Clustered continuous-time random walks: diffusion and relaxation consequences.

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

Experimental evidence of the role of compound counting processes in random walk approaches to fractional dynamics.

We present dielectric spectroscopy data obtained for gallium-doped Cd(0.99)Mn(0.01)Te:Ga mixed crystals, which exhibit a very special case of the two-power-law relaxation pattern with the high-frequency power-law exponent equal to 1. We explain this behavior, which cannot be fitted by any of the well-known empirical relaxation functions, in a subordinated diffusive framework. We propose a diffusion scenario based on a renormalized clustering of a random number of spatio-temporal steps in the continuous-time random walk. Such a construction substitutes the renewal counting process, which is used in the classical continuous time random walk methodology, with a compound counting one. As a result, we obtain an appropriate relaxation function governing the observed nonstandard pattern, and we show the importance of the compound counting processes in studying fractional dynamics of complex systems.

Overshooting and undershooting subordination scenario for fractional two-power-law relaxation responses.

In this paper, we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation. We investigate properties of a coupled continuous-time random walk that follows from modeling the occurrence of jumps with compound counting processes. As a result, two different diffusion processes corresponding to over- and undershooting operational times, respectively, have been found. We show that within the proposed framework, all empirical two-power-law relaxation patterns may be derived. This work is motivated by the so-called "less typical" relaxation behavior observed, e.g., for gallium-doped Cd0.99Mn0.01Te mixed crystals.

The frequency-domain relaxation response of gallium doped Cd(1-x)Mn(x)Te.

In this paper the complex dielectric permittivity of gallium doped Cd(0.99)Mn(0.01)Te mixed crystals is studied at different temperatures. We observe a two-power-law relaxation pattern with m and n, the low- and high-frequency power-law exponents respectively, satisfying the relation m < 1-n. To interpret the empirical result we propose a correlated-cluster relaxation mechanism. This approach allows us to find origins of both power-law exponents, m and n.

Generalized Mittag-Leffler relaxation: clustering-jump continuous-time random walk approach.

A stochastic generalization of renormalization-group transformation for continuous-time random walk processes is proposed. The renormalization consists in replacing the jump events from a randomly sized cluster by a single renormalized (i.e., overall) jump. The clustering of the jumps, followed by the corresponding transformation of the interjump time intervals, yields a new class of coupled continuous-time random walks which, applied to modeling of relaxation, lead to the general power-law properties usually fitted with the empirical Havriliak-Negami function.

Hopping models of charge transfer in a complex environment: coupled memory continuous-time random walk approach.

Charge transport processes in disordered complex media are accompanied by anomalously slow relaxation for which usually a broad distribution of relaxation times is adopted. To account for those properties of the environment, a standard kinetic approach in description of the system is addressed either in the framework of continuous-time random walks (CTRWs) or fractional diffusion. In this paper the power of the CTRW approach is illustrated by use of the probabilistic formalism and limit theorems that allow one to rigorously predict the limiting distributions of the paths traversed by charges and to derive effective relaxation properties of the entire system of interest. In particular, the standard CTRW scenario is generalized to a new class of coupled memory CTRWs that effectively can lead to the well known Havriliak-Negami response. Application of the method is discussed for nonexponential electron-transfer processes controlled by dynamics of the surrounding medium.