Finite element solution of the Fokker–Planck equation for single domain particles

Abstract

The Fokker–Planck equation derived by Brown for the probability density function of the orientation of the magnetic moment of single domain particles is one of the basic equations in the theory of superparamagnetism. Brown’s equation has an analytical solution, which is represented as a series in spherical harmonics with time-dependent coefficients. This analytical solution is commonly used when studying problems related to Brown’s equation. However, for particles with complex magnetic anisotropy, calculating the coefficients of the analytical solution can be a rather difficult task. In this paper, we propose an algorithm for the numerical solution of Brown’s equation based on the finite element method. The algorithm allows numerically solving Brown’s equation for particles with anisotropy of a fairly general form for constant and variable magnetic fields and with time-dependent temperature and other model parameters. In particular, an example of the numerical solution of an equation for particles with cubic anisotropy accounting two anisotropy constants and variable temperature is presented.