Continuous-time random walk theory

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. 

Levy flights are the central topic of this review. For a homogeneous environment the central relation of continuous time random walk theory is given by [14,45]... 

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... 

As far as the physical mechanism underlying the Cole-Cole equation is concerned, we first remark that Eq. (9) arises from the diffusion limit of a continuous-time random walk (CTRW) [17] (see Section II.A). In this context one should recall that the Einstein theory of the Brownian motion relies on the diffusion limit of a discrete time random walk. Here the random walker makes a jump of a fixed mean-square length in a fixed time, so that the only random... 

Relaxation functions for fractal random walks are fundamental in the kinetics of complex systems such as liquid crystals, amorphous semiconductors and polymers, glass forming liquids, and so on [73]. Relaxation in these systems may deviate considerably from the exponential (Debye) pattern. An important task in dielectric relaxation of complex systems is to extend [74,75] the Debye theory of relaxation of polar molecules to fractional dynamics, so that empirical decay functions for example, the stretched exponential of Williams and Watts [76] may be justified in terms of continuous-time random walks. 

Neither the electron density dependence nor the shape (which is approximately stretched exponential) of the kinetics can be explained with second order reaction kinetics, where it is assumed that the reaction is controlled only by the concentrations of electrons and dye cations, nor are they consistent with simple electron transfer theory. An explanation was proposed by Nelson based on the continuous time random walk [109]. In the CTRW, electrons perform a random walk on a lattice, which contains trap sites distributed in energy, according to some distribution function, g E). In contrast to normal diffusion, where the mean time taken for each step is a constant, in the CTRW the time taken for each electron to move is determined by the time for thermal escape from the site currently occupied. 

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. 

Note that the Poisson process plays a very important role in random walk theory. It can be defined in two ways (1) as a continuous-time Markov chain with constant intensity, i.e., as a pure birth process with constant birth rate k (2) as a renewal process. In the latter case, it can be represented as (3.25) with T = Here... 

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