Thermodynamic Entropy and Chaos in a Discrete Hydrodynamical System

TL;DR

This paper establishes a proportional relationship between thermodynamic entropy density and the largest Lyapunov exponent in a discrete hydrodynamical system, linking thermodynamics and chaos theory.

Contribution

It introduces a method to define the Lyapunov exponent for cellular automata and demonstrates its proportionality to entropy density in a lattice gas automaton.

Findings

Thermodynamic entropy density is proportional to the LLE.

The LLE is sensitive to traveling waves in the system.

Boltzmann's H function relates linearly to the LLE expansion factor.

Abstract

We show that the thermodynamic entropy density is proportional to the largest Lyapunov ex- ponent (LLE) of a discrete hydrodynamical system, a deterministic two-dimensional lattice gas automaton. The definition of the LLE for cellular automata is based on the concept of Boolean derivatives and is formally equivalent to that of continuous dynamical systems. This relation is jus- tified using a Markovian model. In an irreversible process with an initial density difference between both halves of the system, we find that Boltzmann's H function is linearly related to the expansion factor of the LLE, although the latter is more sensitive to the presence of traveling waves.