Agnieszka Jurlewicz

Relaxation of dynamically correlated clusters

Abstract In the frame of a new probabilistic approach to relaxation, the scenario of relaxation leading to the Havriliak–Negami and Kohlrausch–Williams–Watts responses of complex systems is presented. In this approach the macroscopic laws are related to the micro/mesoscopic stochastic characteristics of the relaxing systems. This provides a rigorous formulation of the energy-criterion argument, introduced by Jonscher to explain the commonly observed high-frequency fractional power law. The presented considerations reinforce the physical significance of the empirically found forms of relaxation, and open a new line of analysis of relaxation phenomena.

Fractional governing equations for coupled random walks

In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW is coupled if the waiting time and the subsequent jump are dependent random variables. The CTRW is used in physics to model diffusing particles. Its scaling limit is governed by an anomalous diffusion equation. Some applications require an overshoot continuous time random walk (OCTRW), where the waiting time is coupled to the previous jump. This paper develops stochastic limit theory and governing equations for CTRW and OCTRW. The governing equations involve coupled space-time fractional derivatives. In the case of infinite mean waiting times, the solutions to the CTRW and OCTRW governing equations can be quite different.

A relationship between asymmetric Lévy-Stable distributions and the dielectric susceptibility

This paper, as a complement to the work of Montroll and Bendler, is concerned with the Lévy-stable distributions and their connection to the dielectric response of dipolar materials in the frequency domain. The necessary and sufficient condition for this connection is found. The presented probabilistic analysis is based on the mathematically correct representation of the meaning of the relaxation function of a system of dipoles and shows why the same form of a distribution of relaxation rates, namely, the completely asymmetric Lévy-stable distribution, should apply in all different relaxing systems. This is in contrast to the traditional definition of the relaxation function, expressed as a weighted average of exponential relaxation functions, which does not explain the universality of the dielectric relaxation law. It also follows from the present considerations that not only is the imaginary part χ″(ω) of the dielectric susceptibility directly related to the Lévy-stable distribution (as was found by Montroll and Bendler), but so is the real partχ′(ω). As a consequence the relationχ″(ω)/χ′(ω)=cot(nπ/2) forω>ωp and 0

Energy criterion in interacting cluster systems

The high-frequency fractional power law of relaxation, seen in a wide range of materials, yields a constant ratio of the macroscopic energy lost per radian to the energy stored in the system, in the corresponding frequency range. For almost two decades, the above energy criterion has been supposed to imply the existence of similar microscopic properties which determine the observed power-law exponent. Here, a rigorous formulation of the energy-criterion argument is proposed in the frame of a new probabilistic approach to derive the Havriliak-Negami (HN) and Kohlraush-Williams-Watts (KWW) responses. In this approach the commonly observed macroscopic laws are related to the microscopic scenario of relaxation, and the energy-criterion interpretation is applied to the physical basis of the relation. The presented considerations reinforce the physical significance of the empirically found forms of relaxation, and open a new line of analysis of relaxation phenomena.

The light scattering relaxation function of glass-forming molecules: a general probabilistic approach

Depolarized photon correlation spectra of two fragile glass-forming epoxy molecules of different molecular weights are reported. The correlation function of the simplest molecule shows stretched-exponential behaviour, while deviations from this form have been observed for the larger molecule, suggesting the presence of a second, long-time power law. The relaxation dynamics is suitably described for both systems in terms of a probabilistic approach based on the tool of the limit theorems of probability theory. On the basis of this theory, the presence of the long-time power law can be related to the existence of intermolecular interactions, while the stretched-exponential behaviour of simple molecules can be obtained in the case of negligible long-range interactions.

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.

The frequency-domain relaxation response of gallium doped Cd1-xMnxTe

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.

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.

Relaxation dynamics of the fastest channel in multichannel parallel relaxation mechanism

Abstract Empirical evidence accumulated over the years shows that the time-dependent change of macroscopic properties of physical systems evolving to equilibrium exhibits a great degree of universality. We address the question of the origins of the universal relaxation laws in terms of probabilistic approach based on the multichannel parallel relaxation mechanism. We present a clear stochastic scheme uniquely leading to the whole class of experimentally observed relaxation responses. The proposed scheme results from the common assumption that only the fastest channel contributes to relaxation dynamics.

Frequency-Independent Rules for the Dielectric Susceptibility Derived from Two Forms of Self-Similar Dynamical Behavior of Dipolar Systems

This paper provides the frequency domain analysis of the probabilistic representation of the cluster model for dielectric relaxation in dipolar systems. It is proved that the restriction (0.1) experimentally found for both the powerlaw coefficientsn andm is the necassary and sufficient condition to obtain the low- and high-frequency power-law behavior. Consequently, in both frequency regions the Kramers-Krönig-compatible frequency-independent rules are fulfilled. Moreover, in contrast to the empirical functions proposed to fit the experimental data, the dielectric susceptibility derived from the stochastioc considerations does cover the full range of the observed dielectric responses.