Long time behaviour of a stochastic nano particle

TL;DR

This paper investigates the long-term behavior of a stochastic model of a ferromagnetic nanoparticle under external and thermal influences, analyzing stability, reversibility, and hysteresis effects through mathematical proofs and simulations.

Contribution

It introduces a stochastic differential equation model for a ferromagnetic nanoparticle and provides new insights into its long-term dynamics and hysteresis phenomena.

Findings

Magnetization converges to a stable equilibrium at a quantifiable rate.

Stochastic perturbation induces non-reversibility in the magnetic behavior.

Hysteresis effects are sharply estimated under stochastic influences.

Abstract

In this article, 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 the first step toward the mathematical understanding of thermal effects on a ferromagnet. In a first part, we present the stochastic model and prove that the associated stochastic differential equation is well defined. The second part is dedicated to the study of the long time behaviour of the magnetic moment and in the third part we prove that the stochastic perturbation induces a non reversibility phenomenon. Last, we illustrate these results through numerical simulations of our stochastic model. The main results presented in this article are the rate of convergence of the magnetization toward the unique stable equilibrium of the deterministic model. The second result is a sharp estimate of the hysteresis phenomenon induced by the stochastic perturbation (remember that with no perturbation, the magnetic moment remains constant).