Stochastic tools hidden behind the empirical dielectric relaxation laws.

The paper is devoted to recent advances in stochastic modeling of anomalous kinetic processes observed in dielectric materials which are prominent examples of disordered (complex) systems. Theoretical studies of dynamical properties of 'structures with variations' (Goldenfield and Kadanoff 1999 Science 284 87-9) require application of such mathematical tools-by means of which their random nature can be analyzed and, independently of the details distinguishing various systems (dipolar materials, glasses, semiconductors, liquid crystals, polymers, etc), the empirical universal kinetic patterns can be derived. We begin with a brief survey of the historical background of the dielectric relaxation study.

Moving average process underlying the holographic-optical-tweezers experiments.

We study the statistical properties of recordings that contain time-dependent positions of a bead trapped in optical tweezers. Analysis of such a time series indicates that the commonly accepted model, i.e.

Anomalous diffusion with transient subordinators: a link to compound relaxation laws.

This paper deals with a problem of transient anomalous diffusion which is currently found to emerge from a wide range of complex processes. The nonscaling behavior of such phenomena reflects changes in time-scaling exponents of the mean-squared displacement through time domain - a more general picture of the anomalous diffusion observed in nature. Our study is based on the identification of some transient subordinators responsible for transient anomalous diffusion.

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.

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.

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.

Anomalous diffusion with under- and overshooting subordination: a competition between the very large jumps in physical and operational times.

In this paper we present an approach to anomalous diffusion based on subordination of stochastic processes. Application of such a methodology to analysis of the diffusion processes helps better understanding of physical mechanisms underlying the nonexponential relaxation phenomena. In the subordination framework we analyze a coupling between the very large jumps in physical and two different operational times, modeled by under- and overshooting subordinators, respectively.

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.

Diffusion and relaxation controlled by tempered alpha-stable processes.

We derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered alpha-stable processes. The tempering results in the existence of all moments of operational time. The subordination by the inverse tempered alpha-stable process provides diffusion (relaxation) that occupies an intermediate place between subdiffusion (Cole-Cole law) and normal diffusion (exponential law).

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.

Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation.

In this paper we attack the challenging problem of modeling subdiffusion with an arbitrary space-time-dependent driving. Our method is based on a combination of the Langevin-type dynamics with subordination techniques. For the case of a purely time-dependent force, we recover the death of linear response and field-induced dispersion -- two significant physical properties well-known from the studies based on the fractional Fokker-Planck equation.

Wait-and-switch relaxation model: relationship between nonexponential relaxation patterns and random local properties of a complex system.

The wait-and-switch stochastic model of relaxation is presented. Using the "random-variable" formalism of limit theorems of probability theory we explain the universality of the short- and long-time fractional-power laws in relaxation responses of complex systems. We show that the time evolution of the nonequilibrium state of a macroscopic system depends on two stochastic mechanisms: one, which determines the local statistical properties of the relaxing entities, and the other one, which determines the number (random or deterministic) of the microscopic and mesoscopic relaxation contributions.

Fractional Fokker-Planck dynamics: stochastic representation and computer simulation.

A computer algorithm for the visualization of sample paths of anomalous diffusion processes is developed. It is based on the stochastic representation of the fractional Fokker-Planck equation describing anomalous diffusion in a nonconstant potential. Monte Carlo methods employing the introduced algorithm will surely provide tools for studying many relevant statistical characteristics of the fractional Fokker-Planck dynamics.

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.

Complete description of all self-similar models driven by Lévy stable noise.

A canonical decomposition of H-self-similar Lévy symmetric alpha-stable processes is presented. The resulting components completely described by both deterministic kernels and the corresponding stochastic integral with respect to the Lévy symmetric alpha-stable motion are shown to be related to the dissipative and conservative parts of the dynamics. This result provides stochastic analysis tools for study the anomalous diffusion phenomena in the Langevin equation framework.

Random walk models of electron tunneling in a fluctuating medium.

A modified approach to the electron transfer theory in disordered media is discussed by use of continuous time random walk models accounting for medium fluctuations. The models apply to the situations when the bridging medium between the donor and acceptor pair fluctuates changing coupling between intermediate transferring states. Effect of the latter on the long distance electron transfer is discussed pointing out emergence of nonexponential decay kinetics.