In this chapter, we will discuss briefly some simple models of fluid systems that are designed to exhibit many of the nonequilibrium properties of a real fluid, and to be very suitable for precise computer studies of fluid flows since only binary arithmetic is used to simulate these models. The models were devised by Prisch, Hasslacher, and Pomeau, among others, and are generally called cellular automata lattice gases. The corresponding one-dimensional Lorentz gas, studied in great detail by Ernst and co-workers, may be viewed as a ‘modern-day’ Kac ring model. The interest of these models for us consists in the fact that it is rather straightforward to compute both the transport as well as the chaotic properties of these systems, and the thermodynamic formalism is especially useful here. After introducing the general class of cellular automata lattice gases (CALGs) we will turn our attention to the special case of the one-dimensional Lorentz lattice gas (LLG) to outline how its dynamical quantities can be calculated.

Cellular automata lattice gases

Consider a two-dimensional hexagonal or square lattice with bonds connecting the nearest-neighbor lattice sites. A CALG is constructed by (i) putting indistinguishable particles on this lattice with velocities that are aligned along the bond directions, (ii) considering that the time is discretized, and (iii) stating that in one time step a particle goes from one site to the next in the direction of its velocity. The number of possible velocities for any particle is then equal to the coordination number, b, of the lattice, although models with rest particles (zero velocity), or with other velocities, are often considered.