abstract

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by distributed-order equations. In the present paper we consider different forms of distributed-order fractional kinetic equations and investigate the effects described by different classes of such equations. In particular, the equations describing accelerating and decelerating subdiffusion, as well as those describing accelerating and decelerating superdiffusion are presented.

abstract

We employ an ergodic theory argument to demonstrate the foundations of ubiquity of Lévy stable self-similar processes in physics and present a class of models for anomalous and nonextensive diffusion. A relationship between stationary and self-similar models is clarified. The presented stochastic integral description of all Lévy stable processes could provide new insights into the mechanism underlying a range of self-similar natural phenomena. Finally, this effect is illustrated by self-similar approach to financial modelling.

abstract

We show a procedure for determining the class of diffusion in systems governed by a generalized Langevin equation with memory. The analysis holds for one-dimensional systems. We provide a simple answer for the diffusive exponent and its relation with noise and memory. We discuss as well limits for mixing, ergodicity and of the fluctuation–dissipation theorem.

abstract

Using a numerical library for arbitrary precision arithmetic I study the irregular dependence of the diffusion coefficient on the slope of a piecewise linear map defining a dynamical system. I find that the graph of the diffusion coefficient as a function of the slope has the fractal dimension 1, but the convergence to this limit is slowed down by logarithmic corrections. The exponent controlling this correction depends on the slope and is either 1 or 2 depending on existence and properties of a Markov partition.

abstract

A novel approach to the Tsallis’ superstatistics is discussed. On the basis of limit theorems of probability theory we have shown that the Tsallis generalization of the classical Boltzmann–Gibbs statistics can be represented by a distribution of an appropriately constructed scaled minima of a random variables sequence. This formalism provides a natural framework of construction of even more generalized statistics in which the Tsallis and the Boltzmann–Gibbs ones are special cases. It leads also to a new interpretation of the entropic index \(q\).

abstract

We study effects of noisy and deterministic perturbations on oscillatory solutions to delay differential equations. We develop the multiscale technique and derive amplitude equations for noisy oscillations near a critical delay. We investigate effects of additive and multiplicative noise. We show that if the magnitudes of noise and deterministic perturbations are balanced, then the oscillatory behavior persists for long times being sustained by the noise. We illustrate the technique and its results on linear and logistic delay equations.

abstract

We consider a Brownian particle whose motion is confined to a “meandering” pathway and which is driven away from thermal equilibrium by an alternating external force. This system exhibits absolute negative mobility, i.e. when an external static force is applied the particle moves in the direction opposite to that force. We reveal the physical mechanism behind this “donkey-like” behavior, and derive analytical approximations that are in excellent agreement with numerical results.

abstract

The noise can stabilize a fluctuating or a periodically driven metastable state in such a way that the system remains in this state for a longer time than in the absence of white noise. This is the noise enhanced stability phenomenon, observed experimentally and numerically in different physical systems. After shortly reviewing all the physical systems where the phenomenon was observed, the theoretical approaches used to explain the effect are presented. Specifically the conditions to observe the effect in systems: (a) with periodical driving force, and (b) with random dichotomous driving force, are discussed. In case (b) we review the analytical results concerning the mean first passage time and the nonlinear relaxation time as a function of the white noise intensity, the parameters of the potential barrier, and of the dichotomous noise.

abstract

We generalize recent studies of particle transport to the case of quasiperiodic potentials with quasiperiodic driving. We obtain the relevant set of space-time symmetries and the way of violating them in order to obtain directed transport. Numerical results confirm the predicted rectification for the dissipationless case.

abstract

The exact formulae for spectra of equilibrium diffusion in a fixed bistable piecewise linear potential and in a randomly flipping monostable potential are derived. Our results are valid for arbitrary intensity of driving white Gaussian noise and arbitrary parameters of potential profiles. We find: (i) an exponentially rapid narrowing of the spectrum with increasing height of the potential barrier, for fixed bistable potential; (ii) a nonlinear phenomenon, which manifests in the narrowing of the spectrum with increasing mean rate of flippings, and (iii) a nonmonotonic behaviour of the spectrum at zero frequency, as a function of the mean rate of switchings, for randomly switching potential. The last feature is a new characterisation of resonant activation phenomenon.

abstract

A diffusion equation with a functional drift (generalized Smoluchowski equation) has been derived from the clannish random walk (nonlinear discrete master equation) for both probability density and velocity fields, in case of 1D. A relation between Burgers and generalized Smoluchowski equations as well as between concentration and velocity fields, has been discussed.

abstract

We report on a method to segregate granular mixtures which consist of two kinds of particles by an oscillating “ratchet” mechanism. The segregation system has an asymmetrical sawtooth-shaped base which is vertically oscillating. Such a ratchet base produces a directional current of particles owing to its transport property. It is a counterintuitive and interesting phenomenon that a vertically vibrated base transports particles horizontally. This system is studied with numerical simulations, and it is found that we can apply such a system to segregation of mixtures of particles with different properties (radius or mass). Furthermore, we find out that an appropriate inclination of the ratchet-base makes the quality of segregation high.

abstract

We analyze the spatio-temporal patterns of two competing species in the presence of two white noise sources: an additive noise acting on the interaction parameter and a multiplicative noise which affects directly the dynamics of the species densities. We use a coupled map lattice (CML) with uniform initial conditions. We find a nonmonotonic behavior both of the pattern formation and the density correlation as a function of the multiplicative noise intensity.

abstract

The noise-induced pattern formation in a population dynamical model of three interacting species in the coexistence regime is investigated. A coupled map lattice of Lotka–Volterra equations in the presence of multiplicative noise is used to analyze the spatiotemporal evolution. The spatial correlation of the species concentration as a function of time and of the noise intensity is investigated. A nonmonotonic behavior of the area of the patterns as a function of both noise intensity and evolution time is found.

abstract

A concept of “topological” atom–atom neighborhood relation in a strongly fluctuating solid is introduced. The divergence of metrical and topological definitions of a cluster of atoms for a sufficiently high level of atom’s displacement \(\xi \gt \xi _{\rm tr}\), and its consequences for an analysis of local structure in locally solid-like ordered liquids are discussed. The threshold amplitude \(\xi _{\rm tr}\) is calculated for a two-dimensional (2D) close-packed lattice. The Monte Carlo simulations of a 2D system of Lennard–Jones atoms lead to a hypothesis, closely related to Lindemann’s melting criterion: melting occurs for \(\xi = \xi _{\rm m} \simeq \xi _{\rm tr}\), i.e. when metrical and topological approaches diverge.

abstract

We study the nonequilibrium spatial correlations of the density fluctuations for a selected reagent as well as the spatial correlations of its energetic states in a stationary state of a model chemical system. In the considered model the reagent of interest appears as a product of a binary process and vanishes as the result of a fast unimolecular decay. We present analytical formulae for all correlation functions for the case when the lifetime of the reagent is short enough to treat the motion of its molecules as a ballistic rather than a diffusive one. We demonstrate that the results obtained from these formulae agree well with correlations directly measured in molecular dynamics simulations.

abstract

Lacunarity can work as a supplement for describing texture of self-similar objects. We propose a systematic study of this measure based on a set of carefully designed prototype structures, providing also a comparision to already existing measures (generalized fractal dimension). We also illustrate its applications to material science (describing changes in polymer surface during gold dispersion) and cellular biology (describing stains in cancer cells from two cell lines in two conditions). To measure lacunarity the gliding box method was used.

abstract

A model of linear and star-branched polymer chains confined between two parallel and impenetrable surfaces was built. The polymer chains were restricted to a simple cubic lattice. Two macromolecular architectures of the chain: linear and star branched (consisted of \(f\) = 3 branches of equal length) were studied. The excluded volume was the only potential introduced into the model (the athermal system). Monte Carlo simulations were carried out using a sampling algorithm based on chain’s local changes of conformation. The simulations were carried out at different confinement conditions: from light to high chain’s compression. The scaling of chain’s size with the chain length was studied and discussed. The influence of the confinement and the macromolecular architecture on the shape of a chain was studied. The differences in the shape of linear and star-branched chains were pointed out.

abstract

We studied a simplified model of a polymer brush. It consisted of star-branched chains, which were restricted to a simple cubic lattice. Each star-branched macromolecule consisted of three linear arms of equal length emanating from a common origin (the branching point). The chains were grafted to an impenetrable surface, i.e. they were terminally attached to the surface with one arm. The number of chains was varied from low to high grafting density. The model system was studied at good solvent conditions because the excluded volume effect was the only potential of interaction included in the model. The properties of this model system were studied by means of Monte Carlo simulation. The sampling algorithm was based on local changes of chain conformations. The dynamic properties of the polymer brush were studied and correlated with its structure. The differences in relaxation times of particular star arms were shown. The short-time mobility of polymer layers was analyzed. The lateral self-diffusion of chains was also studied and discussed.

abstract

Experimental data of neutron super monitors, solar wind velocity and components of the interplanetary magnetic field (IMF) have been used to study a relationship between the temporal changes of the energy spectrum of the Forbush effects of galactic cosmic rays (GCR) and the power spectral density (PSD) of the IMF’s strength fluctuations. Based on the energy spectrum of the Forbush effects of GCR a structure of the IMF’s fluctuations is determined in the disturbed vicinity of the interplanetary space when the direct (in situ) measurements of the IMF are absent. In order to study anisotropic diffusion propagation of GCR a second order four dimensional Fokker–Plank’s type partial differential equation has been numerically solved. Diffusion, convection, drift due to the regular component of the IMF and adiabatic energy changes of the GCR particles because of the interaction with the diverged solar wind inhomogeneities are included in the transport equation. The spatial distributions of the density, radial, heliolatitudinal and heliolongitudinal gradients during the Forbush effect of GCR intensity have been found for the positive \((qA \gt 0)\) period of solar magnetic cycle. It is shown that a stationary diffusion–convection–drift approximation of GCR transport is an acceptable model for describing the recurrent Forbush effects of GCR associated with the established corotating disturbances in the inner heliosphere.

abstract

Data of neutron monitors for different solar magnetic cycles have been used to study the role of the regular and stochastic components of the interplanetary magnetic field (IMF) on the diffusion propagation of galactic cosmic rays (GCR). Two classes of GCR variations are considered. The first one is the 11-year variation of GCR related with the similar periodic changes in solar activity and solar wind parameters; the second one is the quasi-periodic 27-day variation stipulated by the heliolongitudinal asymmetry of the electro-magnetic conditions (e.g. solar wind velocity and diffusion) in the inner heliosphere caused by the Sun’s rotation. Transport equation of GCR particles has been numerically solved for two and three dimensional IMF including diffusion, convection, drift due to gradient and curvature of the regular IMF and the energy change of GCR particles because of interaction with the irregularities of solar wind. It is shown that a significant changes in the structure of the IMF’s irregularities from the minima to the maxima epochs of solar activity reflecting in the dependences of the diffusion coefficient on the GCR particles’ rigidity is one of the general reasons of the 11-year variation of GCR. The heliolongitudinal asymmetries of the solar wind velocity and diffusion processes in the inner heliosphere cause the GCR 27-day variation with the larger amplitude in the minima and near minima epochs of the \(qA\gt 0\) solar magnetic cycle, than that in the \(qA\lt 0\) cycles due to existence of the oppositely directed drift streams of GCR. An interpretation of this phenomenon has been proposed based on the modern theory of GCR propagation.