Thermodynamic Entropy and Chaos in a Discrete Hydrodynamical System

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.