A Cellular Automaton Model for Diffusive and Dissipative Systems

TL;DR

This paper introduces a cellular automaton model that simulates energy diffusion and dissipation, capturing different state behaviors and applying to physical phenomena like X-ray bursters, with comparisons to the forest-fire model.

Contribution

The model uniquely incorporates energy diffusion and dissipation thresholds, providing a new framework for analyzing physical systems and their statistical behaviors.

Findings

System exhibits localized, power-law, or exponential dissipation distributions.

Model describes burst size and rate statistics in astrophysical phenomena.

Comparative analysis with the forest-fire model highlights differences and similarities.

Abstract

We study a cellular automaton model, which allows diffusion of energy (or equivalently any other physical quantities such as mass of a particular compound) at every lattice site after each timestep. Unit amount of energy is randomly added onto a site. Whenever the local energy content of a site reaches a fixed threshold $E_{c1}$, energy will be dissipated. Dissipation of energy propagates to the neighboring sites provided that the energy contents of those sites are greater than or equal to another fixed threshold $E_{c2} (\leq E_{c1})$. Under such dynamics, the system evolves into three different types of states depending on the values of $E_{c1}$ and $E_{c2}$ as reflected in their dissipation size distributions, namely: localized peaks, power laws, or exponential laws. This model is able to describe the behaviors of various physical systems including the statistics of burst sizes and burst rates in type-I X-ray bursters. Comparisons between our model and the famous forest-fire model (FFM) are made.