Fokker-Planck equation inertial effects

If inertial effects were of interest, then we would introduce a distribution function in the phase space of the polymer chain, and we would be led to a Fokker-Planck equation of the Kramers type. 

KRAMERS EQUATION—ARBITRARY FRICTION REGIME. In the presence Of inertial effects, the one-dimensional motion is determined by a bivariate Fokker-Planck equation. 

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. 

In order to generalize the normal Fokker-Planck equation excluding inertial effects to fractional diffusion, we first recall the general form of that equation for normal diffusion in operator representation [49]... 

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. 

The fractional Fokker-Planck equation (22) which ignores inertial effects can be solved exactly for an harmonic potential (Qmstein-Uhlenbeck process),... 

If a 0, we have the noninertial response. This is treated by Shliomis and Stepanov [9], who were able to factorize the joint distribution of the dipole and easy axis orientations in the Fokker Planck equation into the product of the two separate distributions. Thus as far as the internal relaxation process is concerned, the axially symmetric treatment of Brown [50] applies. Hence no intrinsic coupling between the transverse and longitudinal modes exists that is, the eigenvalues of the longitudinal relaxation process are independent of a. The distribution function of the easy axis orientations n is simply that of a free Brownian rotator excluding inertial effects. 

This concept, which is based on a random walk with a well-defined characteristic time and which applies when collisions are frequent but weak [13], leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 71, due to Fiirth), we obtain the Fokker Planck equation for the evolution of the distribution function in phase space which describes normal diffusion. 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

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