Analytic solutions of the modified Langevin equation in a mean-field model

TL;DR

This paper derives simple, accurate analytical solutions to the modified Langevin equation within a mean-field model of interacting superparamagnetic particles, providing formulas for magnetic properties and their temperature dependencies.

Contribution

It presents novel approximate analytical solutions to the modified Langevin equation and applies them to a mean-field model of superparamagnetic particles, including formulas for key magnetic properties.

Findings

Derived analytical formulas for magnetization and coercive force.

Found similarity between approximate and exact equations.

Provided formulas applicable to various models using the modified Langevin equation.

Abstract

Approximate analytical solutions of the modified Langevin equation are obtained. These solutions are relatively simple and enough accurate. They are illustrated by considering a mean-field model of a system with interacting superparamagnetic particles. Within the framework of this model system we derived analytical approximate formulas for the temperature dependencies of the saturation and remnant magnetization, coercive force, initial magnetic susceptibility as well as for the law of approach to saturation. We obtained also some exact analytical relationships for the coercive force. We found remarkable similarity between the approximate cubic equation, which is resulted from the modified Langevin equation, and the exact equation resulting from the divergence condition of a solution derivative. The analytical formulas obtained in this work can be used in various models (not only magnetic ones), where the modified Langevin equation is applied.