Dynamics in a kinetic model of oriented particles with phase transition

TL;DR

This paper analyzes a kinetic model of self-propelled particles with phase transitions, proving global existence, characterizing steady-states, and establishing convergence rates to equilibrium across different noise regimes.

Contribution

It provides a rigorous analysis of the Doi equation with dipolar potential, including global well-posedness, steady-state characterization, and convergence rates in any dimension.

Findings

Existence of a phase transition at a critical noise threshold.

Convergence to equilibrium is exponential in supercritical and subcritical cases.

Algebraic convergence rate at the critical noise threshold.

Abstract

Motivated by a phenomenon of phase transition in a model of alignment of self-propelled particles, we obtain a kinetic mean-field equation which is nothing else than the Doi equation (also called Smoluchowski equation) with dipolar potential. In a self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global well-posedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady-states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, there is also a family of non-isotropic equilibria. We give a rigorous prove of convergence of the solution to a steady-state as time goes to infinity. In particular we show that in the supercritical case, the only initial conditions leading to the uniform distribution in large time are those with vanishing momentum. For any positive value of the noise parameter, and any initial condition, we give rates of convergence towards equilibrium, exponentially for both supercritical and subcritical cases and algebraically for the critical case.