Fractional dynamics

ANOMALOUS STOCHASTIC PROCESSES IN THE FRACTIONAL DYNAMICS FRAMEWORK FOKKER-PLANCK EQUATION, DISPERSIVE TRANSPORT, AND NON-EXPONENTIAL RELAXATION... 

V. The Fractional Klein-Kramers Equation Fractional Dynamics in Phase Space... 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

Among the most striking changes brought about by fractional dynamics is the substitution of the traditionally obtained exponential system equilibration of time-dependent system quantities by the Mittag-Leffler pattern [44-46]... 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

V. THE FRACTIONAL KLEIN-KRAMERS EQUATION FRACTIONAL DYNAMICS IN PHASE SPACE... 

Fractional dynamics emerges as the macroscopic limit of the combination of the Langevin and the trapping processes. After straightforward calculations based on the continuous-time version of the Chapman-Kolmogorov equation [75, 114] which are valid in the long-time limit t max r, t, one obtains the fractional Klein-Kramers equation... 

Fractional dynamics is a made-to-measure approach to the description of temporally nonlocal systems, the kinetics of which is governed by a selfsimilar memory. Fractional kinetic equations are operator equations that are mathematically close to the well-studied, analogous Brownian evolution equations of the Klein-Kramers, Rayleigh, or Fokker-Planck types. Consequently, methods such as the separation of variables can be applied. More-... 

The characteristic changes brought about by fractional dynamics in comparison to the Brownian case include the temporal nonlocality of the approach manifest in the convolution character of the fractional Riemann-Liouville operator. Initial conditions relax slowly, and thus they influence the evolution of the system even for long times [62, 116] furthermore, the Mittag-Leffler behavior replaces the exponential relaxation patterns of Brownian systems. Still, the associated fractional equations are linear and thus extensive, and the limit solution equilibrates toward the classical Gibbs-B oltzmann and Maxwell distributions, and thus the processes are close to equilibrium, in contrast to the Levy flight or generalised thermostatistics models under discussion. 

In our presentation, we concentrated on the modeling of subdiffusive phenomena—that is, modeling of processes whose mean squared displacement in the force-free limit follows the power-law dependence (x2 )) oc tK for 0 < k < 1. The extension of fractional dynamics to systems where the transport is subballistic but superdiffiisive, 1 < k < 2, is presently under discussion [77, 78], (compare also Ref. 117). 

The Mittag-Leffler function [44-46] can be viewed as a natural generalization of the exponential function. Within fractional dynamics, it replaces the traditional exponential relaxation patterns of moments, modes, or of the Kramers survival. It is an entire function that decays completely monotoni-cally for 0 < a < 1. It is the exact relaxation function for the underlying multiscale process, and it leads to the Cole-Cole behavior for the complex... 

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