磁化动力学：随机Landau-Lifshitz-Gilbert方程的路径积分形式
Magnetization dynamics: path-integral formalism for the stochastic Landau-Lifshitz-Gilbert equation
摘要：
我们构建了一个路径积分表示的经典磁矩的耗散动力学的生成泛函所描述的随机推广的Landau-Lifshitz-Gilbert方程提出的布朗，与可能增加的自旋扭矩条款。在笛卡尔坐标系中构造该泛函的过程中，我们批判地重新审视了该随机方程。我们提出了它的形式，适应任何离散化计划感谢包括漂移项。广义方程确保了在没有非势和时间依赖力的情况下磁化模量的守恒和吉布斯-玻尔兹曼平衡的方法。漂移项仅在使用中点Stratonovich公式时才消失。接下来，我们在更自然的球面坐标系中重新设置问题。我们表明，噪声变换非平凡的球坐标获得一个非零的平均值在这个坐标系，一个事实，在文献中经常被忽视。接下来，我们构造生成功能的形式主义在这个坐标系中的任何离散化处方。在笛卡尔坐标或球坐标的功能形式主义应作为一个出发点，研究不同方面的磁铁的非平衡动力学。扩展到色噪声、微磁性和无序问题是很简单的。
英文摘要：
We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau-Lifshitz-Gilbert equation proposed by Brown, with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized equation ensures the conservation of the magnetization modulus and the approach to the Gibbs-Boltzmann equilibrium in the absence of non-potential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.
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