abstract

We analyze the diffusive transport of Brownian particles in narrow channels with periodically varying cross-section. The geometrical confinements lead to entropic barriers, the particle have to overcome in order to proceed in transport direction. The transport characteristics exhibit peculiar behaviors which are in contrast to what is observed for the transport in potentials with purely energetic barriers. By adjusting the geometric parameters of the channel one can effectively tune the transport and diffusion properties. A prominent example is the maximized enhancement of diffusion for particular channel parameters. The understanding of the role of channel-shape provides the possibility for a design of stylized channels wherein the quality of the transport can be efficiently optimized.

abstract

We review various features of the statistics of random paths on graphs. The relationship between path statistics and Quantum Mechanics (QM) leads to two canonical ways of defining random walk on a graph, which have different statistics and hence different entropies. Generic random walk (GRW) is in correspondence with the field-theoretical formalism, whereas maximal entropy random walk (MERW), introduced by us in a recent work, is motivated by the Feynman path-integral formulation of QM. GRW maximizes entropy locally (neighbors are chosen with equal probabilities), in contrast to MERW which does so globally (all paths of given length and endpoints are equally probable). The stationary distribution for MERW is given by the ground state of a quantum-mechanical problem where nodes whose degree is smaller than average act as repulsive impurities. We investigate static and dynamical properties GRW and MERW in a variety of examples in one and two dimensions. The most spectacular difference arises in the case of weakly diluted lattices, where a particle performing MERW gets eventually trapped in the largest nearly spherical region which is free of impurities. We put forward a quantitative explanation of this localization effect in terms of a classical Lifshitz phenomenon.

abstract

We consider boundary value problems for a subdiffusion-degradation problem within an approach based on reaction-subdiffusion equations derived earlier. Thus, boundary value problems for subdiffusion with degradation can be solved within the Green functions approach, where the new Green functions have the structure of a product of the Green functions for mere subdiffusion with an exponential. Although the equations with non-decoupling reaction and transport appear complicated at first glance, it turns out that the methods for the solution of the corresponding linear reaction-diffusion equations can easily be adopted to the anomalous case. In particular, the solutions can be expressed in terms of solutions to the problem without degradation.

abstract

We show that a turnover from the classical Debye to the two-power-law relaxation behavior, observed in the majority of physical systems, is associated with a new type of a coupled memory continuous-time random walk driving a fractional dynamics. We derive a general class of the two-power-law relaxation responses which is able to reproduce all of the observed relaxation patterns, given by the low- and high-frequency power-law exponents falling in the range (0,1].

abstract

The ergometric theory of the ergodic hypothesis is a physical theory for studying ergodicity in \(quantum\) many-body systems. It is based on the recurrence relations method, an exact dynamical formalism, well established and applied to numerous models both classical and quantum mechanical. In this work we show that the frequency spectra, in particular the distributions of the frequency, have properties which appear equivalent to the invariant measure and transitivity of phase space in classical ergodicity. We also show that the ergometric statement of the ergodic hypothesis can be reduced to Boltzmann’s statement cast in microcanonical ensembles. This reduction indicates that the ergometric theory is a general theory of the ergodic hypothesis.

abstract

Detection method of long-range dependences based on multifractal analysis in time series is proposed. A short description of multifractal analysis and estimator construction (based on Multifractal Detrended Fluctuation Analysis) are given. This method gives accurate results when applied to large scale analysis of fractional Brownian motions (fBm) and describes consistently the mixture of two fBm processes. Finally, this method has been applied to series corresponding \(K^{\,+}\) ionic current through the cellular membrane. For times shorter than 1 sec, a similarity between ionic \(K^{\,+}\) current and the mixture of two antipersistent processes has been found with self-similarity parameter of this mixture less than 0.30.

abstract

We study the effects of the coloured noise on the dynamics of three interacting species, namely two preys and one predator, in a two-dimensional lattice with \(N\) sites. The three species are affected by multiplicative time correlated noise, which accounts for the effects of environment on the species evolution. Moreover, the interaction parameter between the two preys is a dichotomous stochastic process, which determines two dynamical regimes corresponding to different biological conditions. Preliminarily, we study the noise effect on the three species dynamics in single site. Then, we use a mean field approach to obtain, in Gaussian approximation, the moment equations for the species densities. We find that the multiplicative noise does not affect the time behaviour of the 1st order moments. Conversely, the 2nd order moments are strongly dependent both on the intensity and correlation time of the multiplicative noise. Finally, we compare our results with those obtained from a discrete time approach based on a model of coupled map lattice.

abstract

An introduction to the old controversy about a relativistic transformation of thermodynamical quantities is presented from a personal point of view. New formulas, derived for an ideal gas, are advocated on the basis of Clausius–Caratheodory axiomatic thermodynamics.

abstract

The idea of using a mathematical model to describe the behaviour of a physical phenomenon is well established, but in many problems we have to consider a time-dependent phenomenon for which it is not possible to write a deterministic model; nevertheless, it may be possible to derive a stochastic model. The models for time series that are needed for example to achieve optimal forecasting and control are in fact stochastic models, but the choice of a proper model is never straightforward. In particular, this paper is concerned with the problem of forecasting a time series that possibly exhibits long-memory features. It results that the fractionally integrated ARMA processes may provide an adequate representation of the actual process, but do not yield a satisfactory forecasting performance.

abstract

Diffusion of an overdamped Brownian particle on a symmetric periodic substrate is investigated in the presence of pulsated perturbations of two kinds: (i) stepwise lateral displacements (flashing substrate), and (ii) instantaneous tilts (shot noise). Pulses are applied in either periodic or random sequences with assigned mean (bias) and average waiting time (time constant). At zero bias, the diffusion coefficient of the particle can be greatly enhanced by tuning the time constant. Such a diffusion resonance should not be mistaken for the excess diffusion peaks earlier reported for finite biases.

abstract

In the present work I try to develop a model that predicts the development time of a newborn that suffers a development delay. The paper should be helpful for parents, who apply therapy to their children and often need to know how long will it take to complete the development.

abstract

A model of cardiorespiratory synchronization, i.e. the synchronization between the heart rate and the breathing rate, is proposed. From a mathematical point of view it is a delayed nonlinear system — a pair of pulse coupled rotators. The model is studied theoretically and numerically. Classification of synchronization states is supplied by theoretical analysis and verified by numerical experiment. The constitutive role of phase resetting and refraction in synchronization is discussed. As the model is minimalistic it is discussed how well can it mimic the original physiological phenomena.

abstract

We study a motion of an anomalous random walker on finite intervals restricted by two absorbing boundaries. The competition between anomalously long jumps and long waiting times leads to a very general kind of behavior. Trapping events distributed according to the power-law distribution result in occurrence of the Mittag–Leffler decay pattern which in turn is responsible for universal asymptotic properties of escape kinetics. The presence of long jumps which can be distributed according to non-symmetric heavy tailed distributions does not affect asymptotic properties of the survival probability. Therefore, the probability of finding a random walker within a domain of motion decays asymptotically according to the universal pattern derived from the Mittag–Leffler function, which describes decay of single modes in subdiffusive dynamics.

abstract

We study bond and site percolation in four real social networks: two Internet society of friends consisting of over \(10^6\) and \(10^7\) people, over \(10^6\) users of music community website and over \(5 \times 10^6\) users of gamers community server. We study the properties of those systems (e.g. the network components size distribution) in function of fraction \(p\) of nodes or links that retained in network. We have calculated critical fraction \(p_{\rm c}\) at which the percolation transition takes place and giant component emerges.

abstract

This paper presents results of a research conducted on a simple model of a peer-to-peer network (a network in which users exchange files directly, without any central server involved). The conditions necessary for the file exchange process to be efficient and stable are investigated through numerical simulations and analytical calculations based on the master equation. Ways of preventing free-riding (selfish behavior, when users download files without sharing them) are also discussed.

abstract

In this paper we discuss subdiffusive mechanism for the description of some stock markets. We analyse the fractional Black–Scholes model in which the price of the underlying instrument evolves according to the subdiffusive geometric Brownian motion. We show how to efficiently estimate the parameters for the subdiffusive Black–Scholes formula i.e. parameter \(\alpha \) responsible for distribution of length of constant stock prices periods and \(\sigma \) — volatility parameter. A simple method how to price subdiffusive European call and put options by using Monte Carlo approach is presented.

abstract

A simulative study of cancer growth dynamics in patients affected by Chronic Myeloid Leukemia (CML), under the effect of a targeted dose-dependent continuous or pulsed therapy, is presented. We have developed a model for the dynamics of CML in which the stochastic evolution of white blood cell populations are simulated by adopting a Monte Carlo approach. Several scenarios in the evolutionary dynamics of white blood cells, as a consequence of the efficacy of the different modelled therapies, pulsed or continuous, are described. The best results, in terms of a permanent disappearance of the leukemic phenotype, are achieved with a continuous therapy and higher dosage. However, our findings demonstrate that an intermittent therapy could represent a valid choice in patients with high risk of toxicity, when a long-term therapy is considered. A suitably tuned pulsed therapy can enhance the treatment efficacy and reduce the percentage of patients developing resistance.

abstract

The influence of temperature and transport conditions on the electron spin relaxation in lightly doped n-type GaAs semiconductors is investigated. A Monte Carlo approach is used to simulate electron transport, including the evolution of spin polarization and relaxation, by taking into account intravalley and intervalley scattering phenomena of the hot electrons in the medium. Spin relaxation lengths and times are computed through the D’yakonov–Perel process, which is the more relevant spin relaxation mechanism in the regime of interest (\(10\lt T\lt 300\) K). The decay of the initial spin polarization of the conduction electrons is calculated as a function of the distance in the presence of a static electric field varying in the range 0.1–2 kV/cm. We find that the electron spin depolarization lengths and times have a nonmonotonic dependence on both the lattice temperature and the electric field amplitude.

abstract

Molecular engines are nano-scale machines operating under far-from-equilibrium conditions. In living cells they transform chemical energy into mechanical work while acting under randomly fluctuating forces in the form of thermal noise. In this paper we discuss a previously introduced model of a stepper motor [M. Żabicki, E. Gudowska-Nowak, W. Ebeling, Chem. Phys. in press], which is able to drive the system uphill at the cost of the energy inflow from an energy reservoir. The efficiency of the motor is defined as the ratio of the power exerted in the uphill motion with respect to the energy influx from the depot. We analyze the efficiency of this system by adapting the motor model in which only the internal motion includes inertia, whereas the motion of its center of mass becomes overdamped. Based on the numerical simulations of the center of mass trajectories and analysis of directed fluxes of motor particles moving along a one-dimensional track, we derive thermodynamic estimates of the motor efficiency as a function of the opposing force.