Fokker-Planck equation ferrofluids

We have mentioned that the question posed above was answered in part by Shliomis and Stepanov [9]. They showed that for uniaxial particles, for weak applied magnetic fields, and in the noninertial limit, the equations of motion of the ferrofluid particle incorporating both the internal and the Brownian relaxation processes decouple from each other. Thus the reciprocal of the greatest relaxation time is the sum of the reciprocals of the Neel and Brownian relaxation times of both processes considered independently that is, those of a frozen Neel and a frozen Brownian mechanism In this instance the joint probability of the orientations of the magnetic moment and the particle in the fluid (i.e., the crystallographic axes) is the product of the individual probability distributions of the orientations of the axes and the particle so that the underlying Fokker Planck equation for the joint probability distribution also... 

The Shliomis Stepanov approach [9] to the ferrofluid relaxation problem, which is based on the Fokker Planck equation, has come to be known in the literature on magnetism as the egg model. Yet another treatment has recently been given by Scherer and Matuttis [42] using a generalized Lagrangian formalism however, in the discussion of the applications of their method, they limited themselves to a frozen Neel and a frozen Brownian mechanism, respectively. 

In Section V we are concerned with Gilbert s equation as applied to the Debye relaxation of a ferrofluid particle with the inertia of the particle included. It is shown, by averaging Gilbert s equation for Debye relaxation corrected for inertia and proceeding to the noninertial limit, how analytic expressions for the transverse and longitudinal relaxation times for Debye relaxation may be obtained directly from that equation thus bypassing the Fokker-Planck equation entirely. These expressions coincide with the previous results of the group of Shliomis [16]. 

In this section we summarize the approach used by previous authors [8, 16-19] to find expressions for the relaxation times of single domain ferromagnetic and ferrofluid particles. We begin with the Fokker-Planck equation obtained from Gilbert s equation, in spherical polar coordinates, augmented by a random field term, that is, with Brown s equation. We then expand the probability density of orientations of M, that is. 

---