Degond, P; Liu, J-G, *Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation*,
Mathematical Models & Methods in Applied Sciences, vol. 22 no. SUPPL.1
(2012),
pp. 1114001-18 [doi] .
**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 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 LandauLifschitzGilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological LandauLifschitzGilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics. © 2012 World Scientific Publishing Company.*