Master Equation approach to Reversible and Conservative Discrete Systems

TL;DR

This paper applies a master equation approach to a reversible cellular automaton model (Q2R), revealing macroscopic irreversible dynamics and providing a tractable probabilistic description validated through explicit examples.

Contribution

It introduces a novel application of master equations to reversible cellular automata, enabling coarse-grained analysis of complex dynamics.

Findings

Master equation captures macroscopic irreversible behavior.

Probability transfer matrix is manageable for small systems.

Method validated with explicit examples.

Abstract

A master equation approach is applied to a reversible and conservative cellular automata model (Q2R). The Q2R model is a dynamical variation of the Ising model for ferromagnetism that possesses quite a rich and complex dynamics. The configurational space is composed by a huge number of cycles with exponentially long periods. Following Nicolis and Nicolis [Phys. Rev. A 38, 427-433 (1988)], a coarse-graining approach is applied to the time series of the total magnetization leading to a master equation that governs the macroscopic irreversible dynamics of the Q2R automata. The methodology is replicated for various lattice sizes. In the case of small systems, it is shown that the master equation leads to a tractable probability transfer matrix of moderate size which provides a master equation for a coarse-grained probability distribution. The method is validated and some explicit examples are discussed.