Zbigniew Czechowski

On One-to-One Dependence of Rebound Parameters on Statistics of Clusters: Exponential and Inverse-Power Distributions Out of Random Domino Automaton

We introduce the stochastic domino cellular automaton model exhibiting avalanches. Depending of the choice of the parameters, the model covers wide range of properties: various types of exponential and long tail (up to inverse-power) distributions of avalanches are observed. The stationary state of automaton is described by a set of nonlinear discrete equations derived in an exact way from elementary combinatorial arguments. These equations allow to derive formulas explaining both various exponential and inverse power distributions relating them to values of the parameters. The exact relations between the state variable of the model (moments) are derived in two ways: from direct arguments and from the set of equations. Excellent agreement of the obtained analytical results with numerical simulations is observed.The stochastic cellular automaton — a slowly driven system with avalanches — is proposed and an explicit one-to-one dependence between rebound parameters and statistics of clusters is investigated. Depending on the choice of the parameters determining the rule of the automaton, the model covers a wide range of properties: various types of exponential and long-tail (up to inverse-power) distributions of avalanches are observed. The stationary state of automaton is described by a set of nonlinear discrete equations derived from elementary combinatorial arguments. Formulas for average size of clusters and avalanches as well as for moments of arbitrary order are presented. Application of the automaton as a toy model of earthquakes and forest fires is presented. Utility as a test model for Ito equation is also pointed out.

Three-level description of the domino cellular automaton

Motivated by the approach of kinetic theory of gases, a three-level description (microscopic, mesoscopic and macroscopic) of cellular automaton is presented. To provide an analytical treatment, a simple domino cellular automaton with avalanches is constructed. Formulas concerning exact relations for density, clusters, avalanches and other parameters in a stationary state are derived. Some relations approximately valid for deviations from the stationary state are found, and the adequate Ito equation is constructed. It provides the time evolution description of the density on the macroscopic level. The results support an idea of application of the Ito equation to some natural time series.

Ito equations out of domino cellular automaton with efficiency parameters

Ito equations are derived for simple stochastic cellular automaton with parameters describing efficiencies for avalanche triggering and cell occupation. Analytical results are compared with the numerical one obtained from the histogram method. Good agreement for various parameters supports the wide applicability of the Ito equation as a macroscopic model of some cellular automata and complex natural phenomena which manifest random energy release. Also, the paper is an example of effectiveness of histogram procedure as an adequate method of nonlinear modeling of time series.

The construction of an Ito model for geoelectrical signals

The Ito stochastic differential equation governs the one-dimensional diffusive Markov process. Geoelectrical signals measured in seismic areas can be considered as the result of competitive and collective interactions among system elements. The Ito equation may constitute a good macroscopic model of such a phenomenon in which microscopic interactions are adequately averaged. The present study shows how to construct an Ito model for a geoelectrical time series measured in a seismic area of southern Italy. Our results reveal that the Ito model describes the whole time series quite well, but it performs better when one considers fragments of the data set with lower variability range (absent or rare large fluctuations). Our findings show that generally detrended geoelectrical time series can be considered as approximations of Markov diffusion processes.

On a Simple Stochastic Cellular Automaton with Avalanches: Simulation and Analytical Results

Cellular automata (CA) models are widely used in many natural and human sciences. The rule that defines CA, which may be very simple, can lead to a very complicated evolution of a system and rich structure of produced patterns. It often comes from nonlinearity present in the system. The rule of the model encodes the crucial features of the phenomenon under investigation. It contains the information about the behaviour of the automaton and usually is suggestive (convincing) reference point for explanations of its properties. Instead of equations, the rule often plays a central role in description of automata. CA are also convenient and hence attractive tools for making computer simulations; being completely discrete, in principle, CA do not require any approximation procedure for machine implementation.

On reconstruction of the Ito-like equation from persistent time series

The Langevin equation with finite-range persistence was introduced as a macroscopic model of various geophysical phenomena. The modified histogram procedure (MHP) of reconstruction of the equation from time series was proposed. An efficiency of MHP was tested on artificial persistent time series (with short and long-tail distributions) generated by different Ito-like equations. For an exemplary geophysical time series, the appropriate Ito-like equation was reconstructed.

Multifractal analysis of time series generated by discrete Ito equations.

In this study, we show that discrete Ito equations with short-tail Gaussian marginal distribution function generate multifractal time series. The multifractality is due to the nonlinear correlations, which are hidden in Markov processes and are generated by the interrelation between the drift and the multiplicative stochastic forces in the Ito equation. A link between the range of the generalized Hurst exponents and the mean of the squares of all averaged net forces is suggested.

Random Domino Automaton: Modeling Macroscopic Properties by Means of Microscopic Rules

A stochastic cellular automaton called Random Domino Automaton (RDA) was set up to model basic properties of earthquakes. This review presents definition of the automaton as well as investigates its properties in detail. It emphasizes transparent structures of the automaton and their relations to seismology, statistical physics and combinatorics. In particular, a role of RDA for modeling time series with Ito equation is emphasized.

Detrended fluctuation analysis of the Ornstein-Uhlenbeck process: Stationarity versus nonstationarity

The stationary/nonstationary regimes of time series generated by the discrete version of the Ornstein-Uhlenbeck equation are studied by using the detrended fluctuation analysis. Our findings point out to the prevalence of the drift parameter in determining the crossover time between the nonstationary and stationary regimes. The fluctuation functions coincide in the nonstationary regime for a constant diffusion parameter, and in the stationary regime for a constant ratio between the drift and diffusion stochastic forces. In the generalized Ornstein-Uhlenbeck equations, the Hurst exponent H influences the crossover time that increases with the decrease of H.

On Microscopic Mechanisms Which Elongate the Tail of Cluster Size Distributions: An Example of Random Domino Automaton

On the basis of simple cellular automaton, the microscopic mechanisms, which can be responsible for elongation of tails of cluster size distributions, were analyzed. It was shown that only the appropriate forms of rebound function can lead to inverse power tails if densities of the grid are small or moderate. For big densities, correlations between clusters become significant and lead to elongation of tails and flattening of the distribution to a straight line in log–log scale. The microscopic mechanism, given by the rebound function, included in simple 1D RDA can be projected on the geometric mechanism, which favours larger clusters in 2D RDA.