In this thesis we have collected a panoramic of experiments, theories and models concerning the dynamics of granular systems. The rapid development in the study of granular gases in the last ten years has been made easier by the strong advancements in the kinetic theories of molecular systems achieved in the past decades. Many conclusions concerning correlations, breakdown of molecular chaos, the necessity of going beyond the Navier-Stokes order in hydrodynamics, finite size effects, scaling laws, boundary layers and so on, have been pointed out in the investigation of molecular systems under extreme conditions (high shear rates, high Reynolds number, etc.). Turbulence is another field of Physics where fluids show strong fluctuations, absence of scale separation and consequent scale dependent constitutive relations, and much more. The study of all these systems has put in evidence that standard tools of kinetic theory must be replaced by different kinds of analysis: often the study of stochastic processes or that of dynamical systems has led to important results. All techniques of statistical mechanics can be of relevance for granular gases: we have used Monte Carlo algorithms, as well as master equations for simple stochastic processes, probabilistic considerations, mean field models, kinetic or Fokker-Planck equations, coarsening models and diffusion equations, analysis of structure factors or persistence; even renormalization group (or mode-coupling) approaches have been used by other authors. More remarkably, the ability of reproduce essential features of a system with minimal models which can be more easily investigated (through analytical or numerical but also experimental efforts) has proven to have a key role in the approach to granular matter, as well as in the general scientific methodology.
The models that we have studied are in some case near to reality, in some other they must be considered first steps in a more general comprehension of physical mechanisms. We have probed the validity of some assumptions (Gaussian equilibrium and lack of short-range correlations) in the theory of driven granular fluids, using a model of hard objects with inelastic collisions coupled with an external heat bath. The analogy between the idealized heat bath and the more realistic driving plates or air fluidizations mechanisms should be understood better, but the agreement between the properties of the model and the experimental observation is not questioned: a real granular stationary flow presents a non-Gaussian equilibrium and short-range density correlations which can induce a lack of scale separation. We have obtained similar results using Molecular Dynamics as well as Direct Simulation Monte Carlo simulations: this means that the results are robust even if the short-range velocity correlations are ruled out by force. Measurement of large scale velocity correlations in real systems [71] have been obtained recently and could give hints on the correctness of some assumptions in the models: the randomly heated model with the external friction (as introduced by us) does not predict velocity structures, while the absence of friction induces structures but poses questions on the fluctuations of momentum and energy. Probably there is not a typical real situation and things can change from an experimental setup to another. Moreover, the inelastic hard object model with imposed Molecular Chaos (i.e. using DSMC) has proven to reproduce all the features of typical non-homogeneous steady granular flows, for example when driven by gravity: here again we have obtained a fair agreement with experiments, where non-Gaussian distributions of velocities and density correlations are found. The study of velocity correlations (which always suffer from the use of DSMC algorithm) is currently at work, in order to be compared with recent experiments. We have also checked the limits of hydrodynamic theories: there is usually, in the same setup, a region of the system that is well described by Navier-Stokes-like equations, and at the same time there are boundary layers that need different (more fundamental) approaches: ballistic flights or strong space-time gradients characterize these kinetic regions. The problem of boundary conditions is probably one of the more difficult and opened subjects of investigation in the physics of Granular Flows: it surely deserves more attention.
Cooling granular gases are apparently less connected to real
situations, but in turns are more rich of intriguing phenomena and
connections with other field of Physics (such as physics of systems
quenched under the critical temperature, aging systems and so
on). More remarkably, they are strongly connected with driven granular
gases: a realistic steady flow should be understood as an alternation
of cooling regions (in space-time) and driven regions, it is very rare
that a granular system is subjected to a truly continuous and
truly homogeneous external driving. Therefore some of the
correlations expected in a cooling gas could be found also in a driven
gas. As a matter of fact, the physics of cooling granular gases has
still many things to understand: it happens that the homogeneous
phase, early or later, ends and is substituted by a correlated
regime. The positions and velocities of the grains organize in
structures that reduce dissipation. A serious conjecture that involves
some kind of principle of minimum dissipation has not yet been
proposed, but it seems a question that needs to be asked. Our main
contribution in this field is the introduction of a lattice gas model
with inelastic interactions, which in spite of its simplicity
reproduces many properties of the inelastic hard objects model in its
homogeneous phase as well as in its velocity correlated phase. The
model is not capable of going through the density correlated phase but
may be considered in good agreement with the predictions for the
incompressible cooling gas. More remarkably, it shows the expected
energy decay and the expected growth of structure factors by means of
a microscopic dynamics, while the existing theories are based on
stability analysis of hydrodynamics (i.e. they are mesoscopic
approaches). We also put in evidence the appearance and
disappearance of the separation between kinetic and hydrodynamic
scales. This point deserves a deeper analysis: we could ask if such a
scale separation is stationary and, in that case, if there is a
transition in the parameter 
(the restitution coefficient) from
``quasi-elastic'' systems that have it and
``quasi-perfectly-inelastic'' systems that do not have. Or otherwise,
it could be that the later formation of new structures (in density),
breaks this scale separation for every value of . The study of
pseudo-Maxwell models without spatial coordinates (where we have
discovered an important analytical solution) perhaps can be useful for
the Physics of granular gases, even if they have been given results
different from MD simulations. The lattice model that we introduced, in
fact, is a sort of lattice version of the pseudo-Maxwell kinetic
kernel, that in the cross section disregards the relative velocity
between colliding particles. The goodness of the latter in spite of
the badness of the last could be explained as follows: in the lattice
model the particles have the possibility of correlate, i.e. there can
appear regions of neighboring particles with very similar velocities;
this means that, apart from interfaces among these regions, the relative velocities of
candidates for collisions are all very similar and therefore the
kinetic term 
of Boltzmann equation can be disregarded. In the
pseudo-Maxwell model without spatial coordinates, instead, the
colliding candidates can have very different velocities and the effect
of disregarding their relative velocity is dramatic.
We are now considering some other issues of the physics of granular gases, a brief list of our actual studies follows: