Fokker-Planck and Landau-Lifshitz-Bloch Equations for Classical Ferromagnets

TL;DR

This paper derives a macroscopic equation for ferromagnet magnetization that seamlessly transitions between low and high temperature regimes, based on stochastic models and mean-field approximations.

Contribution

It provides an exact derivation of the Landau-Lifshitz-Bloch equation from the Fokker-Planck equation for classical ferromagnets under slow external changes.

Findings

Derived the LLB equation from the Fokker-Planck equation.

Formulated a macroscopic LLB equation interpolating between T_C and low temperatures.

Validated the approach within the mean-field approximation for weakly anisotropic ferromagnets.

Abstract

A macroscopic equation of motion for the magnetization of a ferromagnet at elevated temperatures should contain both transverse and longitudinal relaxation terms and interpolate between Landau-Lifshitz equation at low temperatures and the Bloch equation at high temperatures. It is shown that for the classical model where spin-bath interactions are described by stochastic Langevin fields and spin-spin interactions are treated within the mean-field approximation (MFA), such a ``Landau-Lifshitz-Bloch'' (LLB) equation can be derived exactly from the Fokker-Planck equation, if the external conditions change slowly enough. For weakly anisotropic ferromagnets within the MFA the LLB equation can be written in a macroscopic form based on the free-energy functional interpolating between the Landau free energy near T_C and the ``micromagnetic'' free energy, which neglects changes of the magnetization magnitude |{\bf M}|, at low temperatures.