  Kinetic properties of a granular gas of viscoelastic particles in a
homogeneous cooling state are studied analytically and numerically. We employ
the most recent expression for the velocity-dependent restitution coefficient
for colliding viscoelastic particles, which allows to describe systems with
large inelasticity. In contrast to previous studies, the third coefficient a3
of the Sonine polynomials expansion of the velocity distribution function is
taken into account. We observe a complicated evolution of this coefficient.
Moreover, we find that a3 is always of the same order of magnitude as the
leading second Sonine coefficient a2; this contradicts the existing hypothesis
that the subsequent Sonine coefficients a2, a3 ..., are of an ascending order
of a small parameter, characterizing particles inelasticity. We analyze
evolution of the high-energy tail of the velocity distribution function. In
particular, we study the time dependence of the tail amplitude and of the
threshold velocity, which demarcates the main part of the velocity distribution
and the high-energy part. We also study evolution of the self-diffusion
coefficient D and explore the impact of the third Sonine coefficient on the
self-diffusion. Our analytical predictions for the third Sonine coefficient,
threshold velocity and the self-diffusion coefficient are in a good agreement
with the numerical finding.
