  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.
