Three-level description of the domino cellular automaton

  title={Three-level description of the domino cellular automaton},
  author={Zbigniew Czechowski and Mariusz Białecki},
  journal={Journal of Physics A: Mathematical and Theoretical},
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… 

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

Bi-SOC-states in one-dimensional random cellular automaton.

Two statistically stationary states with power-law scaling of avalanches are found in a simple 1 D cellular automaton and the migration of states of the automaton between these two self-organized criticality states is demonstrated during evolution of the system in computer simulations.

Properties of a Finite Stochastic Cellular Automaton Toy Model of Earthquakes

A way to achieve a quasi- periodic like behaviour of RDA is presented and applicability of the model is extended to quasi-periodic phenomena, based on the same microscopic rule which produces exponential and inverse-power like distributions.

Finite Random Domino Automaton

A way to achieve a quasi-periodic like behaviour of RDA is presented and based on the same microscopic rule - which produces exponential and inverse-power like distributions - it is extended applicability of the model to quasi- periodic phenomena.

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

A role of RDA for modeling time series with Ito equation is emphasized and transparent structures of the automaton and their relations to seismology, statistical physics and combinatorics are emphasized.

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

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.

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

Catalan numbers out of a stochastic cellular automaton

  • M. Białecki
  • Computer Science
    Journal of Mathematical Physics
  • 2019
A stochastic cellular automaton is defined, whose stationary state is characterized by Catalan numbers’ recurrence, which provides a new interpretation of Catalan numbers in terms of stochastically discrete dynamical systems.

Cellular automata to describe seismicity: A review

Cellular Automata have been used in the literature to describe seismicity. We first historically introduce Cellular Automata and provide some important definitions. Then we proceed to review the most



Exact results for the one-dimensional self-organized critical forest-fire model.

The analytic solution of the self-organized critical forest-fire model is presented in one dimension proving SOC in systems without conservation laws by analytic means and the critical exponent describing the size distribution of forest clusters is exactly 2.

Exactly solved model of self-organized critical phenomena.

This work defines a variant of the model of Bak, Tang, and Wiesenfeld of self-organized critial behavior by introducing a preferred direction and characterize the critical state and determines the critical exponents exactly in arbitrary dimension d.

Exactly solvable sandpile with fractal avalanching.

A simple one-dimensional sandpile model is constructed which possesses exact analytical solvability while displaying both scale-free behavior and fractal properties. The sandpile grows by avalanching

Exact solution of a deterministic sandpile model in one dimension.

We present an exact solution of a one-dimensional sandpile model for which sand is dropped along the wall and N=2 grains of sand fall over the neighboring downhill sites when the critical slope is

The privilege as the cause of power distributions in geophysics

SUMMARY Power laws are known to be associated with dynamic systems residing near the critical point in the state space of the system. However, both models, that of phase transitions reached by the

Height correlations in the Abelian sandpile model

The authors study the distribution of heights in the self-organized critical state of the Abelian sandpile model on a d-dimensional hypercubic lattice. They calculate analytically the concentration


The authors consider a variant of directed percolation in which the set of nodes is itself dependent on the progress of the percolation process. For the mean-field model studied, the system is driven

On the reconstruction of Ito models on the basis of time series with long-tail distributions

Two methods of reconstruction of Ito equations on the basis of time series were analysed. The Sequin method appeared to be completely inadequate in cases of considerable noise. The histogram method

Theory of the one-dimensional forest-fire model.

  • PaczuskiBak
  • Physics, Environmental Science
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1993
Turbulent cascade processes are studied in terms of a one-dimensional forest-fire model and the exact hole distribution function is found to be [ital N][sub [ital H]]([ital s])=4[ital N]/[[ital s]([ital s]+1)( [ital s]-2)], where [ital S] is the number of forests.