A note on phase transitions for the Smoluchowski equation with dipolar potential

TL;DR

This paper investigates phase transitions in a modified Smoluchowski equation with dipolar potential, analyzing equilibrium states, convergence rates, and hysteresis phenomena through theoretical and numerical methods.

Contribution

It introduces a modification to the Smoluchowski equation where alignment and diffusion depend on the order parameter, and characterizes the resulting phase transition behaviors.

Findings

Identified stable and unstable equilibrium states.

Derived exponential convergence rates for stable equilibria.

Illustrated first and second order phase transitions and hysteresis effects.

Abstract

In this note, we study the phase transitions arising in a modified Smoluchowski equation on the sphere with dipolar potential. This equation models the competition between alignment and diffusion, and the modification consists in taking the strength of alignment and the intensity of the diffusion as functions of the order parameter. We characterize the stable and unstable equilibrium states. For stable equilibria, we provide the exponential rate of convergence. We detail special cases, giving rise to second order and first order phase transitions, respectively. We study the hysteresis diagram, and provide numerical illustrations of this phenomena.