# Singular behavior of the velocity moments for inelastic Maxwell mixtures in
the homogeneous cooling state

Constantino Sánchez Romero111Electronic address<EMAIL_ADDRESS>Departamento
de Física, Universidad de Extremadura, Avda. de Elvas s/n, E-06006 Badajoz,
Spain  Vicente Garzó111Electronic address<EMAIL_ADDRESS>URL:
http://www.unex.es/eweb/fisteor/vicente/ Departamento de Física and Instituto
de Computación Ciéntífica Avanzada Universidad (ICCAEx), Universidad de
Extremadura, Avda. de Elvas s/n, E-06006 Badajoz, Spain

###### Abstract

The Boltzmann equation for $d$-dimensional inelastic Maxwell models is
considered to determine the collisional moments of second, third and fourth
degree in a granular binary mixture. These collisional moments are exactly
evaluated in terms of the velocity moments of the distribution function of
each species when diffusion is absent (mass flux of each species vanishes).
The corresponding associated eigenvalues as well as cross coefficients are
obtained as functions of the coefficients of normal restitution and the
parameters of the mixture (masses, diameters and composition). The results are
applied to the analysis of the time evolution of the moments (scaled with a
thermal speed) in the homogeneous cooling state problem. It is shown that, for
given values of the parameters of the system, the third and fourth degree
moments could diverge in time. An exhaustive study on the influence of the
parameter space of the mixture on the above singularity is carried out. The
present work extends to binary mixtures previous theoretical results derived
for _monocomponent_ inelastic Maxwell models [V. Garzó and A. Santos, Third
and fourth degree collisional moments for inelastic Maxwell models, _J. Phys.
A: Math. Theor._ 40 (2007), 14927–14943].

##  I Introduction

It is well established that when granular matter is externally excited it can
be modeled as a gas of _inelastic_ hard spheres (IHS). In the simplest version
of the model, the spheres are assumed to be completely _smooth_ (i.e., with no
rotational degrees of freedom) so that the inelasticity of collisions is
characterized by a (positive) constant coefficient of normal restitution
$\alpha\leq 1$. The case $\alpha=1$ corresponds to elastic collisions
(molecular gases). In the low-density regime, all the relevant information on
the state of the gas is given by the one-particle velocity distribution
function. This distribution obeys the Boltzmann equation properly adapted to
account for the inelastic nature of collisions BP04 .

One of the main difficulties for obtaining the collisional moments of the
Boltzmann equation for hard spheres comes from the form of the collision rate,
which is proportional to the relative velocity of the two colliding particles.
This difficulty (which is also present for molecular gases) prevents the
possibility of expressing the associated collisional moments in terms of a
finite number of velocity moments, thus precluding the derivation of exact
analytical results. In the case of elastic collisions, the above problem can
be overcome by assuming that the gas is made of Maxwell molecules, namely,
particles interacting via a repulsive potential inversely proportional to the
fourth power of the distance. For this interaction potential, the collision
rate is independent of the relative velocity and so, any collisional moment of
degree $k$ can be expressed in terms of velocity moments of a degree smaller
than or equal to $k$ TM80 . Thanks to this property, nonlinear transport
properties can be exactly obtained TM80 ; GS03  and, when properly reduced,
they present a good agreement with results derived for other interaction
potentials.

When the collisions are inelastic ($\alpha<1$), one can still introduce the
so-called inelastic Maxwell models (IMM) (see for instance Refs. BCG00 ; CCG00
; BK00  for some of the first papers where IMM were introduced). These models
share with elastic Maxwell molecules the property that the collision rate is
independent of the relative velocity but, on the other hand, their collision
rules are the same as for IHS. In this sense, although IMM do not correspond
to any microscopic interaction potential it has been widely shown in the past
that the cost of sacrificing physical realism is in part compensated by the
amount of _exact_ analytical results. In particular, the IMM keeps the
qualitatively correct structure and properties of the nonlinear macroscopic
equations and obey Haff’s law Haff . It must be also remarked that in the
framework of the Boltzmann equation one can introduce Maxwell models at the
level of the cross section, without any reference to a specific interaction
potential E81 . Recently KS22 , an inelastic _rough_ Maxwell model has been
also introduced.

The simplifications introduced by IMM in the kernel of the Boltzmann collision
operator has allowed in some cases to determine exact properties. For this
reason, the Boltzmann equation for IMM has received a great attention of
physicists and mathematicians in the last few years, specially in the study of
overpopulated high energy tails in homogeneous states EB02 ; BMP02 ; NK02a
and in the evaluation of the transport coefficients S02 ; GA05 . The existence
of high energy tails of the Boltzmann equation is common for IHS and IMM,
although this general qualitative agreement fails at a more quantitative
level. The study of this problem has been mainly focused on the monodisperse
gas case. Much less is known for inelastic Maxwell mixtures. In this case,
Marconi and Puglisi have studied the high velocity moments in the free cooling
MP02a  and driven MP02b  states for a one-dimensional gas ($d=1$). For
arbitrary dimensions and in the tracer limit, Ben-Naim and Krapivsky NK02
have analyzed the velocity statistics of an impurity in a uniform granular gas
while the fourth cumulant of the velocity distribution in the homogenous
cooling state (HCS) has been also obtained GA05 . In addition, as for
molecular gases, non-Newtonian transport properties have been exactly
determined G03 ; SG07 ; G07 ; GT10 ; GT11 ; GT12 ; KGS14 ; GT15 ; GG19
showing in some cases an excellent agreement with computer simulations for IHS
MG02 .

Beyond the second degree velocity moments (which are directly related with the
transport properties), Garzó and Santos GS07  have computed all the third and
fourth degree velocity moments of the Boltzmann collision operator for a
monocomponent granular gas. In addition, the collisional rates associated with
the isotropic velocity moments $\langle v^{2r}\rangle$ and the anisotropic
moments $\langle v^{2r}v_{i}\rangle$ and $\langle
v^{2r}\left(v_{i}v_{j}-d^{-1}v^{2}\delta_{ij}\right)\rangle$ have been
independently evaluated in Refs. EB02 ; BK03 ; SG12 . Here, $\langle
h(\mathbf{v})\rangle=\int d\mathbf{v}h(\mathbf{v})f(\mathbf{v})$, where
$h(\mathbf{v})$ is an arbitrary function of the velocity $\mathbf{v}$ and
$f(\mathbf{v})$ is the one-particle velocity distribution function. All the
above calculations have been made for an arbitrary number of dimensions $d$.
To the best of our knowledge, the above papers are the only works where the
computation of higher-degree collisional moments of IMM has been carried out.

The objective of the present paper is to extend the results displayed in Ref.
GS07  to the case of granular binary mixtures. Due to the long and complex
algebra involved in the general problem, here we will consider situations
where the mean flow velocity ${\bf U}_{r}$ ($r=1,2$) of each species is equal
to the mean flow velocity ${\bf U}$ of the mixture and so, no diffusion
processes are present in the mixture (i.e., ${\bf U}_{1}={\bf U}_{2}={\bf
U}$). Although this limitation restricts the applicability of the present
results to general nonequilibrium situations, they are still useful for
contributing to the advancement in the knowledge of exact properties of IMM.
Among the different problems, we can mention shearing problems and/or the
relaxation of the third and fourth degree moments toward the HCS, starting
from arbitrary initial conditions. This is likely one of the main motivations
of the present work.

Some previous results derived for IMM in the monodisperse case EB02 ; NK02
have shown that for $d\geq 2$ the (scaled) velocity distribution function
$\phi(c)$ has a high-velocity tail of the form $\phi(c)\sim
c^{-d-\beta(\alpha)}$, where $c$ is the (scaled) velocity of the particle. The
exponent $\beta(\alpha)$ obeys a transcendental equation whose solution is
always larger than four ($\beta(\alpha)>4$), except for $d=1$ BMP02 .
Consequently, for any value of $\alpha$ and $d\geq 2$, the corresponding
(scaled) velocity moments of degree $k$ equal to or less than four are
_convergent_ and tend towards well-defined values in the long-time limit. An
interesting issue is explore whether or not the convergence of moments of
degree $k\leq 4$ for the single gas case is also present for inelastic binary
mixtures and, if so, to what extent. An indirect way of answering this
question is through the knowledge of the high degree velocity moments (beyond
the second ones) of the distribution function of each species. These moments
play a prominent role in the high velocity region and can show in a clean way
the possible existence of a continuos spectrum of exponents characterizing the
inverse-power-law decay of the tails of the distribution functions.
Surprisingly, our results for binary mixtures show that the (anisotropic)
third and fourth degree moments could diverge in time for given values of the
parameters of the mixture. Therefore, in contrast to the findings for the
monocomponent gases for $d\geq 2$, only the (scaled) moments of degree equal
to or smaller than 2 are always convergent in the HCS for arbitrary values of
the parameters of the mixture.

The plan of the paper is as follows. In section II the Boltzmann kinetic
equation for inelastic Maxwell mixtures is presented. Next, the so-called
Ikenberry polynomials TM80  $Y_{2p|i_{1}i_{2}\ldots i_{q}}(\mathbf{V})$ of
degree $k=2p+q$ are introduced and their collisional moments
$J_{2p|i_{1}i_{2}\ldots i_{q}}^{rs}$ with $k=2,3$ and 4 associated with the
Boltzmann collision operators $J_{rs}[f_{r},f_{s}]$ evaluated in section III.
Some technical details involved in the calculations are relegated to the
Appendix A. The time relaxation problem of the (scaled) moments toward their
asymptotic values in the HCS is studied in section IV while an study of the
regions of the parameter space where the third- and fourth-degree moments can
be divergent is presented in section V. We close the paper in section VI with
a brief discussion of the results derived in this paper.

##  II Boltzmann kinetic equation for inelastic Maxwell mixtures

We consider a granular binary mixture made of particles of diameters
$\sigma_{r}$ and masses $m_{r}$ ($r=1,2$). In the absence of external forces
and assuming molecular chaos, the one-particle velocity distribution function
$f_{r}(\mathbf{r},\mathbf{v};t)$ of species $r$ obeys the Boltzmann equation

$\frac{\partial f_{r}}{\partial t}+\mathbf{v}\cdot\nabla
f_{r}=\sum_{s=1}^{2}J_{rs}[\mathbf{v}|f_{r},f_{s}],\quad(r=1,2),$ (1)

where $J_{rs}[f_{r},f_{s}]$ is the Boltzmann collision operator for collisions
between particles of species $r$ and $s$. In the case that the granular
mixture is modeled as a gas of IHS then any collisional moment of
$J_{rs}[f_{r},f_{s}]$ involves the full distributions $f_{r}$ and $f_{s}$
(namely, all their higher degree moments). This means that one has to resort
to approximate forms of the distributions $f_{r}$ and $f_{s}$ to estimate the
collisional moments of $J_{rs}$. Usually the lowest order in a Sonine
polynomial expansion of these distributions is considered G19 . This problem
is also present in the conventional case of molecular binary mixtures (elastic
collisions). However, if one assumes that the collision rate of the two
colliding spheres is constant (IMM), the collisional moments of the operator
$J_{rs}$ can be expressed in terms of velocity moments of $f_{r}$ and $f_{s}$
without to know their explicit forms.

In the case of IMM, the form of the operator $J_{rs}$ can be obtained from its
corresponding form for IHS by replacing the collision rate of IHS by an
effective mean-field collision rate independent of the velocity. With this
simplification, the Boltzmann collision operator $J_{rs}[f_{r},f_{s}]$ for IMM
is G19

$J_{rs}\left[{\bf
v}_{1}|f_{r},f_{s}\right]=\frac{\omega_{rs}}{n_{s}\Omega_{d}}\int d{\bf
v}_{2}\int d\widehat{\boldsymbol{\sigma}}\left[\alpha_{rs}^{-1}f_{r}({\bf
v}_{1}^{\prime\prime})f_{s}({\bf v}_{2}^{\prime\prime})-f_{r}({\bf
v}_{1})f_{s}({\bf v}_{2})\right]\;.$ (2)

Here,

$n_{r}=\int d\mathbf{v}f_{r}(\mathbf{v})$ (3)

is the number density of species $r$, $\omega_{rs}$ is an effective collision
frequency (to be chosen later), $\Omega_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the
total solid angle in $d$ dimensions, and $\alpha_{rs}\leq 1$ refers to the
constant coefficient of restitution for $r$-$s$ collisions. In addition, the
double primes on the velocities denote the initial values $\\{{\bf
v}_{1}^{\prime\prime},{\bf v}_{2}^{\prime\prime}\\}$ that lead to $\\{{\bf
v}_{1},{\bf v}_{2}\\}$ following a binary collision:

${\bf v}_{1}^{\prime\prime}={\bf
v}_{1}-\mu_{sr}\left(1+\alpha_{rs}^{-1}\right)(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})\widehat{\boldsymbol{\sigma}},\quad{\bf v}_{2}^{\prime\prime}={\bf
v}_{2}+\mu_{rs}\left(1+\alpha_{rs}^{-1}\right)(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})\widehat{\boldsymbol{\sigma}}\;,$ (4)

where $\mu_{rs}=m_{r}/(m_{r}+m_{s})$, ${\bf g}={\bf v}_{1}-{\bf v}_{2}$ is the
relative velocity of the colliding pair and $\widehat{\boldsymbol{\sigma}}$ is
a unit vector directed along the centers of the two colliding spheres.

Apart from the densities $n_{r}$, the granular temperature $T$ is defined as

$T=\sum_{r}\;x_{r}T_{r},$ (5)

where $x_{r}=n_{r}/n$ is the mole fraction of species $r$ ($n=n_{1}+n_{2}$ is
the total number density) and

$T_{r}=\frac{1}{dn_{r}}\int d\mathbf{v}\;m_{r}V^{2}f_{r}(\mathbf{v})$ (6)

is the partial temperature of species $r$. In Eq. (6), we have introduced the
peculiar velocity $\mathbf{V}=\mathbf{v}-\mathbf{U}$, $\mathbf{U}$ being the
mean flow velocity defined as

$\rho\mathbf{U}=\sum_{r}\rho_{r}\mathbf{U}_{r}=\int
d\mathbf{v}m_{r}\mathbf{v}f_{r}(\mathbf{v}).$ (7)

Here, $\rho_{r}=m_{r}n_{r}$ is the mass density of species $r$ and
$\rho=\rho_{1}+\rho_{2}$ is the total mass density. The second identity in Eq.
(7) defines the partial mean flow velocities $\mathbf{U}_{r}$. In addition,
the mass flux of species $r$ is given by
$\mathbf{j}_{r}=\rho_{r}\left(\mathbf{U}_{r}-\mathbf{U}\right)$. As said in
the Introduction section, for the sake of simplicity, we will assume in this
paper that the mass fluxes vanish (i.e., $\mathbf{U}_{r}=\mathbf{U}$).

To evaluate the collisional moments of the Boltzmann operator
$J_{rs}[f_{r},f_{s}]$, a useful identity for an arbitrary function $h({\bf
v})$ is

$\int
d\mathbf{v}_{1}h(\mathbf{v}_{1})J_{rs}[\mathbf{v}_{1}|f_{r},f_{s}]=\frac{\omega_{rs}}{n_{s}\Omega_{d}}\int
d\mathbf{v}_{1}\,\int
d\mathbf{v}_{2}f_{r}(\mathbf{v}_{1})f_{s}(\mathbf{v}_{2})\int
d\widehat{\boldsymbol{\sigma}}\,\left[h(\mathbf{v}_{1}^{\prime})-h(\mathbf{v}_{1}\right],$
(8)

where

$\mathbf{v}_{1}^{\prime}=\mathbf{v}_{1}-\mu_{rs}(1+\alpha_{rs})(\widehat{\boldsymbol{\sigma}}\cdot\mathbf{g})\widehat{\boldsymbol{\sigma}}$
(9)

denotes the post-collisional velocity.

It is worthwhile mentioning that there is another more refined version of IMM
where the collision rate has the same dependence on the scalar product
$(\widehat{\boldsymbol{\sigma}}\cdot\mathbf{g})$ as in the case of IHS. The
corresponding Boltzmann operator can be proved to be equivalent to the one
given by Eq. (2) except that the scattering rules (4) and (9) must be changed
BCG00 ; CCG00 ; BC02 . However, both versions of IMM yield similar results in
problems such as the high energy tails. In this paper, for the sake of
simplicity, we will consider the version provided by the collision rules (4)
and (9).

###  II.1 Ikenberry polynomials

In the case of Maxwell models (both elastic and inelastic), it is convenient
to introduce the Ikenberry polynomials $Y_{2p|i_{1}i_{2}\ldots
i_{q}}(\mathbf{V})$ of degree $k=2p+q$ TM80 . The Ikenberry polynomials are
defined as $Y_{2p|i_{1}i_{2}\ldots
i_{q}}(\mathbf{V})=V^{2p}Y_{i_{1}i_{2}\ldots i_{q}}(\mathbf{V})$, where
$Y_{i_{1}i_{2}\ldots i_{q}}(\mathbf{V})$ is obtained by subtracting from
$V_{i_{1}}V_{i_{2}}\ldots V_{i_{q}}$ that homogeneous symmetric polynomial of
degree $q$ in the components of $\mathbf{V}$ such as to annul the result of
contracting the components of $Y_{i_{1}i_{2}\ldots i_{q}}(\mathbf{V})$ on any
pair of indices. The polynomials functions $Y_{2p|i_{1}i_{2}\ldots
i_{q}}(\mathbf{V})$ of degree smaller than or equal to four are

$Y_{0|0}(\mathbf{V})=1,\quad Y_{0|i}(\mathbf{V})=V_{i},$ (10)
$Y_{2|0}(\mathbf{V})=V^{2},\quad
Y_{0|ij}(\mathbf{V})=V_{i}V_{j}-\frac{1}{d}V^{2}\delta_{ij},$ (11)
$Y_{2|i}(\mathbf{V})=V^{2}V_{i},\quad
Y_{0|ijk}(\mathbf{V})=V_{i}V_{j}V_{k}-\frac{1}{d+2}V^{2}\left(V_{i}\delta_{jk}+V_{j}\delta_{ik}+V_{k}\delta_{ij}\right),$
(12) $Y_{4|0}(\mathbf{V})=V^{4},\quad
Y_{2|ij}(\mathbf{V})=V^{2}\left(V_{i}V_{j}-\frac{1}{d}V^{2}\delta_{ij}\right),$
(13) $\displaystyle Y_{0|ijk\ell}(\mathbf{V})$ $\displaystyle=$ $\displaystyle
V_{i}V_{j}V_{k}V_{\ell}-\frac{1}{d+4}V^{2}\left(V_{i}V_{j}\delta_{k\ell}+V_{i}V_{k}\delta_{j\ell}+V_{i}V_{\ell}\delta_{jk}+V_{j}V_{k}\delta_{i\ell}\right.$
(14)
$\displaystyle\left.+V_{j}V_{\ell}\delta_{ik}+V_{k}V_{\ell}\delta_{ij}\right)+\frac{1}{(d+2)(d+4)}V^{4}\left(\delta_{ij}\delta_{k\ell}+\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk}\right)$
$\displaystyle=$ $\displaystyle
V_{i}V_{j}V_{k}V_{\ell}-\frac{1}{d+4}\left[Y_{2|ij}(\mathbf{V})\delta_{k\ell}+Y_{2|ik}(\mathbf{V})\delta_{j\ell}+Y_{2|i\ell}(\mathbf{V})\delta_{jk}\right.$
$\displaystyle\left.+Y_{2|jk}(\mathbf{V})\delta_{i\ell}+Y_{2|j\ell}(\mathbf{V})\delta_{ik}+Y_{2|k\ell}(\mathbf{V})\delta_{ij}\right]$
$\displaystyle-\frac{1}{d(d+2)}Y_{4|0}(\mathbf{V})\left(\delta_{ij}\delta_{k\ell}+\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk}\right).$

Let us introduce here the notation

$M_{2p|i_{1}i_{2}\ldots i_{q}}^{(r)}=\int d{\bf V}Y_{2p|i_{1}i_{2}\ldots
i_{q}}({\bf V})f_{r}({\bf V}),$ (15) $J_{2p|i_{1}i_{2}\ldots
i_{q}}^{(rs)}=\int d{\bf V}Y_{2p|i_{1}i_{2}\ldots i_{q}}({\bf
V})J_{rs}[f_{r},f_{s}].$ (16)

Equation (15) gives the definition of the velocity moments of the distribution
$f_{r}$ while Eq. (16) provides the definition of the collisional moments of
the Boltzmann operator $J_{rs}$.

Note that $M_{0|0}^{(r)}=n_{r}$, $J_{0|0}^{(rs)}=0$ (conservation of mass),
$M_{0|i}^{(r)}=0$ (since we have assumed that $\mathbf{U}_{r}=\mathbf{U}$) and

$M_{2|0}^{(r)}=d\frac{p_{r}}{m_{r}}=d\frac{n_{r}T_{r}}{m_{r}},$ (17)

where $p_{r}=n_{r}T_{r}$ is the partial pressure of species $r$. Moreover,
$M_{0|ij}^{(r)}=(P_{r,ij}-p_{r}\delta_{ij})/m_{r}$, where

$P_{r,ij}=\int d{\bf V}m_{r}V_{i}V_{j}f_{r}({\bf v})$ (18)

is the partial pressure tensor of species $r$ and
$M_{2|i}^{(r)}=2q_{r,i}/m_{r}$, where

$\mathbf{q}_{r}=\int d{\bf V}\frac{m_{r}}{2}V^{2}{\bf V}f_{r}({\bf V})$ (19)

is the partial contribution to the total heat flux due to species $r$.

The remaining third degree moments $M_{0|ijk}^{(r)}$ and the moments of degree
$k\geq 4$ are not directly related to hydrodynamic quantities. However, they
provide indirect information on the velocity distribution function $f_{r}$.

##  III Collisional moments for inelastic Maxwell mixtures

As in the elastic case TM80 ; GS03 , the mathematical structure of the Maxwell
collision kernel allows one to express a collisional moment of degree $k$ of
$J_{rs}[f_{r},f_{s}]$ in terms of velocity moments of $f_{r}$ and $f_{s}$ of
degree less than or equal to $k$. Let us now display the explicit expressions
for the collisional moments $J_{2p|i_{1}i_{2}\ldots i_{q}}^{(rs)}$ for
$k=2p+q\leq 4$. Some technical details to obtain those collisional moments are
provided in the Appendix A.

###  III.1 Second degree collisional moments

The second degree collisional moments were already evaluated in Refs. G03 ;
GT10 . They are given by

${J}_{2|0}^{(rs)}\equiv-\frac{dp_{r}}{m_{r}}\zeta_{rs}=-\frac{\omega_{rs}}{4dn_{s}}(1+\beta_{rs})\left[(3-\beta_{rs})n_{s}M_{2|0}^{(r)}-(1+\beta_{rs})n_{r}M_{2|0}^{(s)}\right],$
(20)
${J}_{0|ij}^{(rs)}=-\frac{\omega_{rs}}{2d(d+2)n_{s}}(1+\beta_{rs})\left[(2d+3-\beta_{rs})n_{s}M_{0|ij}^{(r)}-(1+\beta_{rs})n_{r}M_{0|ij}^{(s)}\right],$
(21)

where

$\beta_{rs}=2\mu_{sr}(1+\alpha_{rs})-1.$ (22)

The quantity $\zeta_{rs}$ measures the rate of change of the partial
temperature $T_{r}$ of species $r$ due to collisions with particles of species
$s$. The total cooling rate $\zeta$ is

$\zeta=T^{-1}\sum_{r}\;x_{r}T_{r}\zeta_{r},\quad\zeta_{r}=\sum_{s}\;\zeta_{rs}.$
(23)

According to Eq. (20), the parameters $\zeta_{rs}$ are given by

$\zeta_{rs}=\frac{\omega_{rs}}{d}(1+\beta_{rs})\left[1-\frac{(1+\beta_{rs})(\theta_{r}+\theta_{s})}{4\theta_{s}}\right],$
(24)

where

$\theta_{r}=\frac{m_{r}T}{\overline{m}T_{r}},\quad\overline{m}=\frac{m_{1}+m_{2}}{2}.$
(25)

Note that Eq. (20) yields the result

$\displaystyle\sum_{r,s}m_{r}J_{2|0}^{(rs)}$ $\displaystyle=$ $\displaystyle-
dnT\zeta=-\frac{\omega_{11}}{2}n_{1}T_{1}(1-\alpha_{11}^{2})-\frac{\omega_{22}}{2}n_{2}T_{2}(1-\alpha_{22}^{2})$
(26)
$\displaystyle-\omega_{12}n_{1}\mu_{21}\left(\mu_{21}T_{1}+\mu_{12}T_{2}\right)(1-\alpha_{12}^{2}).$

For elastic collisions ($\alpha_{11}=\alpha_{22}=\alpha_{12}=1$), Eq. (26)
shows that the total kinetic energy is conserved by collisions regardless of
the values of the masses and diameters of the mixture. This is the expected
result.

As said before, the collision frequencies $\omega_{rs}$ can be seen as free
parameters in the model. Its dependence on the parameter space of the mixture
can be chosen to optimize the agreement with the results obtained from the
Boltzmann equation for IHS. As in previous papers G03 ; GT10 , we adjust here
the quantities $\zeta_{rs}$ for IMM, Eq. (24), to be the same as those
obtained for IHS. Since $\zeta_{rs}$ is not exactly known for IHS, we will
estimate $\zeta_{rs}$ by considering Maxwellian distributions defined at
temperatures $T_{r}$ and $T_{s}$ for the distributions $f_{r}$ and $f_{s}$,
respectively. In this approximation, $\zeta_{rs}^{\text{IHS}}$ can be written
as G19

$\zeta_{rs}^{\text{IHS}}\to\frac{4}{d}x_{s}\left(\frac{\sigma_{rs}}{\sigma_{12}}\right)^{d-1}\left(\frac{\theta_{r}+\theta_{s}}{\theta_{r}\theta_{s}}\right)^{1/2}(1+\beta_{rs})\nu_{0}\left[1-\frac{(1+\beta_{rs})(\theta_{r}+\theta_{s})}{4\theta_{s}}\right],$
(27)

where $\sigma_{rs}=(\sigma_{r}+\sigma_{s})/2$ and

$\nu_{0}=\frac{\Omega_{d}}{4\sqrt{\pi}}n\sigma_{12}^{d-1}v_{0}$ (28)

is and effective collision frequency. Here, $v_{0}=\sqrt{2T/\overline{m}}$ is
a thermal speed defined in terms of the granular temperature of the mixture
$T$. It must be noted that the expressions obtained for the collisional
moments in this paper are independent of the specific choice of the collision
rates $\omega_{rs}$.

###  III.2 Third degree collisional moments

The evaluation of the third degree collisional moments is performed in the
Appendix A. The results are

$\displaystyle{J}_{2|i}^{(rs)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8d(d+2)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})\left\\{\left[3\beta_{rs}^{2}-2(d+5)\beta_{rs}+10d+11\right]n_{s}M_{2|i}^{(r)}\right.$
(29) $\displaystyle\left.-3(1+\beta_{rs})^{2}n_{r}M_{2|i}^{(s)}\right\\},$
$\displaystyle{J}_{0|ijk}^{(rs)}$ $\displaystyle=$
$\displaystyle-\frac{3}{4d(d+2)(d+4)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})\Big{\\{}\left[\beta_{rs}^{2}-2(d+3)\beta_{rs}+2d^{2}+10d+9)\right]$
(30) $\displaystyle\times
n_{s}M_{0|ijk}^{(r)}-(1+\beta_{rs})^{2}n_{r}M_{0|ijk}^{(s)}\Big{\\}}.$

Equation (29) is consistent with the expression derived in Ref. G03  when the
mass flux $\mathbf{j}_{r}=\mathbf{0}$.

###  III.3 Fourth degree collisional moments

The calculation of the fourth degree collisional moments is more involved. The
collisional moments ${J}_{4|0}^{(rs)}$ and ${J}_{2|ij}^{(rs)}$ can be written
as

$\displaystyle{J}_{4|0}^{(rs)}$ $\displaystyle=$
$\displaystyle-\frac{1}{d(d+2)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})\Bigg{[}\frac{(3-\beta_{rs})(3\beta_{rs}^{2}-6\beta_{rs}+8d+7)}{16}n_{s}M_{4|0}^{(r)}$
(31)
$\displaystyle-\frac{3}{16}(1+\beta_{rs})^{3}n_{r}M_{4|0}^{(s)}-\frac{(1+\beta_{rs})(3\beta_{rs}^{2}-6\beta_{rs}-1)}{4}\frac{P_{r,ij}P_{s,ij}}{m_{r}m_{s}}$
$\displaystyle-\frac{(1+\beta_{rs})(3\beta_{rs}^{2}-6\beta_{rs}+4d+7)}{8}\frac{d^{2}p_{r}p_{s}}{m_{r}m_{s}}\Bigg{]},$
$\displaystyle{J}_{2|ij}^{(rs)}$ $\displaystyle=$
$\displaystyle\frac{1}{d(d+2)(d+4)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})\left\\{\frac{3}{4}(1+\beta_{rs})^{3}n_{r}M_{2|ij}^{(s)}\right.$
(32)
$\displaystyle+\frac{3\beta_{rs}^{3}-3(d+5)\beta_{rs}^{2}+\beta_{rs}(d^{2}+14d+25)-7d^{2}-31d-21}{4}n_{s}M_{2|ij}^{(r)}$
$\displaystyle+\frac{6\beta_{rs}^{3}-3(d+2)\beta_{rs}^{2}-2(d+7)\beta_{rs}+d-2}{4}\frac{\left(P_{r,ik}P_{s,kj}+P_{r,jk}P_{s,ki}\right)}{m_{r}m_{s}}$
$\displaystyle+\frac{(1+\beta_{rs})(3\beta_{rs}^{2}+2d+5)}{4}\frac{dp_{r}}{m_{r}m_{s}}P_{s,ij}$
$\displaystyle+\frac{(1+\beta_{rs})\left[3\beta_{rs}^{2}-3(d+4)\beta_{rs}+d^{2}+7d+9\right]}{4}\frac{dp_{s}}{m_{r}m_{s}}P_{r,ij}$
$\displaystyle-\frac{(1+\beta_{rs})[6\beta_{rs}^{2}-3(d+4)\beta_{rs}+d^{2}+9d+14]}{4}\frac{dp_{r}p_{s}}{m_{r}m_{s}}\delta_{ij}$
$\displaystyle\left.-\frac{(1+\beta_{rs})[6\beta_{rs}^{2}-3(d+4)\beta_{rs}+d-2]}{2d}\delta_{ij}\frac{P_{r,ij}P_{s,ij}}{m_{r}m_{s}}\right\\},$

The expression of ${J}_{0|ijk\ell}^{(rs)}$ is rather large. For the sake of
concreteness, we will display here two representative collisional moments of
this class: ${J}_{0|xxxx}^{(rs)}$ and ${J}_{0|xxxy}^{(rs)}$. They are given by

$\displaystyle{J}_{0|xxxx}^{(rs)}$ $\displaystyle=$
$\displaystyle\frac{1}{2d(d+2)(d+4)(d+6)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})$
(33)
$\displaystyle\times\Big{\\{}\left[3\beta_{rs}^{3}-(6d+27)\beta_{rs}^{2}+3\beta_{rs}(2d^{2}+16d+27)-4d^{3}-42d^{2}\right.$
$\displaystyle\left.-122d-81\right]n_{s}{M}_{0|xxxx}^{(r)}+3(1+\beta_{rs})^{3}n_{r}{M}_{0|xxxx}^{(s)}$
$\displaystyle+6\frac{3\beta_{rs}^{2}-3\beta_{rs}(d+4)+d^{2}+7d+9}{(d+4)}\frac{(1+\beta_{rs})}{m_{r}m_{s}}$
$\displaystyle\times\Big{[}(d+2)^{-1}(2P_{r,ij}P_{s,ij}+d^{2}p_{r}p_{s})-d(p_{s}P_{r,xx}+p_{r}P_{s,xx})$
$\displaystyle-4P_{r,xk}P_{s,kx}+(d+4)P_{r,xx}P_{s,xx}\Big{]}\Big{\\}},$
$\displaystyle{J}_{0|xxxy}^{(rs)}$ $\displaystyle=$
$\displaystyle\frac{1}{2d(d+2)(d+4)(d+6)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})$
(34)
$\displaystyle\times\Big{\\{}\left[3\beta_{rs}^{3}-(6d+27)\beta_{rs}^{2}+3\beta_{rs}(2d^{2}+16d+27)-4d^{3}-42d^{2}\right.$
$\displaystyle\left.-122d-81\right]n_{s}{M}_{0|xxxy}^{(r)}+3(1+\beta_{rs})^{3}n_{r}{M}_{0|xxxy}^{(s)}$
$\displaystyle+3\frac{3\beta_{rs}^{2}-3\beta_{rs}(d+4)+d^{2}+7d+9}{(d+4)}\frac{(1+\beta_{rs})}{m_{r}m_{s}}$
$\displaystyle\times\Big{[}(d+4)\left(P_{r,xx}P_{s,xy}+P_{r,xy}P_{s,xx}\right)-d(p_{s}P_{r,xy}+p_{r}P_{s,xy})$
$\displaystyle-2\left(P_{r,xk}P_{s,ky}+P_{r,yk}P_{s,kx}\right)\Big{]}\Big{\\}}.$

For mechanically equivalent particles ($m_{1}=m_{2}$, $\sigma_{1}=\sigma_{2}$,
$\alpha_{11}=\alpha_{22}=\alpha_{12}$), all the expressions reduce to known
results for molecular ($\alpha=1$) three-dimensional ($d=3$) Maxwell gases
TM80 ; GS03 . Regarding inelastic collisions ($\alpha<1$) and mechanically
equivalent particles, the expressions for the collisional moments are
consistent with previous results GS07  derived for monodisperse gases.
Moreover, in the one-dimensional inelastic case ($d=1$, $\alpha<1$) our
expressions for binary mixtures of the isotropic collisional moments
${J}_{2|0}^{(rs)}$ and ${J}_{4|0}^{(rs)}$ agree with previous results reported
by Marconi and Puglisi MP02a . This shows the consistency of our expressions
with known results published in the granular literature.

##  IV Relaxation to the homogeneous cooling state

The results derived in the preceding section can be applied to several
interesting situations. Here, we will consider the most basic problem in a
granular mixture: the time evolution of the moments of degree less than or
equal to four (both isotropic and anisotropic) in the HCS. The HCS is a
_homogeneous_ state where the granular temperature $T(t)$ monotonically decays
in time. In this case, the set of Boltzmann kinetic equations (1) for $f_{1}$
and $f_{2}$ becomes

$\partial_{t}f_{1}(\mathbf{v},t)=J_{11}[\mathbf{v}|f_{1},f_{1}]+J_{12}[\mathbf{v}|f_{1},f_{2}],$
(35)

and a similar equation for $f_{2}$. Since the collisions are inelastic, the
granular temperature $T(t)$ monotonically decays in time and so a steady state
does not exist. In the context of IMM, it has been proven BC03 ; BCT03  that,
provided that $f_{r}(\mathbf{v},0)$ has a finite moment of some degree higher
than two, $f_{r}(\mathbf{v},t)$ asymptotically tends toward a self-similar
solution of the form

$f_{r}(\mathbf{v},t)\to n_{r}[v_{0}(t)]^{-d}\phi_{r}(V/v_{0}(t)),$ (36)

where $\phi_{r}(c)$ is an isotropic distribution.

Figure 1: Temperature ratio $T_{1}/T_{2}$ versus the (common) coefficient of
restitution $\alpha$ for a three-dimensional binary mixture ($d=3$) with
$\sigma_{1}/\sigma_{2}=1$, $x_{1}=\frac{1}{2}$, and three different values of
the mass ratio: $m_{1}/m_{2}=5$ (circles), $m_{1}/m_{2}=5$ (triangles), and
$m_{1}/m_{2}=0.5$ (squares). The lines correspond to the theoretical results
obtained here for IMM while symbols refer to the results obtained from the
DSMC method for IHS ChGG22 .

At a hydrodynamic level, the only relevant balance equation is that of the
temperature $T(t)$. Its time evolution equation can be easily obtained from
the moments $M_{2|0}^{(1)}$ and $M_{2|0}^{(2)}$ and it is given by

$\partial_{t}T=-\zeta T.$ (37)

However, on a kinetic level, it is also interesting to discuss energy transfer
through the partial temperatures $T_{r}$. The time evolution of the
temperatures $T_{r}$ can be derived from the velocity moment $M_{2|0}^{(r)}$:

$\partial_{t}T_{r}=-\zeta_{r}T_{r},$ (38)

where we recall that $\zeta_{r}=\sum_{s}\zeta_{rs}$ and $\zeta_{rs}$ is given
by Eq. (24). The time evolution of the temperature ratio $\gamma\equiv
T_{1}(t)/T_{2}(t)$ follows from Eq. (38) as

$\partial_{t}\ln\gamma=\zeta_{2}-\zeta_{1}.$ (39)

After a transient period, it is expected that the mixture achieves a
hydrodynamic regime where all the time dependence of $f_{r}$ only occurs
through the granular temperature $T(t)$. This necessarily implies that the
three temperatures $T_{1}(t)$, $T_{2}(t)$ and $T(t)$ are proportional to each
other and their ratios are all constant. This does not mean that all three
temperatures are equal (as in the case of elastic collisions) since both
partial temperatures are in general different GD99 . According to Eq. (39), if
$\gamma\equiv\text{const}$ then $\zeta_{1}(t)=\zeta_{2}(t)$ in the
hydrodynamic regime. The condition of equal partial cooling rates provides the
dependence of the temperature ratio on the parameters of the mixture.

Figure 1 shows the dependence of the temperature ratio $T_{1}/T_{2}$ on the
(common) coefficient of restitution $\alpha_{rs}\equiv\alpha$ for a three-
dimensional binary mixture ($d=3$) for $\sigma_{1}/\sigma_{2}=1$,
$x_{1}=\frac{1}{2}$, and three different values of the mass ratio. Theoretical
results for IMM are compared against Monte Carlo simulations carried out in
Ref. ChGG22  for IHS. Figure 1 highlights one of the most characteristic
features of granular mixtures (as compared with molecular mixtures): the
partial temperatures are different even in homogenous states. We observe that
the departure of $\gamma$ from 1 (breakdown of energy equipartition) increases
with increasing the disparity in the mass ratio. In general, the temperature
of the lighter species is smaller than that of the heavier species. It is also
important to remark the excellent agreement found between theory (developed
for IMM) and computer simulations (performed for IHS), even for quite strong
inelasticity.

###  IV.1 Eigenvalues for inelastic Maxwell mixtures

Figure 2: Panel A: Dependence of the eigenvalues $\nu_{2|0}^{(11)}$ (a),
$\nu_{0|ij}^{(11)}$ (b), $\nu_{2|i}^{(11)}$ (c), and $\nu_{0|ijk}^{(11)}$ (d)
for a three-dimensional binary mixture constituted by particles of the same
mass density [$m_{1}/m_{2}=(\sigma_{1}/\sigma_{2})^{3}$] with
$x_{1}=\frac{1}{2}$ and $m_{1}/m_{2}=2$. Panel B: The same as panel A for the
eigenvalues $\nu_{2|0}^{(12)}$ (a), $\nu_{0|ij}^{(12)}$ (b),
$\nu_{2|i}^{(12)}$ (c), and $\nu_{0|ijk}^{(12)}$ (d).

Figure 3: Panel A: Dependence of the eigenvalues $\nu_{4|0}^{(11)}$ (a),
$\nu_{2|ij}^{(11)}$ (b), and $\nu_{0|ijk\ell}^{(11)}$ (c) for a three-
dimensional binary mixture constituted by particles of the same mass density
[$m_{1}/m_{2}=(\sigma_{1}/\sigma_{2})^{3}$] with $x_{1}=\frac{1}{2}$ and
$m_{1}/m_{2}=2$. Panel B: The same as panel A for the eigenvalues
$\nu_{4|0}^{(12)}$ (a), $\nu_{2|ij}^{(12)}$ (b), and $\nu_{0|ijk\ell}^{(12)}$
(c).

Apart from the partial temperatures, it is worthwhile analyzing the time
evolution of the higher-degree velocity moments in the HCS. To get this
equation, one takes velocity moments in both sides of Eq. (35) and achieves
the equation

$\partial_{t}M_{2p|\bar{q}}^{(1)}=J_{2p|\bar{q}}^{(11)}+J_{2p|\bar{q}}^{(12)},$
(40)

and a similar equation for species $2$. Here, we have introduced the short-
hand notation $\bar{q}\equiv i_{1}i_{2}\ldots i_{q}$. To study the time
evolution of the moments it is convenient to introduce the _scaled_ moments

$M_{2p|\bar{q}}^{*(r)}(t)\equiv
n_{r}^{-1}[v_{0}(t)]^{-(2p+q)}M_{2p|\bar{q}}^{(r)}(t).$ (41)

In accordance with Eq. (36), one expects that after a transient regime the
dimensionless moments $M_{2p|\bar{q}}^{*(r)}$ (scaled with the time-dependent
thermal velocity) reach an asymptotic steady value.

The time evolution of the scaled moments $M_{2p|\bar{q}}^{*(r)}$ can be
obtained when one takes into account the time evolution equation (37) for the
temperature $T(t)$. In that case, from Eqs. (40) and (37) one simply gets

$\partial_{\tau}M_{2p|\bar{q}}^{*(1)}=J_{2p|\bar{q}}^{*(11)}+J_{2p|\bar{q}}^{*(12)}+\frac{2p+q}{2}\zeta^{*}M_{2p|\bar{q}}^{*(1)},$
(42)

where $\zeta^{*}=\zeta/\nu_{0}$,

$J_{2p|\bar{q}}^{*(rs)}=\frac{J_{2p|\bar{q}}^{(rs)}}{\nu_{0}n_{r}v_{0}^{2p+q}},$
(43)

and

$\tau=\int_{0}^{t}dt^{\prime}\nu_{0}(t^{\prime}).$ (44)

Here, $\nu_{0}(t)\propto nT^{1/2}$ is defined by Eq. (28). The parameter
$\tau$ measures time as the number of (effective) collisions per particle.

It is quite apparent from Eqs. (20), (21), and (29)–(34) that a collisional
moment of degree $k$ of the operator $J_{rs}[f_{r},f_{s}]$ can be expressed in
terms of velocity moments of a degree less than or equal to $k$. More
specifically, the combination $J_{2p|\bar{q}}^{*(11)}+J_{2p|\bar{q}}^{*(12)}$
has the structure

$J_{2p|\bar{q}}^{*(11)}+J_{2p|\bar{q}}^{*(12)}=-\nu_{2p|q}^{(11)}M_{2p|\bar{q}}^{*(1)}-\nu_{2p|q}^{(12)}M_{2p|\bar{q}}^{*(2)}+\mathcal{C}_{2p|\bar{q}}^{(11)}+\mathcal{C}_{2p|\bar{q}}^{(12)},$
(45)

where the terms $\mathcal{C}_{2p|\bar{q}}^{(rs)}$ are bilinear combinations of
moments of degree less than $2p+q$. Since the first two terms on the right-
hand side of Eq. (45) are linear, then the quantities $\nu_{2p|q}^{(11)}$ and
$\nu_{2p|q}^{(12)}$ can be considered as the eigenvalues (or _collisional
rates_) of the linearized collision operators corresponding to the
eigenfunctions $Y_{2p|\bar{q}}$. Their explicit forms for velocity moments of
degree less than or equal to four are given in the Appendix B.

As an illustration, the dependence of the eigenvalues (collision rates)
associated with the second, third, and fourth degree moments on the (common)
coefficient of restitution $\alpha_{rs}\equiv\alpha$ is plotted in Figs. 2 and
3 for a binary mixture constituted by particles of the same mass density.
Here, $d=3$, $x_{1}=\frac{1}{2}$, and $m_{1}/m_{2}=2$. While the eigenvalues
$\nu_{0|ij}^{(11)}$ and $\nu_{0|ijk}^{(11)}$ decrease with increasing
inelasticity, the other two eigenvalues ($\nu_{2|0}^{(11)}$ and
$\nu_{2|i}^{(11)}$) exhibit a non-monotonic dependence on $\alpha$. The
eigenvalues of the second- and third-degree moments associated to cross-
collisions ($\nu_{2|0}^{(12)}$, $\nu_{0|ij}^{(12)}$, $\nu_{2|i}^{(12)}$, and
$\nu_{0|ijk}^{(12)}$) are negative and they increase with decreasing $\alpha$.
A similar behavior can be found for the eigenvalues associated with the
fourth-degree moments, as Fig. 3 shows. In general, we can conclude that the
influence of inelasticity on those eigenvalues is in general important,
specially in the case of the ones associated with the self-collisions (i.e.,
those of the form $\nu_{2p|q}^{(11)}$).

###  IV.2 Time evolution of the velocity moments

Let us evaluate the time evolution of the velocity moments. Thus, inserting
the expression (45) into Eq. (42), in matrix form one finally gets

$\left(\delta_{\sigma\sigma^{\prime}}\partial_{\tau}+\mathcal{L}_{\sigma\sigma^{\prime}}\right)\mathcal{M}_{\sigma\sigma^{\prime}}=\mathcal{C}_{\sigma},$
(46)

where $\boldsymbol{\mathcal{M}}$ is the column matrix defined by the set

$\left\\{M_{2p|\bar{q}}^{*(1)},M_{2p|\bar{q}}^{*(2)}\right\\},$ (47)

and the square matrix $\boldsymbol{\mathcal{L}}$ is given by

$\boldsymbol{\mathcal{L}}=\left(\begin{array}[]{cc}\omega_{2p|q}^{(11)}&\nu_{2p|q}^{(12)}\\\
\nu_{2p|q}^{(21)}&\omega_{2p|q}^{(22)}\end{array}\right),$ (48)

where

$\omega_{2p|q}^{(11)}=\nu_{2p|q}^{(11)}-\frac{2p+q}{2}\zeta^{*},\quad\omega_{2p|q}^{(22)}=\nu_{2p|q}^{(22)}-\frac{2p+q}{2}\zeta^{*}.$
(49)

The quantities $\omega_{2p|q}^{(rs)}$ can be seen as _shifted_ collisional
rates associated with the scaled moments $M_{2p|\bar{q}}^{*(r)}$. Moreover,
the column matrix $\boldsymbol{\mathcal{C}}$ is

$\boldsymbol{\mathcal{C}}=\left(\begin{array}[]{c}{\mathcal{C}}_{2p|\bar{q}}^{(11)}+{\mathcal{C}}_{2p|\bar{q}}^{(12)}\\\
{\mathcal{C}}_{2p|\bar{q}}^{(22)}+{\mathcal{C}}_{2p|\bar{q}}^{(21)}\end{array}\right).$
(50)

The solution of Eq. (46) can be written as

$\boldsymbol{\mathcal{M}}(\tau)=e^{-\boldsymbol{\mathcal{L}}\tau}\cdot\left[\boldsymbol{\mathcal{M}}(0)-\boldsymbol{\mathcal{M}}(\infty)\right]+\boldsymbol{\mathcal{M}}(\infty),$
(51)

where the asymptotic steady value $\boldsymbol{\mathcal{M}}(\infty)$ is given
by

$\boldsymbol{\mathcal{M}}(\infty)=\boldsymbol{\mathcal{L}}^{-1}\cdot\boldsymbol{\mathcal{C}}.$
(52)

The long time behavior of $\mathcal{M}_{\sigma}$ $(\sigma=1,2)$ is governed by
the smallest eigenvalue $\ell_{2p|\bar{q}}^{\text{min}}$ of the matrix
$\boldsymbol{\mathcal{L}}$. Given that the eigenvalues $\ell$ of the matrix
$\boldsymbol{\mathcal{L}}$ obey the quadratic equation

$\ell^{2}-(\omega_{2p|q}^{(11)}+\omega_{2p|q}^{(22)})\ell+(\omega_{2p|q}^{(11)}\omega_{2p|q}^{(22)}-\nu_{2p|q}^{(12)}\nu_{2p|q}^{(21)})=0,$
(53)

the smallest eigenvalue $\ell_{2p|\bar{q}}^{\text{min}}$ is

$\ell_{2p|\bar{q}}^{\text{min}}=\frac{\omega_{2p|q}^{(11)}+\omega_{2p|q}^{(22)}-\sqrt{\left(\omega_{2p|q}^{(11)}+\omega_{2p|q}^{(22)}\right)^{2}-4\left(\omega_{2p|q}^{(11)}\omega_{2p|q}^{(22)}-\nu_{2p|q}^{(12)}\nu_{2p|q}^{(21)}\right)}}{2}.$
(54)

If $\ell_{2p|\bar{q}}^{\text{min}}>0$, then all the scaled moments of degree
$2p+q$ tend asymptotically to finite values. Otherwise, for given values of
the parameters of the mixture, if $\ell_{2p|\bar{q}}^{\text{min}}$ becomes
negative for $\alpha_{rs}$ smaller than a certain critical value $\alpha_{c}$,
then the moments of degree $2p+q$ exponentially grow in time for
$\alpha_{rs}<\alpha_{c}$. The critical value $\alpha_{c}$ can be obtained from
the condition $\ell=0$ which implies

$\omega_{2p|q}^{(11)}\omega_{2p|q}^{(22)}-\nu_{2p|q}^{(12)}\nu_{2p|q}^{(21)}=0.$
(55)

##  V Singular behavior of the third and fourth degree moments

Figure 4: Plot of the smallest eigenvalue, $\ell^{\text{min}}_{2|i}$
associated with the long-time evolution of the (scaled) third degree
(anisotropic) moments $\left\\{M_{2|i}^{*(1)},M_{2|i}^{*(2)}\right\\}$ as a
function of the (common) coefficient of restitution $\alpha$ for a three-
dimensional system ($d=3$) with $x_{1}=0.1$, $\sigma_{1}/\sigma_{2}=0.1$ and
three different values of the mass ratio $m_{1}/m_{2}$.  Figure 5: Phase
diagram in the ($\alpha,m_{1}/m_{2}$)–plane for the asymptotic long time
behavior of the third-degree moments for a three-dimensional system ($d=3$)
with $x_{1}=0.1$ and two different values of the size ratio
$\sigma_{1}/\sigma_{2}$.

The purpose of this section is to analyze the relaxation of the second, third
and fourth degree moments to the HCS. However, a full analysis is difficult
due to the many parameters involved in the problem:
$\alpha_{11},\alpha_{12},\alpha_{22},m_{1}/m_{2},x_{1},$ and
$\sigma_{1}/\sigma_{2}$. For the sake of concreteness, henceforth we will
consider the particular case
$\alpha_{11}=\alpha_{12}=\alpha_{22}\equiv\alpha$. With respect to the second
degree moments (those related to the elements of the pressure tensor), as
expected our results show that these moments are convergent and tend
asymptotically to well-defined values. In this context, it is important to
remark that the reliability of the second degree moments of inelastic Maxwell
mixtures have been tested against Monte Carlo simulations for IHS in the
uniform shear flow problem G03 . The comparison between theory and simulations
shows an excellent agreement between both approaches, even for quite strong
dissipation and/or disparate values of the mass and diameter ratios. Let us
analyze now the behavior of the third and fourth degree velocity moments.

###  V.1 Third-degree moments

In the case of the third degree moments, Eqs. (29) and (30) show that all the
moments tend to zero for sufficiently long times provided the corresponding
eigenvalues are negative. However, our analysis shows that while the moments
$M_{2|ijk}^{*(r)}$ appear to be convergent (at least in the cases studied),
the moments $M_{2|i}^{*(r)}$ (which are related to the heat flux) can be
divergent since the corresponding eigenvalue $\ell^{\text{min}}_{2|i}$ can be
negative. As an illustration, in Fig. 4 we plot $\ell^{\text{min}}_{2|i}$ for
$d=3$ with $x_{1}=\sigma_{1}/\sigma_{2}=0.1$ and three different values of the
mass ratio $m_{1}/m_{2}=2,4$ and 8. It is apparent that, for given values of
the parameters of the mixture, $\ell^{\text{min}}_{2|i}$ becomes negative for
values of the coefficient of restitution smaller than a certain critical value
$\alpha_{c}$. This means that, if $\alpha<\alpha_{c}$, the third degree
moments $M_{2|i}^{(r)}$ exponentially grow in time. This singular behavior of
the scaled third degree moments implies that the velocity distribution
function $f_{r}(V)$ develops an algebraic high velocity tail in the long time
limit of the form $f_{r}(V)\sim V^{-d-s}$. The exponent $s$ is quite sensitive
to the values of the parameters of the mixture. In particular, when the
(scaled) moments $M_{2|i}^{*(r)}$ diverge for given values of the control
parameters, then $s\leq 3$.

An study of the convergent/divergent regions of $M_{2|i}^{*(r)}$ is complex
due to the parameter space of the binary system. However, an exhaustive
analysis of the solutions to Eq. (55) shows that the third degree moments
$M_{2|i}^{(r)}$ appear usually to be convergent (i) when
$\sigma_{1}>\sigma_{2}$ and $m_{1}>m_{2}$ , regardless the value of the mole
fraction $x_{1}$, or (ii) when $\sigma_{1}<\sigma_{2}$, $m_{1}>m_{2}$ but the
mole fraction $x_{1}$ is not small enough (say, $x_{1}\gtrsim 0.2$). To
illustrate these trends, Fig. 5 shows a phase diagram associated with the
singular behavior of the third-degree moments $M_{2|i}^{*(r)}$. Here, $d=3$,
$x_{1}=0.1$ and two different values of the size ratio $\sigma_{1}/\sigma_{2}$
are considered. The curves $\alpha_{c}(m_{1}/m_{2})$ split the parameter space
into two regions: the region below the curve corresponds to states
($\alpha,m_{1}/m_{2}$) where the third-degree moments $M_{2|i}^{*(r)}$ diverge
in time while the region above defines the states where those moments are
convergent (and so they go to zero). It is apparent that the region of
divergent moments grows as the size of the defect species decreases with
respect to that of the excess component.

Figure 6: Phase diagram in the ($\alpha,m_{1}/m_{2}$)–plane for the asymptotic
long time behavior of the fourth-degree moments $M_{4|0}^{*(r)}$ and
$M_{2|ij}^{*(r)}$ for a three-dimensional system ($d=3$) with $x_{1}=0.1$ and
$\sigma_{1}/\sigma_{2}=0.2$.  Figure 7: Phase diagram in the
($\alpha,\sigma_{1}/\sigma_{2}$)–plane for the asymptotic long time behavior
of the fourth-degree moments $M_{4|0}^{*(r)}$ for a three ($d=3$) and two
($d=2$) dimensional system with $x_{1}=0.5$ and $m_{1}/m_{2}=2$.  Figure 8:
Phase diagram in the ($\alpha,\sigma_{1}/\sigma_{2}$)–plane for the asymptotic
long time behavior of the fourth-degree moments $M_{4|0}^{*(r)}$ for a three
($d=3$) and two ($d=2$) dimensional system with $x_{1}=0.5$ and
$m_{1}/m_{2}=0.5$.

###  V.2 Fourth-degree moments

In the HCS state, $P_{r,ij}=p_{r}\delta_{ij}$ and so, the inhomogeneous terms
${\mathcal{C}}_{2p|q}^{(rs)}$ appearing in the right hand side of the
expressions (31)–(34) become

$\mathcal{C}_{4|0}^{(rs)}=\frac{\omega_{rs}^{*}}{32}(1+\beta_{rs})^{2}\frac{3\beta_{rs}^{2}-6\beta_{rs}+4d-1}{\theta_{r}\theta_{s}},\quad\mathcal{C}_{2|ij}^{(rs)}=\mathcal{C}_{0|ijk\ell}^{(rs)}=0.$
(56)

Consequently, if the eigenvalues of the matrix $\boldsymbol{\mathcal{L}}$
corresponding to the fourth-degree moments are positive, Eq. (56) shows that
all the moments, except $M_{4|0}^{*(r)}$ tend to zero for $\tau\to\infty$. The
asymptotic expression of $M_{4|0}^{*(r)}$ is

$M_{4|0}^{*(1)}=\frac{\omega_{4|0}^{(22)}(\mathcal{C}_{4|0}^{(11)}+\mathcal{C}_{4|0}^{(12)})-\nu_{4|0}^{(12)}(\mathcal{C}_{4|0}^{(22)}+\mathcal{C}_{4|0}^{(21)})}{\omega_{4|0}^{(11)}\omega_{4|0}^{(22)}-\nu_{4|0}^{(12)}\nu_{4|0}^{(21)}}.$
(57)

The expression for $M_{4|0}^{*(2)}$ can be easily obtained from (57) by just
making the changes $1\leftrightarrow 2$. Note that when the condition (55)
applies, the fourth-degree moments $M_{4|0}^{*(r)}$ tend to infinite as
expected.

An analysis on the possible divergence of the fourth-degree moments clearly
shows that these moments can be also divergent in some regions of the
parameter space of the system, especially the (isotropic) moment
$M_{4|0}^{*(r)}$. Figure 6 illustrates this fact for $d=3$, $x_{1}=0.1$ and
$\sigma_{1}/\sigma_{2}=0.2$. In this case, while the moments
$M_{0|ijk\ell}^{*(r)}$ are convergent (and tend to zero for long times), the
moments $M_{4|0}^{*(r)}$ and $M_{2|ij}^{*(r)}$ can be divergent (regions below
the curves). It is also apparent that for a given value of the mass ratio, the
critical value of the coefficient of restitution of $M_{4|0}^{*(r)}$ is larger
than the one found for $M_{2|ij}^{*(r)}$ so that the divergent region of the
former is bigger than the latter one. Figures 7 and 8 complement the results
shown in Fig. 6. We plot the phase diagram of $M_{4|0}^{*(r)}$ in the
($\alpha,\sigma_{1}/\sigma_{2}$)-plane for spheres ($d=3$) and disks ($d=2$).
We observe that the influence of the mass ratio is not quite significant on
the form of the phase diagram. Moreover, while the divergent region seems to
be more important for disks than for hard spheres when the size ratio is small
(but larger than one), the opposite happens as the size ratio increases.

##  VI Discussion

The use of collision models where the collision rate of two colliding
particles is independent of their relative velocity allows one to determine
exactly the collisional moments without to know the one-particle velocity
distribution function. In the case of molecular gases (i.e., particles
colliding elastically), the model of Maxwell molecules (namely, when particles
collide with a potential inversely proportional to the fourth power of the
distance) is very useful to obtain exactly transport properties in far from
equilibrium states TM80 ; GS03 . When the collisions are inelastic and
characterized by a constant coefficient of normal restitution $\alpha\leq 1$,
the use of IMM opens up the possibility of obtaining exact results (the
elastic limit $\alpha=1$ is a special limit). This is likely the main reason
for which IMM has attracted the attention of physicists and mathematicians in
the past few years.

In the case of monocomponent granular gases, the choice of the Ikenberry
polynomials $Y_{2p|\overline{q}}$ TM80  of degree $2p+q$ allows one to express
the corresponding collisional moment $J_{2p|\overline{q}}$ as an eigenvalue
$-\nu_{2p|q}$ times the velocity moment $M_{2p|\overline{q}}$ plus a bilinear
combination of moments of degree less than $2p+q$. All the third and fourth
degree collisional moments of IMM for monocomponent gases were evaluated in
Ref. GS07 . We have extended in this paper the above results GS07  to the
interesting case of binary granular mixtures. Due to the intricacy of the
general problem, we have considered here situations where diffusion processes
are absent. This means that the mean flow velocities $\mathbf{U}_{r}$ of each
species are equal to the mean flow velocity $\mathbf{U}$ of the mixture
($\mathbf{U}_{1}=\mathbf{U}_{2}=\mathbf{U}$). Apart from this simplification
the results reported here for the third and fourth degree collisional moments
are exact for arbitrary values of the masses $m_{r}$, diameters $\sigma_{r}$,
concentration $x_{r}$, and coefficients of restitution $\alpha_{rs}$. In
addition, all the derived expressions apply for any dimensionality $d$. Known
results for three-dimensional molecular gases TM80 ; GS03  and for
$d$-dimensional monocomponent granular gases GS07  are recovered. In the one-
dimensional case ($d=1$) for binary granular mixtures, our results for the
(isotropic) collisional moments $J_{2|0}^{(rs)}$ and $J_{4|0}^{(rs)}$ agree
with the ones obtained by Marconi and Puglisi MP02a .

As for monocomponent gases GS07 , we have observed that some of the
eigenvalues $\nu_{2p|q}^{(rs)}$ exhibit a non-monotonic dependence on the
coefficients of restitution $\alpha_{rs}$ at a given values of the mass and
diameter ratios and the concentration. We also see that the impact of the
inelasticity in collisions on the eigenvalues is in general important,
specially in the case of the eigenvalues associated with the self-collision
terms. Although the above observations are restricted to the moments of degree
$2p+q\leq 4$, we expect that they extend to moments of higher degree.

As a simple application of the results derived in section III, we have
analyzed the time evolution of the moments of degree equal to or less than 4
in the HCS. In this state, given that the granular temperature $T$ decreases
in time,one has to scale the moments with the thermal speed
$v_{0}(t)=\sqrt{2T(t)/\overline{m}}$ to reach steady values in the long time
limit. Our analysis shows that while all the second degree moments tend toward
finite values in the long time limit, the third degree moments
$M_{2|i}^{*(r)}$ (which are related to the heat flux) can diverge in a region
of the parameter space of the mixture. This sort of divergence also appears in
all (isotropic and anisotropic) fourth degree moments. The above conclusions
contrast with the ones achieved for monocomponent granular gases GS07  where
all the moments of degree $2+p+q\leq 4$ are convergent for $d\geq 2$. The
singular behavior of the third degree moments is consistent with an algebraic
high velocity tail of the form $f_{r}(V)\sim V^{-(d+s)}$, where $s\leq 3$ when
the moments $M_{2|i}^{*(r)}$ are divergent. We plan to explore this
possibility in the near future.

The explicit results provided in this paper can be employed to analyze
different nonequilibrium problems. One of them is the so-called simple or
uniform shear flow; which is intrinsically a non-Newtonian state. Apart from
the rheological properties G03 , it would be interesting to study the time
evolution of the fourth degree velocity moments toward their steady values and
investigate whether these moments can be divergent as occurs for elastic
collisions SG95 . Another interesting application of the present results is to
determine some of the generalized transport coefficients characterizing small
perturbations around the simple shear flow problem. Work along these lines
will be carried out in the near future.

## Acknowledgments

The authors acknowledge financial support from Grant PID2020-112936GB-I00
funded by MCIN/AEI/ 10.13039/501100011033 (V. G.), and from Grants IB20079 (V.
G.) and GR21014 (C. S. R. and V. G.) funded by Junta de Extremadura (Spain)
and by ERDF “A way of making Europe.”

##  Appendix A Some technical details in the evaluation of the collisional
moments

In this Appendix we give some technical details on the derivation of the
collisional moments $J_{2p|i_{1}i_{2}\ldots i_{q}}^{(rs)}$ associated with the
Ikenberry polynomials of third and fourth degree when the mass flux of each
species vanishes.

We consider for the sake of concreteness the (anisotropic) third degree
collisional moment

$\displaystyle J_{0|ijk}^{(rs)}$ $\displaystyle=$ $\displaystyle\int d{\bf
V}\left[V_{i}V_{j}V_{k}-\frac{1}{d+2}V^{2}\left(V_{i}\delta_{jk}+V_{j}\delta_{ik}+V_{k}\delta_{ij}\right)\right]J_{rs}[f_{r},f_{s}]$
(58) $\displaystyle=$ $\displaystyle
J_{ijk}^{(rs)}-\frac{1}{d+2}\left(J_{2|i}^{(rs)}\delta_{jk}+J_{2|j}^{(rs)}\delta_{ik}+J_{2|k}^{(rs)}\delta_{ij}\right),$

where

$J_{ijk}^{(rs)}=\int d{\bf V}V_{i}V_{j}V_{k}J_{rs}[f_{r},f_{s}],$ (59)
$J_{2|i}^{(rs)}=\int d{\bf V}V^{2}V_{i}J_{rs}[f_{r},f_{s}].$ (60)

Let us evaluate first the third degree (canonical) collisional moment
$J_{ijk}^{(rs)}$. Taking into account the property (8), Eq. (59) can be
written as

$J_{ijk}^{(rs)}=\frac{\omega_{rs}}{n_{s}\Omega_{d}}\int\,d{\bf
V}_{1}\,\int\,d{\bf V}_{2}f_{r}({\bf V}_{1})f_{s}({\bf V}_{2})\int
d\widehat{\boldsymbol{\sigma}}\,\left(V_{1i}^{\prime}V_{1j}^{\prime}V_{1k}^{\prime}-V_{1i}V_{1j}V_{1k}\right),$
(61)

where Eq. (9) gives the relationship between the post-collisional velocity
$\mathbf{V}_{1}^{\prime}$ and the pre-collisional velocity $\mathbf{V}_{1}$.
The fact that ${\bf U}_{1}={\bf U}_{2}={\bf U}$ allows one to map some of the
results obtained for monodisperse gases GS07  by making the change

$\alpha\to\beta_{rs}\equiv 2\mu_{sr}(1+\alpha_{rs})-1.$ (62)

In particular, the scattering rule (8) implies that

$\displaystyle
V_{1i}^{\prime}V_{1j}^{\prime}V_{1k}^{\prime}-V_{1i}V_{1j}V_{1k}$
$\displaystyle=$
$\displaystyle-\frac{1+\beta_{rs}}{2}(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})\Big{\\{}V_{1i}V_{1j}\widehat{\sigma}_{k}+V_{1i}V_{1k}\widehat{\sigma}_{j}+V_{1j}V_{1k}\widehat{\sigma}_{i}$
(63)
$\displaystyle-\frac{1+\beta_{rs}}{2}(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})\Big{[}V_{1i}\widehat{\sigma}_{j}\widehat{\sigma}_{k}+V_{1j}\widehat{\sigma}_{i}\widehat{\sigma}_{k}+V_{1k}\widehat{\sigma}_{i}\widehat{\sigma}_{j}$
$\displaystyle-\frac{1+\beta_{rs}}{2}(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})\widehat{\sigma}_{i}\widehat{\sigma}_{j}\widehat{\sigma}_{k}\Big{]}\Big{\\}}.$

To carry out the angular integrations in (61) one needs the results

$\int d\widehat{\boldsymbol{\sigma}}\,(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})^{2k+1}\widehat{{\sigma}}_{i}=B_{k+1}g^{2k}g_{i},$ (64) $\int
d\widehat{\boldsymbol{\sigma}}\,(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})^{2k}\widehat{{\sigma}}_{i}\widehat{{\sigma}}_{j}=\frac{B_{k}}{2k+d}g^{2(k-1)}\left(2kg_{i}g_{j}+g^{2}\delta_{ij}\right),$
(65) $\displaystyle\int
d\widehat{\boldsymbol{\sigma}}\,(\widehat{\boldsymbol{\sigma}}\cdot{\bf
g})^{2k+1}\widehat{\sigma}_{i}\widehat{\sigma}_{j}\widehat{\sigma}_{\ell}$
$\displaystyle=$
$\displaystyle\frac{B_{k+1}}{2(k+1)+d}g^{2(k-1)}\big{[}2kg_{i}g_{j}g_{\ell}+g^{2}\big{(}\delta_{ij}g_{\ell}+\delta_{i\ell}g_{j}$
(66) $\displaystyle+\delta_{j\ell}g_{i}\big{)}\big{]}.$

Here, the coefficients $B_{k}$ are NE98

$B_{k}=\int
d\widehat{\boldsymbol{\sigma}}\,(\widehat{\boldsymbol{\sigma}}\cdot{\widehat{\bf
g}})^{2k}=\Omega_{d}\pi^{-1/2}\frac{\Gamma\left(\frac{d}{2}\right)\Gamma\left(k+\frac{1}{2}\right)}{\Gamma\left(k+\frac{d}{2}\right)}.$
(67)

Making use of Eqs. (63)–(66), one gets

$\displaystyle J_{ijk}^{(rs)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2d(d+2)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})\Big{\\{}(d+2)\langle
V_{1i}V_{1j}g_{k}+V_{1i}V_{1k}g_{j}+V_{1j}V_{1k}g_{i}\rangle$ (68)
$\displaystyle-\frac{(1+\beta_{rs})}{2}\big{[}2\langle
g_{i}g_{j}V_{1k}+g_{i}g_{k}V_{1j}+g_{j}g_{k}V_{1i}\rangle$
$\displaystyle+\langle
g^{2}(V_{1i}\delta_{jk}+V_{1j}\delta_{ik}+V_{1k}\delta_{ij})\rangle\big{]}+\frac{3}{4(d+4)}(1+\beta_{rs})^{2}$
$\displaystyle\times\big{[}2\langle
g_{i}g_{j}g_{k}+g^{2}(g_{i}\delta_{jk}+g_{j}\delta_{ik}+g_{k}\delta_{ij})\rangle\big{]}\Big{\\}},$

where the brackets are defined as

$\langle h({\bf V}_{1},{\bf V}_{2})\rangle\equiv\int d{\bf V}_{1}\int d{\bf
V}_{2}h({\bf V}_{1},{\bf V}_{2})f_{r}({\bf V}_{1})f_{s}({\bf V}_{2}).$ (69)

The integrations over velocities give the relations

$\langle V_{1i}V_{1j}V_{1k}\rangle=\langle
g_{i}g_{j}V_{1k}\rangle=n_{s}M_{ijk}^{(r)},\quad\langle
g^{2}V_{1i}\rangle=n_{s}M_{2|i}^{(r)},$ (70) $\langle
g_{i}g_{j}g_{k}\rangle=n_{s}M_{ijk}^{(r)}-n_{r}M_{ijk}^{(rs)},$ (71)

where

$M_{ijk}^{(r)}=\int\;d{\bf V}V_{i}V_{j}V_{k}f_{r}({\bf V}).$ (72)

Therefore, from Eqs. (68) and (70)–(72) one finally obtains

$\displaystyle J_{ijk}^{(rs)}$ $\displaystyle=$
$\displaystyle-\frac{3}{4d(d+2)(d+4)}\frac{\omega_{rs}}{n_{s}}(1+\beta_{rs})\Big{\\{}[\beta_{rs}^{2}-2(d+3)\beta_{rs}+2d^{2}+10d+9)]$
(73) $\displaystyle\times
n_{s}M_{ijk}^{(r)}-(1+\beta_{rs})^{2}n_{r}M_{ijk}^{(s)}$
$\displaystyle+\frac{(1+\beta_{rs})}{6}(3\beta_{rs}-2d-5)n_{s}\left(M_{2|i}^{(r)}\delta_{kj}+M_{2|j}^{(r)}\delta_{ik}+M_{2|k}^{(r)}\delta_{ij}\right)$
$\displaystyle-\frac{(1+\beta_{rs})^{2}}{2}n_{r}\left(M_{2|i}^{(s)}\delta_{jk}+M_{2|j}^{(s)}\delta_{ik}+M_{2|k}^{(s)}\delta_{ij}\right)\Big{\\}}.$

If one makes $j=k$ and sum over $j$ one obtains the expression (29) for
$J_{2|i}^{(rs)}$. Also, by subtracting
$\left(J_{2|i}^{(rs)}\delta_{jk}+J_{2|i}^{(rs)}\delta_{jk}+J_{2|i}^{(rs)}\delta_{jk}\right)/(d+2)$
from both sides of Eq. (73) one gets Eq. (30) for $J_{0|ijk}^{(rs)}$.

The calculations for the fourth degree collisional moments are similar to
those carried out for the third degree moments. In addition, most of the
mathematical steps followed to get them can be easily mapped from those made
for monodisperse gases in Ref. GS07  by replacing $\alpha\to\beta_{rs}$. After
a long and tedious algebra one obtains the results displayed along the Section
II.

##  Appendix B Explicit forms of the eigenvalues

In this Appendix we give the explicit expressions of the eigenvalues
$\nu_{2p|q}^{(11)}$ and $\nu_{2p|q}^{(12)}$ with $2p+q\leq 4$ of the
collisional moments $J_{2p|\bar{q}}^{*(rs)}$. In the case of the second degree
collisional moments, the eigenvalues are

$\nu_{2|0}^{(11)}=\omega_{11}^{*}\frac{1-\alpha_{11}^{2}}{2d}+\frac{\omega_{12}^{*}}{4d}(1+\beta_{12})(3-\beta_{12}),\quad\nu_{2|0}^{(12)}=-\frac{\omega_{12}^{*}}{4d}(1+\beta_{12})^{2},$
(74)
$\nu_{0|ij}^{(11)}=\frac{\omega_{11}^{*}}{d(d+2)}(1+\alpha_{11})(d+1-\alpha_{11})+\frac{\omega_{12}^{*}}{2d(d+2)}(1+\beta_{12})(2d+3-\beta_{12}),$
(75) $\nu_{0|ij}^{(12)}=-\frac{\omega_{12}^{*}}{2d(d+2)}(1+\beta_{12})^{2},$
(76)

where $\omega_{ij}^{*}=\omega_{ij}/\nu_{0}$. The corresponding eigenvalues in
the case of the third degree collisional moments are given by

$\displaystyle\nu_{2|i}^{(11)}$ $\displaystyle=$
$\displaystyle\frac{\omega_{11}^{*}}{4d(d+2)}(1+\alpha_{11})[4+5d-(d+8)\alpha_{11}]+\frac{\omega_{12}^{*}}{8d(d+2)}(1+\beta_{12})$
(77) $\displaystyle\times[3\beta_{12}^{2}-2(d+5)\beta_{12}+10d+11],$
$\nu_{2|i}^{(12)}=-3\frac{\omega_{12}^{*}}{8d(d+2)}(1+\beta_{12})^{3},$ (78)
$\displaystyle\nu_{0|ijk}^{(11)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\frac{\omega_{11}^{*}}{d(d+2)}(1+\alpha_{11})[d+1-\alpha_{11}]+3\frac{\omega_{12}^{*}}{4d(d+2)(d+4)}(1+\beta_{12})$
(79) $\displaystyle\times[\beta_{12}^{2}-2(d+3)\beta_{12}+2d^{2}+10d+9],$
$\nu_{0|ijk}^{(12)}=-3\frac{\omega_{12}^{*}}{4d(d+2)(d+4)}(1+\beta_{12})^{3}.$
(80)

Finally, the eigenvalues of the fourth degree collisional moments are

$\displaystyle\nu_{4|0}^{(11)}$ $\displaystyle=$
$\displaystyle\frac{\omega_{11}^{*}}{8d(d+2)}(1+\alpha_{11})[9+12d-(4d+17)\alpha_{11}+3\alpha_{11}^{2}-3\alpha_{11}^{3}]$
(81)
$\displaystyle+\frac{\omega_{12}^{*}}{16d(d+2)}(1+\beta_{12})(3-\beta_{12})[3\beta_{12}^{2}-6\beta_{12}+8d+7],$
$\nu_{4|0}^{(12)}=-3\frac{\omega_{12}^{*}}{16d(d+2)}(1+\beta_{12})^{4},$ (82)
$\displaystyle\nu_{2|ij}^{(11)}$ $\displaystyle=$
$\displaystyle\frac{\omega_{11}^{*}}{4d(d+2)(d+4)}(1+\alpha_{11})\big{[}7d^{2}+31d+18-(d^{2}+14d+34)\alpha_{11}$
(83)
$\displaystyle+3(d+2)\alpha_{11}^{2}-6\alpha_{11}^{3}\big{]}+\frac{\omega_{12}^{*}}{4d(d+2)(d+4)}(1+\beta_{12})$
$\displaystyle\times\big{[}7d^{2}+31d+21-(d^{2}+14d+25)\beta_{12}+3(d+5)\beta_{12}^{2}-3\beta_{12}^{3}\big{]},$
$\nu_{2|ij}^{(12)}=-3\frac{\omega_{12}^{*}}{4d(d+2)(d+4)}(1+\beta_{12})^{4},$
(84) $\displaystyle\nu_{0|ijk\ell}^{(11)}$ $\displaystyle=$
$\displaystyle\frac{\omega_{11}^{*}}{d(d+2)(d+4)(d+6)}(1+\alpha_{11})\big{[}2d^{3}+21d^{2}+61d+39$
$\displaystyle-3(d+3)(d+5)\alpha_{11}+3(d+3)\alpha_{11}^{2}-3\alpha_{11}^{3}\big{]}$
$\displaystyle+\frac{\omega_{12}^{*}}{2d(d+2)(d+4)(d+6)}(1+\beta_{12})\big{[}4d^{3}+42d^{2}+122d+81$
$\displaystyle-3(2d^{2}+16d+27)\beta_{12}+(6d+27)\beta_{12}^{2}-3\beta_{12}^{3}\big{]},$
$\nu_{0|ijk\ell}^{(12)}=-3\frac{\omega_{12}^{*}}{2d(d+2)(d+4)(d+6)}(1+\beta_{12})^{4}.$
(86)

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