The binary collision operator
, for inelastic
particles, must be changed [210] according to the inelastic collision rules,
Eqs. (2.41) and Eqs. (2.42). It must be
noted that when
(elastic collisions), the two set of equations
coincide, i.e. the direct or inverse collision are identical
transformation. This is not true if
. Therefore, in the
definition of the inverse binary collision operators at the end of
section 2.2.1, that is
and
, we have put the same operator
that
appears in the direct binary collision operators
and
, while in general it must be used the operator
that replaces velocities with precollisional velocities (using
the transformation given in Eqs. (2.42)). The adjoint of
inverse binary inelastic collision operator(the only one needed in the
following) therefore reads:
![]() |
(2.94) |
Deriving from this the BBGKY hierarchy (analogue of (2.94)) and putting in the first equation of it the Molecular Chaos assumption, the Boltzmann Equation for granular gases is obtained [41,210]:
where the primed velocities are defined in Eqs. (2.42).
This equation has been studied in the spatially homogeneous case (no
spatial gradients, ), with the Enskog correction (i.e. a
multiplying factor
in front of the collision integral)
by Goldshtein and Shapiro [97] and by Ernst and van Noije [211]. The equation is
where
. Goldshtein and Shapiro [97] have shown that
this equation admits an isotropic scaling solution which describes the
so-called ``Homogeneous Cooling State'', depending on time only
through the temperature
defined by the relation
:
with defined by the relation
.
The scaling function solves a complicated integro-differential equation. This can be studied using an expansion in Sonine polynomials:
![]() |
(2.96) |
where
,
is the Maxwellian
and
is the second Sonyne polynomial. The
calculations of Ernst and van Noije [211] have shown that for
the coefficient
is very near to zero, precisely
in three dimensions and
in two
dimensions. The decay of the temperature has also been calculated:
![]() |
(2.97) |
with
(dimensionless damping rate),
, and the initial Maxwellian mean free time
(
is the collision frequency
calculated with a Maxwellian with temperature
), while the true
collision frequency reads
. Monte
Carlo simulations (DSMC, see Appendix A) by Brey et al. [42] of the
Eq. (2.99) have shown that
the temperature decay and the fourth cumulant
qualitatively
agree with the results of Ernst and van Noije. Moreover the agreement
between very dilute Molecular Dynamics simulations and Monte Carlo
solutions of the Boltzmann equation in 2D have shown
agreement up to moderate inelasticities (
).
Ernst and van Noije [211] have also given estimates for the tails of the velocity distribution, using an asymptotic method employed by Krook and Wu [129]. This method assumes that for a fast particle the dominant contributions to the collision integral come from collisions with thermal (bulk) particles and that the gain term of the integral can be neglected with respect to the loss term. They found an exponential behavior for the tails of the scaling distribution function of velocities:
![]() |
(2.98) |
with a constant depending on
.
It is relevant that Ernst and van Noije have also studied Enskog-Boltzmann equation in the presence of a random forcing, i.e. a source of energy that acts randomly, uncorrelated and isotropically on all the particles. The homogeneous equation for this system reads:
where the velocity diffusion coefficient appears, which is
proportional to the rate of energy input per unit mass
. This equation admits a stationary homogeneous solution
with
defined as
above in terms of the granular temperature
. It is immediate to
obtain a temperature balance
From Eq. (2.105)
follows the value for the stationary temperature (if the system
remains homogeneous!). The authors have also given estimates for the
coefficient of the Sonine expansion. Molecular dynamics by van
Noije et al. [211] have shown that the predictions of the homogeneous
Boltzmann equation work for
and low densities. The
Krook and Wu method for the asymptotic behavior of the tails of the
velocity distribution.function gives
![]() |
(2.100) |