Granular systems are almost always out of thermodynamic equilibrium: they are open systems that need an external source to be kept in a steady state. This means that even in a homogeneous granular gas a flow of heat is present, from the environment into the bulk, to keep the gas in a stationary regime; more often the gas is unstable toward the formation of macroscopic flows, e.g. self-induced shear and vortices. In this chapter we review the basics of transport theory for elastic fluids in order to recall all the assumptions needed to follow its derivation. The general transport theory is the arrival point of a path that begins at the microscopic (we will say ``molecular dynamics'', MD) level and goes along through the definition of a space-time scale which separates equilibrium from non-equilibrium together with many other hypothesis. This chapter is a review of the many delicate passages which demand strong requirements usually not fulfilled by inelastic gases.
The main distinction between granular gases and the other ``states'' of granular matter comes from the fact that collisions are always considered binary. The so-called inelastic collapse observed in simulations is the signature of a departure from the gas behavior toward a solid-vapor coexistence (ordered clusters surrounded by the gas phase). This transition is usually avoided by means of a physical regularization (e.g. by assuming a velocity dependent restitution coefficient), or restricting the simulations to a limited region of the parameter space such that this instability cannot arise. We have observed that the presence of a randomizing source is sufficient to prevent the inelastic collapse.
In the first section a brief description of the elastic and inelastic binary collision event is given. Many widely known statistical concepts as mean free time, mean free path and collisional cross section will be defined in the case of elastic collisions. The derivation of the Boltzmann equation from the Liouville equation is sketched in the second section, going through the BBGKY hierarchy. We will put the accent on the assumption of molecular chaos which concerns correlations of pre-collisional velocities. Absence of pre-collisional correlations is a delicate assumption even for elastic systems. Molecular chaos, however, gives important simplification in numerical work, allowing the use of Monte Carlo methods which can be useful to catch qualitative pictures of granular physics. From the Boltzmann equation one can derive, by means of the Chapman-Enskog or the Grad methods the equations of transport for the out of equilibrium behavior of the gas. This derivation will be summarized in the third section. In the fourth section we review the existing closure theories for granular hydrodynamics (i.e. hydrodynamics of inelastic hard spheres). This program was begun in the early 80s and many exact results have been obtained in recent years: the renewal of interest is motivated by the speed up of computer simulations and by the consequent ease of testing the predictions of the theory. In section five, finally, we will present the criticism that has been put forward by different authors concerning the difficulties of an rigorous foundation of granular hydrodynamics: the path from microscopic to mesoscopic description is full of assumptions that could restrict the limits of validity of the existing kinetic theories. All of the concepts reviewed in the first three sections of this chapter will be of aid in order to establish the deep differences between elastic and inelastic gases.