Aleksander Stanislavsky

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.

Twist of fractional oscillations

Using the method of the Laplace transform, we consider fractional oscillations. They are obtained by the time-clock randomization of ordinary harmonic vibrations. In contrast to sine and cosine, the functions describing the fractional oscillations exhibit a finite number of damped oscillations with an algebraic decay. Their fractional differential equation is derived.

Long-term memory contribution as applied to the motion of discrete dynamical systems

We consider the evolution of logistic maps under long-term memory. The memory effects are characterized by one parameter, alpha. If it equals to zero, any memory is absent. This leads to the ordinary discrete dynamical systems. For alpha=1 the memory becomes full, and each subsequent state of the corresponding discrete system accumulates all past states with the same weight just as the ordinary integral of first order does in the continuous space. The case with 0<alpha<1 has the long-term memory effects. The characteristic features are also observed for the fractional integral depending on time, and the parameter alpha is equivalent to the order index of the fractional integral. We study the evolution of the bifurcation diagram among alpha=0 and alpha=0.15. The main result of this work is that the long-term memory effects make difficulties for developing the chaos motion in such logistic maps. The parameter alpha resembles a governing parameter for the bifurcation diagram. For alpha>0.15 the memory effects win over chaos.

Fractional dynamics from the ordinary Langevin equation.

We consider the usual Langevin equation depending on an internal time. This parameter is substituted by a first passage time of a self-similar Markov process. Then the Gaussian process is parent, and the hitting time process is directing. The probability to find the resulting process at the real time is defined by the integral relationship between the probability densities of the parent and directing processes. The corresponding master equation becomes the fractional Fokker-Planck equation. We show that the resulting process has non-Markovian properties, all its moments are finite, the fluctuation-dissipation relation and the H-theorem hold.

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. We derive the corresponding fractional diffusion equation and provide links to the corresponding compound relaxation laws supported by this case generalizing many empirical dependencies well-known in relaxation investigations.

Black-Scholes model under subordination

In this paper, we consider a new mathematical extension of the Black–Scholes (BS) model in which the stochastic time and stock share price evolution is described by two independent random processes. The parent process is Brownian, and the directing process is inverse to the totally skewed, strictly α-stable process. The subordinated process represents the Brownian motion indexed by an independent, continuous and increasing process. This allows us to introduce the long-term memory effects in the classical BS model.

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

In this paper we analyze a coupling between the very large jumps in physical and operational times as applied to anomalous diffusion. The approach is based on subordination of a skewed Lévy-stable process by its inverse to get two types of operational time – the spent and the residual waiting time, respectively. The studied processes have different properties which display both subdiffusive and superdiffusive features of anomalous diffusion underlying the two-power-law relaxation patterns.

Numerical scheme for calculating of the fractional two-power relaxation laws in time-domain of measurements

Abstract A numerical approximation of the three-parameter Mittag-Leffler function, useful for description of the non-exponential relaxation laws, is presented. Our approach is based on the Dirichlet average of the two-parameter Mittag-Leffler function for which the numerical approximation is well developed. We provide estimates for accuracy of the computation.

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.