Granular gases can be kept in a statistically stationary state by means of an external forcing. In experiments, this forcing is realized with shear strains, shaking, air fluidization, and so on. Usually it is not enough to simply give ``energy'' to the granular gas: for example a granular gas falling under the force of gravity (in the void) continues to cool, even if its kinetic energy increases as every grain accelerates. To obtain a stationary granular gas it is necessary to provide it with an ``internal energy'' input, i.e. a randomization (temperature) source. The stationary flow observed on inclined planes is due to the combined effect of gravity and random collisions with the inclined bottom. When a granular layer (monolayers are often used in experimental situation to reproduce a quasi-2d behavior) stays on a horizontal plate which is rapidly vibrated, the grains get random kicks (abrupt changes of moment) even if the vibration is periodic; moreover, while the vertical component of the kicks is not symmetrically distributed, the components parallel to the plate are isotropic. In all experimental situations the temperature source is also responsible for a systematic (non random) dissipation: in the examples given above this is simply understood as an effective friction due to collisions with the bottom. In such a situation, therefore, it is quite natural to model the evolution of velocity as a stochastic process composed of three ingredients:
It is important to keep separated the first two ingredients in this idealized picture, even if they seems similar in the sense that both are random changes of velocity: we impose all the statistical properties of the random kicks (i.e. distribution and spectral properties) but give very few constraints on the statistics of the collisions between grains. This situation is very different from that of the Brownian motion, where all the effects of the collisions between the observed particle and the fluid particles are embedded in the statistics of the random kicks (and in the form of the systematic dissipation). As a consequence of this, while the Brownian motion is essentially reduced to a one-particle problem, the randomly driven granular gas remains a many-body problem, where spatial correlations (in the density and in the velocity) can emerge.