Fractional Fokker-Planck equation model

The second model of Debye or the Debye-Frohlich model may also be generalized to fractional diffusion [8,25] (including inertial effects [26]). Moreover, it has been shown [25] that the Cole-Cole equation arises naturally from the solution of a fractional Fokker-Planck equation in the configuration space of orientations derived from the diffusion limit of a CTRW. The broadening of the dielectric loss curve characteristic of the Cole-Cole spectrum may then be easily explained on a microscopic level by means of the relation [8,24]... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

Thus we have demonstrated how the empirical Havriliak-Negami equation [Eq. (11)] can be obtained from a microscopic model, namely, the fractional Fokker-Planck equation [Eq. (101)] applied to noninteracting rotators. This model can explain the anomalous relaxation of complex dipolar systems, where the anomalous exponents ct and v differ from unity (corresponding to the classical Debye theory of dielectric relaxation) that is, the relaxation process is characterized by a broad distribution of relaxation times. Hence, the empirical Havriliak-Negami equation of anomalous dielectric relaxation which has been... 

This potential has two potential minima on the sites at <(> = 0 and = n as well as two energy barriers located at < ) = jt/2 and <[) = 3n/2. This model has been treated in detail for normal diffusion in Refs. 8,61, and 62. Here we consider the fractional Fokker-Planck equation [Eq. (55)] for a fixed axis rotator with dipole moment p moving in a potential [Eq. (163)]. 

We remark that Eq. (262), unlike the form of the Rocard equation of the Levy sneaking model, Eq. (248), has an inertial term similar to the Rocard equation for normal diffusion, Eq. (249). This has an important bearing on the high-frequency behavior because return to transparency can now be achieved, as we shall demonstrate presently. The exact solution, Eq. (260), also has satisfactory high-frequency behavior. We further remark that, on neglecting inertial effects (y —> 0), Eq. (261) yields the Cole-Cole formula [Eq. (9)]—that is, the result predicted by the noninertial fractional Fokker-Planck equation. 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

The above stochastic collision model then leads to a generalization, Eq. (253), of the Fokker-Planck equation for the evolution of the phase distribution function for mechanical particles, where the velocities acquire a fractional character [30], rather than both the displacements and the velocities as in Eq. (235). In the present context, all these comments apply, of course, to rotational Brownian motion. 

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