Thermal Fluctuations of a Single-Domain Particle

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Abstract
A statistical ensemble of particles, with moment orientations (θ, φ), can be represented by a surface density W (θ, φ, t) of points on the unit sphere. The corresponding surface density J satisfies a continuity equation ∂W/∂t=−∇·J. With no thermal agitation, J=WṀ/Ms, where M is the vector magnetization (| M | = const = Ms); its rate of change Ṁ is assumed to be given by Gilbert's equation. To include thermal agitation, we may add to J a diffusion term −k′∇W; this gives directly the ``Fokker‐Planck'' equation of a previous, more laborious calculation. When ∂/∂φ=0, the equation simplifies and can be replaced by a minimization problem, susceptible to approximate treatment. In the case of a free‐energy function with deep minima at θ=0 and π, such treatment leads again to a result derived previously by a method adapted from Kramers and valid when v(Vmax−Vmin)/kT is at least several times unity (v=particle volume, Vmax and Vmin=maximum and minimum free energy per unit volume, k=Boltzmann's constant, T=Kelvin temperature). When the minima are not deep, a different treatment is necessary; this leads to a formula valid when v(Vmax−Vmin)/kT<<1.

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