Consider a Bravais lattice of vertices with bonds connecting the vertices. Now put point particles on the vertices, with velocities in the directions of the bonds, such that there can be at most one particle with a given velocity at each vertex. Next suppose that at discrete times, each particle moves from one vertex to the next along the direction of its velocity, and when it arrives at the next vertex, its velocity is adjusted according to a set of deterministic or stochastic ‘‘collision rules,’’ depending on the number and velocities of the other particles arriving at the same vertex at the same time.