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

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 nonlocal 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.