Characteristic times, elastic limit, collisionless limit, cooling limit: the two stationary regimes

In the dynamics of the $N$ particles, as defined in Eqs. (3.3), (3.4), (3.5), the most important parameters are:

• the coefficient of normal restitution $r$, which determines the degree of inelasticity;

• the ratio $\rho=\tau_b/\tau_c$ between the characteristic time of the bath and the ``global'' mean free time between collisions;

On the basis of these two parameters, we can define three fundamental limits of the dynamics of our model:

• the elastic limit: $r \to 1^-$;

• the collisionless limit: $\rho \to 0$ ( $\tau_c \gg \tau_b$);

• the cooling limit: $\rho \to \infty$ ( $\tau_c \ll \tau_b$);

The elastic limit is smooth in dimensions $d>1$ (see for example the discussion in paragraph 2.5), so that we can consider it equivalent to put $r=1$. In this case the collisions mix up the components leaving constant the energy (in the center of mass frame as well in the absolute frame). We can assume that, in this limit, the effect of the collisions is that of homogenizing the positions of the particles and making their velocity distribution relax toward the Maxwellian with temperature $T_g=\langle v^2 \rangle/d=\langle v_x^2 \rangle$ (this temperature is equal to the starting kinetic energy, but is modified by the relaxation toward $T_b$ due to the Langevin Eqs. (3.3)). In one dimension this mixing effect (toward a ``Maxwellian'') is no more at work, as the elastic collisions exactly conserve the starting velocity distribution (the collisions can be viewed as exchanges of labels and the particles as non-interacting walkers).

In the collisionless limit we have $\tau_c \gg \tau_b$ and therefore, the collisions are very rare events with respect to the characteristic time of the bath. In this case we can consider the model as an ensemble of non-interacting Brownian walkers, each following the Eqs. (3.3). Therefore, whatever $r$ is, and in any dimension, the distribution of velocities relaxes in a time $\tau _b$ toward a Maxwellian with temperature $T_g=\langle v^2 \rangle/d=T_b$ with a homogeneous density.

Finally, in the cooling limit, the collisions are almost the only events that act on the distribution of velocities, while between collisions the particles move almost ballistically. In this limit (if $r < 1$) the gas can be considered stationary only on observation times very long with respect to the time of the bath $\tau _b$, where the effect of the external driving (the Langevin equation) emerges. For observation times larger than the mean free time $\tau_c$ but shorter than $\tau _b$, the gas appears as a cooling granular gas. This limit is considered in Chapter V of this thesis.

To conclude this brief discussion on the expected behavior of the randomly driven granular gas model, we sketch a scenario with the presence of two fundamental stationary regimes:

• the ``collisionless'' stationary regime: when $\rho \ll 1$, i.e. approaching the collisionless limit; in this regime we expect, after a transient time of the order of $\tau _b$, the stationary statistics of an ensemble of non-interacting Brownian particles (homogeneous density and Maxwell distribution of velocities, absence of correlations);

• the ``colliding'' stationary regime: when $\rho \gg 1$, i.e. approaching the cooling limit, but observing the system on times larger than $\tau _b$; here we expect to see anomalous statistical properties.