Exact homogeneous steady-state distribution for confined granular mixtures

Determine the exact zeroth-order homogeneous steady-state velocity distribution functions for the species of a confined granular mixture in the Enskog kinetic theory, so that the first-order cooling-rate contribution can be evaluated without the Maxwellian approximation.

Background

The first-order contribution to the cooling rate, denoted by ζ(1,0)\zeta^{(1,0)}, depends on the zeroth-order distribution functions fi(0)f_i^{(0)} of the granular-mixture species. Because these distributions are not known exactly, the paper replaces them with Maxwellian distributions to obtain an estimate. An exact determination of the homogeneous steady-state distributions would remove this approximation and yield a more fundamental evaluation of the cooling-rate contribution.

References

According to Eq. 3.19.1, the coefficient $\zeta{(1,0)}$ is given in terms of the distribution $f_i{(0)}$, whose exact form is not known to date.

— Enskog kinetic theory in a model of confined granular mixtures. Heat flux and some applications  (2609.16868 - Méndez et al., 15 Sep 2026) in Section 3.2, immediately after Eq. (3.19.2)