Stochastic Dynamics of Time Correlations in Complex Systems with Discrete Time

In this paper we present the concept of description of random processes in complex systems with the discrete time. It involves the description of kinetics of discrete processes by means of the chain of finite-difference non-Markov equations for time correlation functions (TCF). We have introduced the dynamic (time dependent) information Shannon entropy $S_i(t)$ where i=0,1,2,3,... as an information measure of stochastic dynamics of time correlation $(i=0)$ and time memory (i=1,2,3,...). The set of functions $S_i(t)$ constitute the quantitative measure of time correlation disorder $(i=0)$ and time memory disorder (i=1,2,3,...) in complex system. Harnessing the infinite set of orthogonal dynamic random variables on a basis of Gram-Shmidt orthogonalization procedure tends to creation of infinite chain of finite-difference non-Markov kinetic equations for discrete TCF and memory function.The solution of the equations above thereof brings to the recurrence relations between the TCF and MF of senior and junior orders. The results obtained offer considerable scope for attack on stochastic dynamics of discrete random processes in a complex systems. Application of this technique on the analysis of stochastic dynamics of RR-intervals from human ECG's shows convincing evidence for a non-Markovian phenomemena associated with a peculiarities in short and long-range scaling. This method may be of use in distinguishing healthy from pathologic data sets based in differences in these non-Markovian properties.Comment: 20 pages, RevTeX, 1 table, 2 eps figure

Relaxation time scales in collective dynamics of liquid alkali metals

In this paper the investigation of the dynamical processes of liquid alkali metals is executed by analyzing the time scales of relaxation processes in liquids. The obtained theoretical dynamic structure factor $S(k,\omega)$ for the case of liquid lithium is found to be in excellent agreement with the recently received inelastic X-ray scattering data. The comparison and interrelation with other theories are given here. Finally, an important part of this paper is the confirmation of the scale uniformity of the dynamic processes in liquid alkali metals predicted by some previous molecular dynamic simulation studies

Universal Approach to Overcoming Nonstationarity, Unsteadiness and Non-Markovity of Stochastic Processes in Complex Systems

In present paper we suggest a new universal approach to study complex systems by microscopic, mesoscopic and macroscopic methods. We discuss new possibilities of extracting information on nonstationarity, unsteadiness and non-Markovity of discrete stochastic processes in complex systems. We consider statistical properties of the fast, intermediate and slow components of the investigated processes in complex systems within the framework of microscopic, mesoscopic and macroscopic approaches separately. Among them theoretical analysis is carried out by means of local noisy time-dependent parameters and the conception of a quasi-Brownian particle (QBP) (mesoscopic approach) as well as the use of wavelet transformation of the initial row time series. As a concrete example we examine the seismic time series data for strong and weak earthquakes in Turkey ($1998,1999$) in detail, as well as technogenic explosions. We propose a new way of possible solution to the problem of forecasting strong earthquakes forecasting. Besides we have found out that an unexpected restoration of the first two local noisy parameters in weak earthquakes and technogenic explosions is determined by exponential law. In this paper we have also carried out the comparison and have discussed the received time dependence of the local parameters for various seismic phenomena

Non-Markov stochastic dynamics of real epidemic process of respiratory infections

The study of social networks and especially of the stochastic dynamics of the diseases spread in human population has recently attracted considerable attention in statistical physics. In this work we present a new statistical method of analyzing the spread of epidemic processes of grippe and acute respiratory track infections (ARTI) by means of the theory of discrete non-Markov stochastic processes. We use the results of our last theory (Phys. Rev. E 65, 046107 (2002)) to study statistical memory effects, long - range correlation and discreteness in real data series, describing the epidemic dynamics of human ARTI infections and grippe. We have carried out the comparative analysis of the data of the two infections (grippe and ARTI) in one of the industrial districts of Kazan, one of the largest cities of Russia. The experimental data are analyzed by the power spectra of the initial time correlation function and the memory functions of junior orders, the phase portraits of the four first dynamic variables, the three first points of the statistical non-Markov parameter and the locally averaged kinetic and relaxation parameters. The received results give an opportunity to provide strict quantitative description of the regular and stochastic components in epidemic dynamics of social networks taking into account their time discreteness and effects of statistical memory. They also allow to reveal the degree of randomness and predictability of the real epidemic process in the specific social network.Comment: 18 pages, 8figs, 1 table

Diffusion Time-Scale Invariance, Markovization Processes and Memory Effects in Lennard-Jones Liquids

We report the results of calculation of diffusion coefficients for Lennard-Jones liquids, based on the idea of time-scale invariance of relaxation processes in liquids. The results were compared with the molecular dynamics data for Lennard-Jones system and a good agreement of our theory with these data over a wide range of densities and temperatures was obtained. By calculations of the non-Markovity parameter we have estimated numerically statistical memory effects of diffusion in detail.Comment: 10 pages, 3 figure

Stratification of the phase clouds and statistical effects of the non-Markovity in chaotic time series of human gait for healthy people and Parkinson patients

In this work we develop a new method of diagnosing the nervous system diseases and a new approach in studying human gait dynamics with the help of the theory of discrete non-Markov random processes. The stratification of the phase clouds and the statistical non-Markov effects in the time series of the dynamics of human gait are considered. We carried out the comparative analysis of the data of four age groups of healthy people: children (from 3 to 10 year olds), teenagers (from 11 to 14 year oulds), young people (from 21 up to 29 year oulds), elderly persons (from 71 to 77 year olds) and Parkinson patients. The full data set are analyzed with the help of the phase portraits of the four dynamic variables, the power spectra of the initial time correlation function and the memory functions of junior orders, the three first points in the spectra of the statistical non-Markov parameter. The received results allow to define the predisposition of the probationers to deflections in the central nervous system caused by Parkinson's disease. We have found out distinct differencies between the five submitted groups. On this basis we offer a new method of diagnostics and forecasting Parkinson's disease.Comment: 15 pages, 5 figs, 3 Table

The study of dynamic singularities of seismic signals by the generalized Langevin equation

Analytically and quantitatively we reveal that the GLE equation, based on a memory function approach, in which memory functions and information measures of statistical memory play fundamental role in determining the thin details of the stochastic behavior of seismic systems, naturally conduces to a description of seismic phenomena in terms of strong and weak memory. Due to a discreteness of seismic signals we use a finite - discrete form of GLE. Here we studied some cases of seismic activities of Earth ground motion in Turkey with consideration of complexity, nonergodicity and fractality of seismic signals.Comment: 12 pages, 7 figures. submitted to "Physica A

A simple measure of memory for dynamical processes described by the generalized Langevin equation

Memory effects are a key feature in the description of the dynamical systems governed by the generalized Langevin equation, which presents an exact reformulation of the equation of motion. A simple measure for the estimation of memory effects is introduced within the framework of this description. Numerical calculations of the suggested measure and the analysis of memory effects are also applied for various model physical systems as well as for the phenomena of ``long time tails'' and anomalous diffusion