Mapping Monte Carlo to Langevin dynamics: A Fokker-Planck approach

TL;DR

This paper introduces a Fokker-Planck-based method to rigorously connect Monte Carlo simulations with Langevin dynamics in micromagnetics, enabling accurate time quantification and validation across damping regimes.

Contribution

It derives the FPE terms for MC methods, establishing their equivalence to Langevin equations, and demonstrates improved accuracy over previous methods across damping factors.

Findings

Analytical mapping between MC and Langevin dynamics via FPE.

Close numerical agreement between MC and LLG simulations.

Enhanced accuracy of MC at low damping factors.

Abstract

We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte-Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and disusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also found that our Metropolis MC is accurate for a large range of damping factors $\alpha$, unlike previous time-quantified MC methods which break down at low $\alpha$, where precessional motion dominates.