Chapter 5 - Macrodynamics: Chapman–Enskog method

Summary

In Chapter 4, we gave a statistical mechanical analysis of lattice gases, and discussed the equilibrium properties; we described uniform uncorrelated statistical equilibria, and we showed that they have a Fermi–Dirac probability distribution. The existence of these uniform equilibrium solutions can be established without recourse to the Boltzmann approximation; the only required conditions are the semi-detailed balance and the existence of local invariants. These equilibrium solutions – the analogue of global equilibria in usual statistical mechanics – are uniform by construction: the macroscopic variables, i.e. the ensemble-averages of the local invariants, are space-and time-independent.

Now, we address the problem of space-and time-varying macrostates in the ‘hydrodynamic limit’, that is, macrostates for which macroscopic variables vary over space and time scales much larger than the characteristic microscopic scales (lattice spacing and time-step duration). This scale separation between microscopic and macroscopic variables is a crucial physical ingredient in the theory of lattice gas hydrodynamic equations, and it is at the core of the forthcoming derivation, which rests upon a discrete version of the well-known ‘Chapman–Enskog method’ (see Chapman and Cowling, 1970).

Although the principles of the method described hereafter apply to a wider class of models, we are led, at a certain point of the forthcoming algebraic procedure, to particularize our study to the (still wide) class of ‘single-species thermal models’ defined in Chapter 4, Section 4.5.2. The simpler case of nonthermal models is also treated, but in a less detailed manner (see Section 5.7).