Stochastic dynamics of planar magnetic moments in a three-dimensional environment

TL;DR

This paper develops a covariant stochastic Landau-Lifshitz-Gilbert equation for a 2D magnetic moment in 3D, providing a path integral approach to analyze correlation functions and noise characteristics.

Contribution

It introduces a covariant generalization of the sLLG equation and a path integral formalism applicable to different stochastic calculus prescriptions.

Findings

Equivalence between Cartesian and polar formulations of the stochastic process.

For isotropic fluctuations, the noise is effectively additive.

Anisotropic fluctuations lead to truly multiplicative noise.

Abstract

We study the stochastic dynamics of a two-dimensional magnetic moment embedded in a three-dimensional environment, described by means of the stochastic Landau-Lifshitz-Gilbert (sLLG) equation. We define a covariant generalization of this equation, valid in the "generalized Stratonovich discretization prescription". We present a path integral formulation that allows to compute any $n-$point correlation function, independently of the stochastic calculus used. Using this formalism, we show the equivalence between the cartesian formulation with vectorial noise, with the polar formulation with just one scalar fluctuation term. In particular, we show that, for isotropic fluctuations, the system is represented by an {\em additive stochastic process}, despite of the multiplicative terms appearing in the original formulation of the sLLG equation, but, for anisotropic fluctuations the noise turns out to be truly multiplicative.