Role of particle conservation in self-propelled particle systems

Actively propelled particles undergoing dissipative collisions are
known to develop a state of spatially distributed coherently moving
clusters. For densities larger than a characteristic value,
clusters grow in time and form a stationary well-ordered state of
coherent macroscopic motion. In this work we address two questions.
(i) What is the role of the particles’ aspect ratio in the context
of cluster formation, and does the particle shape affect the
system’s behavior on hydrodynamic scales? (ii) To what extent does
particle conservation influence pattern formation? To answer these
questions we suggest a simple kinetic model permitting us to depict
some of the interaction properties between freely moving particles
and particles integrated in clusters. To this end, we introduce two
particle species: single and cluster particles. Specifically, we
account for coalescence of clusters from single particles, assembly
of single particles on existing clusters, collisions between
clusters and cluster disassembly. Coarse graining our kinetic
model, (i) we demonstrate that particle shape (i.e. aspect ratio)
shifts the scale of the transition density, but does not impact the
instabilities at the ordering threshold and (ii) we show that the
validity of particle conservation determines the existence of a
longitudinal instability, which tends to amplify density
heterogeneities locally, and in turn triggers a wave pattern with
wave vectors parallel to the axis of macroscopic order. If the
system is in contact with a particle reservoir, this instability
vanishes due to a compensation of density heterogeneities.