Stochastic cellular automata simulation of oscillations and autowaves in reaction-diffusion systems

Abstract

In this paper, the investigations on stochastic cellular automata models of stable oscillations and autowaves in active media are summarized to formulate a concept of stochastic cellular automaton (CA) corresponding to asynchronous CAs with probabilistic transition rules. A formal description of a stochastic CA and a stochastic CA model is given. The properties and methods of synthesis of CA models for a specified set of elementary physical and chemical transformations are described. Some possibilities of simulating autowave and oscillatory processes are demonstrated using, as an example, a carbon monoxide oxidation reaction on a platinum catalyst with surface restructuring. The CA-simulation has made it possible to reveal the range of reaction parameter values with stable oscillations of reagent concentrations and to observe autowaves on the platinum surface. Special attention is given to providing a highly efficient parallel implementation of the stochastic CA algorithm. For this, the asynchronous mode must be preliminarily transformed to a block-synchronous one. Its equivalence to the asynchronous mode is justified. This is done for the CA reaction model by a comparative statistical analysis of the simulation results.

Parallelization efficiency versus stochasticity in simulation reaction–diffusion by cellular automata

“…Asynchronous stochastic mode Φ α,γ is an idealization of maximally randomized representation of particles behavior on micro-level. In chemistry [4][5][6] such a model is called " Monte-Carlo simulation method", in [12] this model forms a class of stochastic CA. Global operator of such a CA is computed as follows:…”

“…The synchronization is induced [16] by transforming asynchronous mode into an equivalent block-synchronous one. In [12] it is shown on a real life example of a heterogeneous catalytic simulation, that such a transformation induces no errors in asynchronous CA behavior. This fact provokes the following question: how much of synchronization may be admissible in reaction-diffusion CA.…”

“…Whereas there exists some experience in ACA simulation [12][13][14], there is no systematic methodology for synthesize a model of a complex phenomenon given by a description of involved elementary actions and their interactions. The main problem is in constructing a global operator given elementary local functions, which provides the best adequacy to the process under simulation on one hand, and satisfies computational correctness and efficiency conditions on the other hand.…”

“…Actually, all of them may be considered as asynchronous cellular automata [11], simulating phenomena which are composed of movements and transformations [12] of certain kind of micro entities, such as molecules or nano particles either real or abstract. Accordingly, the model should consist of a number of simple local operators being applied to randomly chosen sites of the discrete space.…”

Simulation Performance Versus Stochasticity in Large-Scale Cellular Automata Models

“…Among the existing computer simulation methodologies of reaction-diffusion phenomena completely or partly based on the above statements, the following methods are most known: kinetic Monte Carlo methods in chemistry [2,3] and in microelectronics [4], probabilistic cellular automata in material science [5][6][7], self-organizing CA in biology [8,9]. Actually, all of them may be considered as asynchronous cellular automata [10] simulating phenomena which are composed of movements and transformations [11] of a certain kind of micro-entities, such as molecules or nano particles either real or abstract. Accordingly, a model should consist of a number of simple local operators being applied to randomly chosen sites of discrete space.…”