When the lattice gas was introduced in statistical physics around 1985 (see Frisch, Hasslacher and Pomeau, 1986), it was originally constructed as a physical model for hydrodynamics. In fact, the concept of the lattice gas is as much a physical concept – and we shall indeed start with intuitive physical ideas – as it is a mathematical concept, as a more formal definition can also be given. We first present the point of view of the physicist (Section 1.1), then we describe the lattice gas automaton from the mathematical viewpoint (Section 1.2), and in Section 1.3 we discuss the two aspects.

The physicist's point of view

A lattice gas can be viewed as a simple, fully discrete microscopic model of a fluid, where fictitious particles reside on a finite region of a regular Bravais lattice. These fictitious particles move at regular time intervals from node to node, and can be scattered by local collisions according to a node-independent rule that may be deterministic or non-deterministic. Thus, time, space coordinates and velocities are discrete at the microscopic scale, that is, at the scale of particles, lattice nodes and lattice links.

The stationary states in statistical equilibrium and thus the large-scale dynamics of a lattice gas will crucially depend on its conservation properties, that is, on the quantities preserved by the microscopic evolution rule of the system.