The problem of granular gases becomes even more subtle when the energy input is suppressed and the system cools down due to dissipative collisions among particles. This is addressed as the problem of ``cooling'' granular gases and historically has preceded the problem of stationary granular gases with energy feeding mechanisms [103,96,157]. It is reasonable that a problem is investigated with a minimum set of ingredients (i.e.: free cooling, only collisions among particles) and then new physical ingredients (e.g: the energy input [77]) are added thereafter. Unfortunately, in this case the presence of an energy input changes drastically the behavior of the gas and in some sense reduces the difficulty of its investigation. A stationary granular gas, in fact, has only one regime to be studied (apart of a transient time necessary to forget initial conditions), while a cooling granular gas has several regimes intervening at different times, most of which are still lacking a complete comprehension.
Moreover, one can prepare a driven granular gas in such a way to obtain an ideal gas-like stationary regime [183] (simply chosing a not too high density or a not too high inelasticity or a strong energy input), while a cooling granular gas is unstable to the formation of liquid-like and solid-like phases, breaking in a non-trivial way the simmetries imposed by the initial condition.
The inelastic hard spheres model without energy input exhibits an
initial regime characterized by homogeneous density and a probability
distribution of velocities that depends on time only through the total
kinetic energy (global granular temperature 
), i.e. a scaling
velocity distribution. This is called the Homogeneous Cooling
State or Haff regime [103]. It has been shown by several
authors [96,157,73,214] that this
state is not stable towards shear and clustering instabilities:
structures can form that seem to minimize dissipation, mainly in the
form of velocity vortices and high density clusters. These
instabilities grow on different space and time scales, so that one can
investigate them separately. Several theories have been proposed to
take into account the emergence of structures in granular gases. Some
of them are more fundamental, obtaining the correlation functions from
the kinetic equations [210]; others are more macroscopic
(they have been called ``mesoscopic'' by the authors), involving the
study of fluctuating hydrodynamics [212]; others are simply
phenomenological theories that suggest analogies with Burgers
equation [22] or spinodal decomposition
models [219]. Some of these theories can grasp the behavior
of the cooling granular gas far deeply into the correlated regime,
giving predictions for the asymptotic decay of energy.
We review these recent theories in the first section of this chapter, focusing our attention on the results of Goldhirsch and Zanetti [96], those of Ernst and co-workers [214,219], and finally to those of Ben-Naim and Redner [22] and some alternative models proposed by Ben-Naim and Krapivisky [24]. In the second section we study the behavior of the 1D cooling granular gas, comparing Molecular Dynamics simulations with a 1D inelastic lattice gas model proposed by us, which seems to reproduce very well the true (MD) phenomenology in the collisional (uncompressible) regime. In the third section we discuss this particular model, which is the inelastic mean-field version of the Maxwell molecules model (also known as Ulam model [208]), obtaining an exact scaling solution for its kinetic equation. In the fourth section we propose a 2D lattice gas model that well reproduces the behavior of the 2D cooling granular gas, providing a tool to study the diffusive character of the shearing instability and its statistical properties. The last section is devoted to a brief review of the results (analytical and numerical) known for 1d and 2d cooling granular gases, with different models considered.
