paper develops a kinetic model for reversal collision dynamics inspired by myxobacteria behavior, proving existence, uniqueness, and exponential convergence of solutions, supported by numerical simulations.

Contribution

It introduces a generic collision model with symmetry properties, characterizes equilibria via graph connectivity, and demonstrates exponential convergence to steady states.

Findings

Existence and uniqueness of measure solutions.

Characterization of equilibria through graph components.

Exponential convergence to equilibrium.

Abstract

Motivated by the study of reversal behaviour of myxobacteria, in this article we are interested in a kinetic model for reversal dynamics, in which particles with directions close to be opposite undergo binary collision resulting in reversing their orientations. To this aim, a generic model for binary collisions between particles with states in a general metric space exhibiting specific symmetry properties is proposed and investigated. The reversal process is given by an involution on the space, and the rate of collision is only supposed to be bounded and lower semi-continuous. We prove existence and uniqueness of measure solutions as well as their convergence to equilibrium, using the graph-theoretical notion of connectivity. We first characterise the shape of equilibria in terms of connected components of a graph on the state space, which can be associated to the initial data of the problem. Strengthening the notion of connectivity on subsets for which the rate of convergence is bounded below, we then show exponential convergence towards the unique steady-state associated to the initial condition. The article is concluded with numerical simulations set on the one-dimensional torus giving evidence to the analytical results.