Study of the generalization of spin semiclassical Langevin equation

The stochastic dynamics of spin semiclassical system at finite temperature is usually described by stochastic Landau-Lifshitz equation. In this work, the stochastic differential equation for spin semiclassical system is studied. The generalized formulation of effective Langevin equation and the corresponding Fokker-Planck equation are derived. The obtained effective Langevin equation offers an accurate description of the distribution in the canonical ensemble for spin semiclassical system. When the damping term and the stochastic term vanish, the effective Langevin equation reduces to the semiclassical equation of motion for spin system. Hence, the effective Langevin equation can be seen as a generalization of the stochastic Landau-Lifshitz equation. The explicit expressions for the effective Langevin equation and the corresponding Fokker-Planck equation are shown in both Cartesian and Spherical coordinates. It is demonstrated that, the longitudinal effect can be easily illustrated from the expressions in Spherical coordinates. The effective Langevin equation is applied to the simple system of a single spin in a constant magnetic field. In choosing an appropriate form, the Langevin equation can be easily solved and the stationary Boltzmann distribution can be obtained. The correctness of the Langevin approach to the spin semiclassical system is thus confirmed.