Fractional Fokker-Planck equation

C. Boundary Value Problems for the Fractional Diffusion Equation HI. The Fractional Fokker-Planck Equation... 

If an external force field acts on the random walker, it has been shown [58, 59] that in the diffusion limit, this broad waiting time process is governed by the fractional Fokker-Planck equation (FFPE) [60]... 

Equation (71) reduces to the telegrapher s-type equation found in the Brownian limit a = 1 [115]. In the usual high-friction or long-time limit, one recovers the fractional Fokker-Planck equation (19). The generalized friction and diffusion coefficients in Eq. (19) are defined by [75]... 

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. 

In previous sections, we have treated anomalous relaxation in the context of the fractional Fokker-Planck equation. As far as the Langevin equation treatment of anomalous relaxation is concerned, we proceed first by noting that Lutz [47] has introduced the following fractional Langevin equation for the translational Brownian motion in a potential V ... 

A. Numerical Solution of the Fractional Fokker—Planck Equation [Eq. (38)] via the Griinwald—Letnikov Method... 

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