Stochastic modelling of thermal effects on a ferromagnetic nano particle

TL;DR

This paper investigates how thermal fluctuations influence the behavior of a ferromagnetic nanoparticle under stochastic perturbations, providing mathematical insights and numerical validation of the system's long-term stability.

Contribution

It introduces a stochastic model for ferromagnetic nanoparticles, analyzes its long-term behavior, and quantifies convergence rates to equilibrium, advancing understanding of thermal effects in magnetism.

Findings

System converges to deterministic equilibrium

Quantified L^p convergence rates

Numerical simulations support theoretical results

Abstract

In this work, we are interested in the behaviour of a single ferromagnetic mono--domain particle submitted to an external field with a stochastic perturbation. This model is a step toward the mathematical understanding of thermal effects on ferromagnets. In a first part, we discuss modelling issues and propose several ways to integrate a random noise in the deterministic model. Then, among all these approaches, we focus on the more natural one and study its long time behaviour. We prove that the system converges to the unique stable equilibrium of the deterministic model and make precise the L^p rate of the convergence. Finally, we illustrate the theoretical results by numerical simulations.