Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation

TL;DR

This paper derives a hydrodynamic model for self-propelled particles with alignment and precession, connecting microscopic interactions to classical micromagnetization equations like Landau-Lifschitz-Gilbert.

Contribution

It introduces a kinetic framework that links particle alignment with precession to macroscopic magnetization models, including diffusive corrections and weakly non-local interactions.

Findings

Derived a hydrodynamic system for particle density and orientation.

Identified conditions under which the system loses hyperbolicity.

Connected the model to classical micromagnetization equations in a special case.

Abstract

We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly non-local interaction, we derive diffusive corrections to the first order system which lead to the combination of a heat flow of the harmonic map and Landau-Lifschitz-Gilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological Landau-Lifschitz-Gilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics.