Nambu mechanics for stochastic magnetization dynamics

TL;DR

This paper introduces a Nambu mechanics framework to model stochastic magnetization dynamics, providing a consistent way to incorporate damping and stochastic effects, with numerical validation and implications for conserved quantities.

Contribution

It formulates the Landau-Lifshitz-Gilbert equation within dissipative Nambu mechanics, enabling a topologically consistent approach to stochastic magnetization dynamics.

Findings

Nambu mechanics elegantly describes damping in magnetization dynamics.

Numerical integrators preserve topological structure of Nambu equations.

Derived a master equation for stochastic magnetization interactions.

Abstract

The Landau-Lifshitz-Gilbert (LLG) equation describes the dynamics of a damped magnetization vector that can be understood as a generalization of Larmor spin precession. The LLG equation cannot be deduced from the Hamiltonian framework, by introducing a coupling to a usual bath, but requires the introduction of additional constraints. It is shown that these constraints can be formulated elegantly and consistently in the framework of dissipative Nambu mechanics. This has many consequences for both the variational principle and for topological aspects of hidden symmetries that control conserved quantities. We particularly study how the damping terms of dissipative Nambu mechanics affect the consistent interaction of magnetic systems with stochastic reservoirs and derive a master equation for the magnetization. The proposals are supported by numerical studies using symplectic integrators that preserve the topological structure of Nambu equations. These results are compared to computations performed by direct sampling of the stochastic equations and by using closure assumptions for the moment equations, deduced from the master equation.