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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A macroscopically homogeneous aggregate contains two materials A, B, with dielectric constants εA, εB, in volume ratio p: (1—p); what is the dielectric constant of the aggregate? By a method used by Yvon and others in molecular theory, this very old problem is formulated rigorously and solved in series form. One form of the solution is ε/ε′ = 1+⅓p(1—p)(δ′/ε′)2+..., where ε′ = pεA+(1—p)εB, δ′ = εA — εB. The cubic and higher terms depend on statistical properties of particle geometry: namely, on such functions as p(3)123, the probability that three specified points 1, 2, 3 are all in material A. The relation to earlier calculations is discussed.

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