Generalized stochastic Landau-Lifshitz-Gilbert equation for yttrium-iron garnet films

We derive a generalization of the well-known stochastic Landau-Lifshitz-Gilbert equation starting from a microscopic Heisenberg model coupled to the lattice degrees of freedom. By integrating out the phonons we obtain a non-Markovian, stochastic equation of motion for the spin degrees of freedom satisfying a Fluctuation-Dissipation theorem. We apply our theory to study the parametric pumping and thermalization of spin excitations in thin yttrium-iron garnet films.