From Kinetic Models to Hydrodynamics

A certain number of techniques have been designed in the kinetic theory of gases to derive macroscopic time evolution equations from the Boltzmann equation. Most of these methods require the single-particle distribution function to be parameterized by a set of distinguished fields, such as the hydrodynamic ones: the number (or mass) density, momentum vector, and temperature. This is a plausible assumption as long as the microscopic dynamics enjoys a vast separation of time scales and local thermodynamic equilibrium exists. Moreover, the derivation of hydrodynamics from kinetic theory is often concerned with the hydrodynamic limit of the Boltzmann equation. Loosely speaking, one is interested, typically, in the scaling of the Boltzmann equation with respect to some reference macroscopic length and time scales, which are expected to largely dominate the intrinsic kinetic scales. Nonetheless, it makes sense to consider the extension of the hydrodynamic description beyond the standard domain, considering reference scales comparable with the kinetic ones. This is the subject dealt with by generalized hydrodynamics. There are several delicate aspects hindering this line of investigation. A first, natural, objection points to the fact that below a certain length scale, the notion itself of “local equilibrium,” which is brought about by a sufficiently large number of particle collisions, is questionable. Moreover, from the technical side, one typically deals, in this context, with perturbative methods, such as Hilbert’s procedure or the Chapman–Enskog (CE) technique, which, at a certain order of truncation, may give rise to artificial instabilities. In particular, the CE method introduces an expansion of the distribution function in terms of a parameter, the Knudsen number, defined as the ratio of the mean free path to a representative macroscopic length. For small values of the Knudsen number, the CE method recovers the standard Navier–Stokes–Fourier (NSF) equations of hydrodynamics. In more refined approximations, referred to as the Burnett and super-Burnett hydrodynamics, the hydrodynamic modes become polynomials of higher order in the wave vector. In such an extension, the resulting hydrodynamic equations may become unstable and violate the H-theorem, as first shown by Bobylev for a particular case of Maxwell molecules. This indicates that the CE theory cannot be immediately trusted away from the hydrodynamic limit. Thus, while the mathematical framework concerning the hydrodynamic limit of the Boltzmann equation is well established, there is no consolidated counterpart addressing the short-wavelength domain.

In this chapter, we will outline the general mathematical framework concerning the statistical description of a many-particle system in the phase space.

In this chapter, we will review some analytical methods that make it possible to determine approximate solutions of the Boltzmann equation. In particular, we will discuss the structure of the Hilbert and Chapman–Enskog perturbation techniques and will also outline the essential features of the invariant manifold method, which stems from the assumption of time scale separation and, unlike the former methods, is also applicable beyond the strict hydrodynamic limit. Before reviewing the wealth of different techniques, it is worth investigating in greater depth the role of the different time scales in a particle system, which is one of the main ingredients underlying the onset of collective behavior.

In this chapter, we will focus on the statistical properties of a fluid from a macroscopic perspective. To this end, we will discuss the properties of the linearized version of the NSF equations of hydrodynamics and will introduce the correlation function formalism, which allows us to characterize the spectrum of fluctuations of the hydrodynamic variables.

In this chapter, we will describe a procedure that allows us to reduce the description from Grad’s moment system [1] to the hydrodynamic level.

In this work, we employed the invariant manifold method to derive closed hydrodynamic equations from some kinetic models. The main novelty of our approach stems from the use of a nonperturbative technique that allows us to sum exactly the classical Chapman–Enskog expansion. The method postulates a separation between slow and fast moments, and allows us to extract the slow invariant manifold in the space of distribution functions.