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有限温度下自旋半经典系统的随机动力学行为通常由随机Landau-Lifshitz方程描述. 本文在朗之万随机微分方程的框架内, 推导出有效朗之万方程的一般形式, 及其对应的Fokker-Planck方程的表达式. 该有效朗之万方程能正确描述正则系综下自旋半经典系统的统计物理性质, 并且在阻尼项和随机项消失时能退化到自旋半经典运动方程, 因此是随机Landau-Lifshitz方程的一种推广. 在笛卡尔坐标系和球坐标系中, 分别给出有效朗之万方程的一般形式和对应的Fokker-Planck方程的显式表达式. 在球坐标系中, 讨论了朗之万方程中的纵场效应, 并从方程采取的形式中给出是否包含纵场效应的判断依据. 最后, 有效朗之万方程在一个单自旋、定值外磁场的体系中进行应用. 对方程采取特定的形式进行简便的求解, 并成功得到玻尔兹曼稳定分布, 该结果也检验了有效朗之万方程的准确性.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 coordinates 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. 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 for the spin semiclassical system is thus confirmed.
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Keywords:
- stochastic Landau-Lifshitz equation/
- Langevin equation/
- Fokker-Planck equation/
- Boltzmann distribution
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