Deviation from Maxwell Distribution in Granular Gases with Constant Restitution Coefficient

We analyze the velocity distribution function of force-free granular gases in the regime of homogeneous cooling when deviations from the Maxwellian distribution may be accounted only by leading term in the Sonine polynomial expansion. These are quantified by the magnitude of the coefficient $a_2$ of the second term of the expansion. In our study we go beyond the linear approximation for $a_2$ and observe that there are three different values (three roots) for this coefficient which correspond to a scaling solution to the Boltzmann equation. The stability analysis performed showed, however, that among these three roots only one corresponds to a stable scaling solution. This is very close to $a_2$, obtained in previous studies in a linear with respect to $a_2$ approximation.