Finite Random Domino Automaton

TL;DR

This paper investigates the finite version of the Random Domino Automaton, deriving equations for its stationary state, comparing with the infinite case, and proposing approximations for larger systems, extending its applicability to quasi-periodic phenomena.

Contribution

It introduces the finite Random Domino Automaton, derives its stationary state equations, compares them with the infinite case, and proposes approximations for larger systems, extending the model's applicability.

Findings

Equations for large systems coincide with RDA equations.

Exact equations do not exist for size N > 4, approximations are proposed.

A method to achieve quasi-periodic behavior in RDA is demonstrated.

Abstract

Finite version of Random Domino Automaton (FRDA) - recently proposed a toy model of earthquakes - is investigated. Respective set of equations describing stationary state of the FRDA is derived and compared with infinite case. It is shown that for the system of big size, these equations are coincident with RDA equations. We demonstrate a non-existence of exact equations for size N bigger then 4 and propose appropriate approximations, the quality of which is studied in examples obtained within Markov chains framework. 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.