Connection between Different Levels of Description

Abstract

One of the major issues raised by the Boltzmann equation is the problem of the reduced description. Equations of hydrodynamics constitute a closet set of equations for the hydrodynamic field (local density, local momentum, and local temperature). From the standpoint of the Boltzmann equation, these quantities are low-order moments of the one-body distribution function, or, in other words, the macroscopic variables. The problem of the reduced description consists in deriving equations for the macroscopic variables from kinetic equations, and predicting conditions under which the macroscopic description sets in. The classical methods of reduced description for the Boltzmann equation are: the Hilbert method, the Chapman–Enskog method, and the Grad moment method, reviewed in [98]. The general approach to the problem of reduced description for dissipative system was recognized as the problem of finding stable invariant manifolds in the space of distribution function. The notion of invariant manifold generalizes the normal solution in the Hilbert and in the Chapman–Enskog method, and the finite-moment sets of distribution function in the Grad method. A generalization of the Grad moment method is the concept of the quasiequilibrium approximation, cf. Sect. 2.6 and [94, 98]. Boltzmann’s kinetic equation has been expressed in GENERIC form [347], cf. Sect. 8.3, demonstrating that no dissipative potential is required for representing these equations.