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

Pierre Degond (1) , Jian-Guo Liu (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Duke Physics

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.

Domaines: Physique mathématique [math-ph]

Pierre Degond, Jian-Guo Liu. Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation. Mathematical Models and Methods in Applied Sciences, 2012, 22 (suppl. 1), pp.1140001. ⟨10.1142/S021820251140001X⟩. ⟨hal-00614664⟩