Mariusz Białecki

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.

Algebro-Geometric Solution of the Discrete KP Equation over a Finite Field out of a Hyperelliptic Curve

We transfer the algebro-geometric method of construction of solutions of the discrete KP equation to the finite field case. We emphasize the role of the Jacobian of the underlying algebraic curve in construction of the solutions. We illustrate in detail the procedure on example of a hyperelliptic curve.

Motzkin numbers out of Random Domino Automaton

Abstract Motzkin numbers are derived from a special case of Random Domino Automaton – recently proposed a slowly driven system being a stochastic toy model of earthquakes. It is also a generalisation of 1D Drossel–Schwabl forest-fire model. A solution of the set of equations describing stationary state of Random Domino Automaton in inverse-power case is presented. A link with Motzkin numbers allows to present explicit form of asymptotic behaviour of the automaton.

The discrete KP and KDV equations over finite fields

We propose an algebro-geometric method for constructing solutions of the discrete KP equation over a finite field. We also perform the corresponding reduction to the finite-field version of the discrete KdV equation. We write formulas that allow constructing multisoliton solutions of the equations starting from vacuum wave functions on an arbitrary nonsingular curve.

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.

From statistics of avalanches to microscopic dynamics parameters in a toy model of earthquakes

A toy model of earthquakes — Random Domino Automaton — is investigated in its finite version. A procedure of reconstruction of intrinsic dynamical parameters of the model from produced statistics of avalanches is presented. Examples of exponential, inverse-power and M-shape distributions of avalanches illustrate remarkable flexibility of the model as well as the efficiency of proposed reconstruction procedure.

Properties of a Finite Stochastic Cellular Automaton Toy Model of Earthquakes

Finite version of Random Domino Automaton — a recently proposed toy model of earthquakes — is investigated in detail. Respective set of equations describing stationary state of the FRDA is derived and compared with infinite case. It is shown that for a system of large size, these equations are coincident with RDA equations. We demonstrate a non-existence of exact equations for size N ≥ 5 and propose appropriate approximations, the quality of which is studied in examples obtained within the framework of Markov chains.We derive several exact formulas describing properties of the automaton, including time aspects. In particular, a way to achieve a quasi-periodic like behaviour of RDA is presented. Thus, based on the same microscopic rule — which produces exponential and inverse-power like distributions – we extend applicability of the model to quasi-periodic phenomena.

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.