Fractional Klein-Kramers Research Articles

Stationarity$ndash$conservation laws for fractional differential equations with variable coefficients

In this paper, we study linear fractional differential equations with variable coefficients. It is shown that, by assuming some conditions for the coefficients, the stationarity–conservation laws can be derived. The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. Applications of the proposed procedure include the fractional Fokker–Planck equation in (1 + 1)- and (d + 1)-dimensional space and the fractional Klein–Kramers equation.

Inertial effects in the anomalous dielectric relaxation of rotators in space.

The linear dielectric response of an assembly of noninteracting linear (needlelike) dipole molecules (each of which is free to rotate in space) is evaluated in the context of fractional dynamics. The infinite hierarchy of differential-recurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional Fokker-Planck (fractional Klein-Kramers) equation. On solving this hierarchy in terms of continued fractions (as in the normal rotational diffusion), the complex dynamic susceptibility is obtained and is calculated for typical values of the model parameters. It is shown that the model can reproduce nonexponential anomalous dielectric relaxation behavior at low frequencies (omega tau< or =1, where tau is the Debye relaxation time) and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies (in the far infrared region) so that Gordon's sum rule for integral dipolar absorption is satisfied.

Inertial effects in anomalous dielectric relaxation.

The inertia corrected Debye model of rotational Brownian motion of polar molecules is generalized to fractional dynamics (anomalous diffusion) in the context of the fractional Klein-Kramers equation. The fractal generalization of the Gross-Sack solution for the complex dielectric susceptibility chi(omega) for an assembly of fixed axis rotators is given. The high-frequency behavior of chi(omega) is controlled by the inertia of a dipole as in normal diffusion, so that the Gordon sum rule for dipolar absorption is satisfied ensuring a return to optical transparency at very high frequencies.

Subdiffusive transport close to thermal equilibrium: from the langevin equation to fractional diffusion

Subdiffusion in the presence or absence of an external force field is established on the basis of an extension of conventional Langevin dynamics to include long-tailed trapping events. It is demonstrated how the presence of the trapping events leads to the macroscopic observation of fractional diffusion, described by a fractional Klein-Kramers equation.