Fluid models with phase transition for kinetic equations in swarming

TL;DR

This paper studies kinetic models for swarming behavior, revealing phase transitions in equilibria based on density and orientation, and derives fluid equations in the high interaction frequency limit.

Contribution

It introduces a new analysis of phase transitions in kinetic swarming models with density and orientation dependence, and derives corresponding fluid equations.

Findings

Identification of phase transitions in equilibria

Characterization of potential functions leading to phase transitions

Derivation of fluid equations in the high interaction limit

Abstract

We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs in the homogeneous kinetic equation. We analyze the profile of equilibria for general potentials identifying a family of potentials leading to phase transitions. Finally, we derive the fluid equations when the interaction frequency becomes very large.