In this chapter we give a full derivation of the Navier-Stokes equation for the lattice gas. The first step, the Boltzmann approximation, is an approximation of the exact Liouville dynamics. The Boltzmann approximation should not be confused with the lattice-Boltzmann method of Chapter 6 and Chapter 7, but results that we obtain here for the Boltzmann approximation and the Navier-Stokes equations are also useful for the Boltzmann method. One of these results is the H-theorem for lattice gases. In this chapter we also reopen the tricky issue of the spurious invariants of lattice-gas or Boltzmann dynamics. We specifically discuss non-uniform global linear invariants for which, unlike the general nonlinear invariants, some theoretical results are known. Among them we find the staggered momentum invariants. We discuss their effect on hydrodynamics, which leads us to corrections to the Euler equation of Chapter 2.

General Boolean dynamics

It is useful to express the Boolean dynamics as a sequence of Boolean calculations, as we did in Section 2.6. In this section we shall denote by s or n local configurations. We further define a field of “rate bits”, which are random, Boolean variables, defined independently on each site and denoted by ass′(x, t). They are equal to one with probability 〈ass′〉 = A(s, s′). Thus if the pre-collision configuration is n the post-collision configuration is that single s′ for which ans′ = 1.