Karina Weron

On the Cole-Cole relaxation function and related Mittag-Leffler distribution

In the framework of the one-dimensional fractal time random walk (FTRW) relaxation model, we rigorously show that the frequency domain response takes, in both nonbiased and biased walks, the only possible Cole-Cole form. The underlying reason for this is the specific form of the relaxation function (the survival probability of a relaxing system) determined in this model by the Mittag-Leffler distribution. We provide also analytical formulas for the propagators of the nonbiased and biased FTRWs.

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.

Subordination model of anomalous diffusion leading to the two-power-law relaxation responses

We derive a general pattern of the nonexponential, two-power-law relaxation from the compound subordination theory of random processes applied to anomalous diffusion. The subordination approach is based on a coupling between the very large jumps in physical and operational times. It allows one to govern a scaling for small and large times independently. Here we obtain explicitly the relaxation function, the kinetic equation and the susceptibility expression applicable to the range of experimentally observed power-law exponents which cannot be interpreted by means of the commonly known Havriliak-Negami fitting function. We present a novel two-power relaxation law for this range in a convenient frequency-domain form and show its relationship to the Havriliak-Negami one.

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). Here we obtain explicitly the Fokker-Planck equation and the Cole-Davidson relaxation function. This model includes subdiffusion as a particular case.

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

On the equivalence of the parallel channel and the correlated cluster relaxation models

The question of the origins of nonexponential relaxation is addressed in terms of the probabilistic approach to relaxation. The interconnection between two differently rooted probabilistic models, i.e., between the parallel channel and the correlated cluster models, is presented. We show that clearly different probabilistic origins yield in both approaches a well-defined class of universally valid two-power-law responses with the stretched-exponential and exponential decay laws as special cases. The equivalence of both models indicates that variations in the local environment of the relaxing configurational units (parallel channel relaxation) can provide a basis for self-similar relaxation dynamics without the need for hierarchically constrained dynamics (correlated clusters relaxation).

Probabilistic basis for the Cole-Cole relaxation law

Abstract Dielectric dispersion studies of real ferroelectric crystals in the phase transition region indicate a deviation from the Debye response of pure crystals. In this paper the question of the origins of this deviation is addressed in terms of a probabilistic approach. An effort to understand the relaxation mechanism underlying the universally valid Cole-Cole response is undertaken. A model of relaxation as a self-similar process directly related to the fractal geometry of ferroelectric systems is proposed.

FARIMA MODELING OF SOLAR FLARE ACTIVITY FROM EMPIRICAL TIME SERIES OF SOFT X-RAY SOLAR EMISSION

A time series of soft X-ray emission observed by the Geostationary Operational Environment Satellites from 1974 to 2007 is analyzed. We show that in the solar-maximum periods the energy distribution of soft X-ray solar flares for C, M, and X classes is well described by a fractional autoregressive integrated moving average model with Pareto noise. The model incorporates two effects detected in our empirical studies. One effect is a long-term dependence (long-term memory), and another corresponds to heavy-tailed distributions. The parameters of the model: self-similarity exponent H, tail index α, and memory parameter d are statistically stable enough during the periods 1977-1981, 1988-1992, 1999-2003. However, when the solar activity tends to minimum, the parameters vary. We discuss the possible causes of this evolution and suggest a statistically justified model for predicting the solar flare activity.

Burr, Lévy, Tsallis

The purpose of this short paper dedicated to Prof Constantin Tsallis on his 60th anniversary is to show how the use of mathematical tools and physical concepts introduced by Burr, Levy and Tsallis open a new line of analysis of the old problem of non-Debye decay and universality of relaxation. We also show how a finite characteristic time scale can be expressed in terms of a q-expectation using the concept of q-escort probability. The comparison with the Weron et al. probabilistic theory of relaxation leads to a better understanding of the stochastic properties underlying the Tsallis entropy concept.

Application of dwell-time series in studies of long-range correlation in single channel ion transport: analysis of ion current through a big conductance locust potassium channel

The time series of successive closed- and open-states durations determined for a voltage-dependent big conductance locust potassium channel (BK channel) have been analyzed. The problem of correlation between the states have been discussed by means of the autocorrelation function, Hurst and detrended fluctuation analysis (DFA) as well as surrogate data tools. We showed that the existence of correlation between closed and open times can be detected only by simultaneous application of several different tools. All techniques pointed at a stronger correlation, therefore longer memory, in the series of closed states. Persistent character of both examined dwell-time series has been shown by the Hurst and DFA analyses, whose exponents were found significantly larger than 0.5—a value typical for random walk. It may originate from the externally applied electric field, which directs ion movement and influences channel kinetics. DFA detected two ranges of different correlations in the series of closed states durations. This result points to the existence of two processes with different correlation characteristics which govern the channel functioning. The DFA exponent equal to 0.86±0.07 for long-range scaling of closed times suggests that the long memory of the BK channel action arising from the dichotomous nature of the signal and the long-tail properties of the closed-times distribution can be also governed by the correlation between closed states. The possible biological implications of the findings have been also discussed.