# On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases

Harsha Hutridurga and Francesco Salvarani

###### Abstract.

Multi-species Boltzmann equations for gaseous mixtures, with analytic cross
sections and under Grad’s angular cutoff assumption, are considered under
diffusive scaling. In the limit, we formally obtain an explicit expression for
the binary diffusion coefficients in the Maxwell-Stefan equations.

##  1\. Introduction

The macroscopic description of diffusive phenomena in mixtures goes back to
the 19th Century, thanks to the work of Maxwell [21] and Stefan [23], who
introduced a coupled system of cross-diffusion equations, nowadays known as
the Maxwell-Stefan system. Since then, a wide literature about multicomponent
diffusive phenomena has been published by the Chemical Engineering community
(we refer to [19] for a review on the state-of-the-art on the subject), since
multicomponent diffusion in fluids plays a crucial role in many (bio)chemical
processes.

However, the mathematical and numerical study of the Maxwell-Stefan system is
relatively new, and solid results on the subject appeared only very recently
(see, for example, [15, 14, 13, 16, 4, 5, 18, 22, 8, 11, 6]).

In particular, [9] is devoted to the formal derivation, under the standard
diffusive scaling, of the Maxwell-Stefan diffusion equations from the non-
reactive elastic Boltzmann system for monatomic gaseous mixtures, in the
vanishing Mach and Knudsen numbers limit. The approach follows the research
line introduced in [1, 2, 3] and allows to deduce asymptotically a
compressible dynamics and to obtain, in the limit, an explicit expression of
the binary diffusion coefficients in the Maxwell-Stefan system, depending on
the reduced mass of the species, on the temperature and on the cross sections
of the kinetic model. A peculiar feature of this approach is the possibility
of obtaining explicit coefficients which could be compared with experimental
results and contribute to a better understanding of the quantitative behaviour
of gaseous mixtures.

The main assumptions of [9] are the following:

  * •

the initial conditions are well prepared, and are given by local Maxwellians,
all with the same temperature;

  * •

the process is supposed to be isothermal;

  * •

the cross sections of the Boltzmann system are of Maxwellian type.

The first two assumptions are very natural, since the presence of temperature
gradients may introduce transport phenomena which could hinder the diffusion
process, as shown in [7]. On the other hand, the third one is not completely
satisfactory. Indeed, even if the hypothesis of Maxwellian cross-sections has
been very popular as it could lead to many explicit calculations (Maxwell and
Boltzmann themselves used them very often [25]), it has been subsequently
noticed that the influence of the collision kernel (or equivalently, of the
cross-section) on the solutions to the Boltzmann equation is far from being
negligible [25].

For this reason, it is important to recover the Maxwell-Stefan diffusion
equations from the non-reactive elastic Boltzmann equations for mixtures with
more realistic cross sections. In this article, we introduce a new strategy
which allows us to handle at the formal level general factorized cross
sections, in which the kinetic collision kernel is an analytic function of its
argument and the angular collision kernel is even and satisfies Grad’s cutoff
assumption [17]. By considering these more physical cross sections, we are
able to obtain explicit quantitative expressions of the binary diffusion
coefficients, whose structure exhibits some nonlinearities which were absent
in the case of Maxwellian cross sections.

This study is also interesting from a physical point of view. Indeed, even
though many results are available in the case of binary mixtures (see, for
example, [24]), many attempts have been made to experimentally determine the
binary diffusion coefficients in the Maxwell-Stefan equations, but with a
limited success [20]. We hope that our computation will give a contribution
for encouraging the experimental activity and be useful for comparison with
experimental data.

We finally point out that the results obtained in this article are valid for
monatomic ideal gases only. The extension of our computations to polyatomic
gases should be possible by starting from kinetic systems taking into account
the internal energy of the molecules (such as the model proposed in [10]). We
will consider this extension in a near future.

The structure of the article is the following: after describing the kinetic
model for non-reactive monatomic gaseous mixtures in Section 2 and the
Maxwell-Stefan system in Section 3, in Section 4 we compute the diffusion
limit and obtain an explicit expression of the binary diffusion coefficients
in the case of analytical cross sections under Grad’s cutoff assumption.
Finally, we give an appendix where we explicitly compute the Gaussian
integrals in the expression for the binary diffusion coefficients.

##  2\. The kinetic model

The mathematical form of the Boltzmann system for non-reactive monatomic gas
mixtures is classical, and has been the starting point of many extensions
(such as, for example, [10, 12]). However, in order to make this article self-
consistent, we briefly describe its form.

The model considers a mixture of ideal monatomic inert gases
$\mathcal{A}_{i}$, $i=1,\dots,I$ with $I\geq 2$. Each of them is described by
a distribution function $f_{i}$, which depends on $t\in{\mathbb{R}}^{*}_{+}$
(time), $x\in{\mathbb{R}}^{3}$ (space position) and $v\in{\mathbb{R}}^{3}$
(velocity).

By supposing that no chemical reactions occur and no external forces act on
the mixture, its time evolution is a consequence of the mechanical collisions
between molecules, which are supposed here to be elastic.

Let us consider two particles belonging to the species $\mathcal{A}_{i}$ and
$\mathcal{A}_{j}$, $1\leq i,j\leq I$, with masses $m_{i}$, $m_{j}$, and pre-
collisional velocities $v^{\prime}$, $v_{*}^{\prime}$. A microscopic collision
is an instantaneous phenomenon which modifies the velocities of the particles,
which become $v$ and $v_{*}$, obtained by imposing the conservation of both
momentum and kinetic energy:

(1)
$m_{i}v^{\prime}+m_{j}v^{\prime}_{*}=m_{i}v+m_{j}v_{*},\qquad\frac{1}{2}m_{i}\,|v^{\prime}|^{2}+\frac{1}{2}m_{j}\,|v_{*}^{\prime}|^{2}=\frac{1}{2}m_{i}\,|v|^{2}+\frac{1}{2}m_{j}\,|v_{*}|^{2}.$

The previous equations allow to write $v^{\prime}$ and $v^{\prime}_{*}$ with
respect to $v$ and $v_{*}$:

(2)
$v^{\prime}=\frac{1}{m_{i}+m_{j}}(m_{i}v+m_{j}v_{*}+m_{j}|v-v_{*}|\,\sigma),\qquad
v^{\prime}_{*}=\frac{1}{m_{i}+m_{j}}(m_{i}v+m_{j}v_{*}-m_{i}|v-v_{*}|\,\sigma),$

where $\sigma\in{\mathbb{S}}^{2}$ describes the two degrees of freedom in (1).

If $f$ and $g$ are nonnegative functions, the operator describing the
collisions of molecules of species $\mathcal{A}_{i}$ with molecules of species
$\mathcal{A}_{j}$ is defined by

(3)
$Q_{ij}(f,g)(v):=\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{S}}^{2}}B_{ij}(v,v_{*},\sigma)\big{[}f(v^{\prime})g(v^{\prime}_{*})-f(v)g(v_{*})\big{]}\,\mathrm{d}\sigma\,\mathrm{d}v_{*},$

where $v^{\prime}$ and $v^{\prime}_{*},$ are given by (2), and the cross
section $B_{ij}$ satisfies the microreversibility assumptions
$B_{ij}(v,v_{*},\sigma)=B_{ji}(v_{*},v,\sigma)$ and
$B_{ij}(v,v_{*},\sigma)=B_{ij}(v^{\prime},v^{\prime}_{*},\sigma)$.

It is clear that, when $i=j$, the previous expressions reduce to the standard
Boltzmann kernel in the mono-species case:

(4)
$Q_{ii}(f,f)(v)=\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{S}}^{2}}B_{ii}(v,v_{*},\sigma)\big{[}f(v^{\prime})f(v^{\prime}_{*})-f(v)f(v_{*})\big{]}\,\mathrm{d}\sigma\,\mathrm{d}v_{*}.$

The operators $Q_{ij}$ can be written in weak form. For example, by using the
changes of variables $(v,v_{*})\mapsto(v_{*},v)$ and
$(v,v_{*})\mapsto(v^{\prime},v^{\prime}_{*})$, we have

(5) $\int_{{\mathbb{R}}^{3}}Q_{ij}(f,g)(v)\,\psi(v)\,\mathrm{d}v\\\
=-\frac{1}{2}\int_{{\mathbb{R}}^{6}}\int_{{\mathbb{S}}^{2}}B_{ij}(v,v_{*},\sigma)\big{[}f(v^{\prime})g(v^{\prime}_{*})-f(v)g(v_{*})\big{]}\big{[}\psi(v^{\prime})-\psi(v)\big{]}\,\mathrm{d}\sigma\,\mathrm{d}v\,\mathrm{d}v_{*}\\\
=\int_{{\mathbb{R}}^{6}}\int_{{\mathbb{S}}^{2}}B_{ij}(v,v_{*},\sigma)\,f(v)g(v_{*})\,\left[\psi(v^{\prime})-\psi(v)\right]\,\mathrm{d}\sigma\,\mathrm{d}v\,\mathrm{d}v_{*},$

or

(6)
$\int_{{\mathbb{R}}^{3}}Q_{ij}(f,g)(v)\,\psi(v)\,\mathrm{d}v+\int_{{\mathbb{R}}^{3}}Q_{ji}(g,f)(v)\,\phi(v)\,\mathrm{d}v=\\\
-\frac{1}{2}\int_{{\mathbb{R}}^{6}}\int_{{\mathbb{S}}^{2}}B_{ij}(v,v_{*},\sigma)\big{[}f(v^{\prime})g(v^{\prime}_{*})-f(v)g(v_{*})\big{]}\big{[}\psi(v^{\prime})+\phi(v^{\prime}_{*})-\psi(v)-\phi(v_{*})\big{]}\,\mathrm{d}\sigma\,\mathrm{d}v\,\mathrm{d}v_{*},$

for any $\psi$, $\phi:\,{\mathbb{R}}^{3}\to{\mathbb{R}}$ such that the first
integrals in (5)–(6) are well defined.

When $\psi(v)=1$ in (5) we deduce the conservation of the total number of
molecules of species $\mathcal{A}_{i}$. Moreover, if $\psi(v)=m_{i}\,v$ and
$\phi(v_{*})=m_{j}\,v_{*}$, and then if $\psi(v)=m_{i}\,|v|^{2}/2$ and
$\phi(v)=m_{j}\,|v_{*}|^{2}/2$, we recover the conservation of the total
momentum and of the total kinetic energy during the collision between a
particle of species $\mathcal{A}_{i}$ and a particle of species
$\mathcal{A}_{j}$:

(7)
$\int_{{\mathbb{R}}^{3}}Q_{ij}(f,g)(v)\,\left(\begin{array}[]{c}m_{i}\,v\\\
m_{i}\,{|v|^{2}}/2\end{array}\right)\,\mathrm{d}v+\,\int_{{\mathbb{R}}^{3}}Q_{ji}(g,f)(v)\,\left(\begin{array}[]{c}m_{j}\,v\\\
m_{j}\,{|v|^{2}}/2\end{array}\right)\,\mathrm{d}v=0.$

The system of equations satisfied by the set of distribution functions
$(f_{i})_{1\leq i\leq I}$ is hence

(8)
$\partial_{t}f_{i}+v\cdot\nabla_{x}f_{i}=\sum_{j=1}^{I}Q_{ij}(f_{i},f_{j})\quad\mbox{on
}{\mathbb{R}}_{+}\times{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}.$

##  3\. The Maxwell-Stefan model

The Maxwell-Stefan model is suitable to describe an ideal gaseous mixture of
$I\geq 2$ species, with molecular masses $m_{i}$, in which no convective
phenomena take place and the system is driven to equilibrium by pure
diffusion.

For each species of the mixture $\mathcal{A}_{i}$, $1\leq i\leq I$, we
consider its concentration $c_{i}$ and its flux $F_{i}$ – which depend on the
macroscopic variables $t\in{\mathbb{R}}^{+}$ (time) and $x\in{\mathbb{R}}^{3}$
(position).

These quantities, which are the unknowns of the system, satisfy the continuity
equation

(9) $\partial_{t}c_{i}+\nabla_{x}\cdot F_{i}=0\quad\mbox{on
}{\mathbb{R}}_{+}\times{\mathbb{R}}^{3},$

for all $1\leq i\leq I$.

Let $c=\sum c_{i}$ be the total concentration of the mixture and let
$n_{i}=c_{i}/c$ the mole fraction of species $\mathcal{A}_{i}$. The Maxwell-
Stefan equations can be written in the following form:

(10) $-c\,\nabla_{x}n_{i}=\frac{1}{c}\sum_{j\neq
i}\frac{c_{j}F_{i}-c_{i}F_{j}}{\textrm{\DH}_{ij}}\quad\textrm{ on
}{\mathbb{R}}_{+}\times{\mathbb{R}}^{3},$

for all $1\leq i\leq I$.

The quantities $\textrm{\DH}_{ij}$ are the binary diffusion coefficients
between the species $\mathcal{A}_{i}$ and $\mathcal{A}_{j}$. They are
symmetric with respect to the particles exchange, in such a way that
$\textrm{\DH}_{ij}=\textrm{\DH}_{ji}$.

Note that the Maxwell-Stefan equations (10) are linearly dependent. Indeed, by
summing (10) with respect to $i$, we obtain an identity. Hence, a
supplementary equation is necessary for assuring the closure of the Maxwell-
Sterfan system (9)–(10).

By assuming that the system is closed and under constant and uniform
temperature and pressure, it is usual to assume that the total diffusive flux
satisfies

(11) $\sum_{i=1}^{I}F_{i}=0\quad\mbox{on
}{\mathbb{R}}_{+}^{*}\times{\mathbb{R}}^{3},$

which physically means that the diffusive fluxes do not create any mass [16].

By summing (9) with respect to $i$, we note that $c$ is uniform in time.
Hence, if we suppose that the molecules of the mixture are initially uniformly
distributed, the quantity $c$ is a pure constant.

##  4\. The Maxwell-Stefan asymptotics

In this section we apply the strategy proposed in [9], with more general cross
sections, satisfying the hypotheses defined in Subsection 4.1.

We work in the standard diffusive scaling, by supposing that the mean free
path (or, equivalently, the Knudsen number) tends to zero. Since temperature
gradients can induce transport phenomena, we will suppose that the temperature
$T$ of the mixture is a constant. Moreover, we assume that the bulk velocity
of the mixture is of the same order of magnitude as the Knudsen and Mach
numbers.

###  4.1. Collision kernels

In this article, we suppose that the collision kernels $B_{ij}$ depend only on
the modulus of the relative velocity, i.e. $|v-v_{*}|$ and on the cosine of
the deviation angle $\theta$, where

$\cos\theta=\frac{v-v_{*}}{|v-v_{*}|}\cdot\sigma.$

More specifically, we work with collision kernels of the form:

(12) $\displaystyle B_{ij}(v,v_{*},\sigma)=\Phi(|v-v_{*}|)b_{ij}(\cos\theta),$

where we assume that the angular collision kernels $b_{ij}\in L^{1}(-1,+1)$
and are even. Observe that, because of the microreversibility assumption on
the collision kernels, we have

$\displaystyle B_{ij}(v,v_{*},\sigma)=B_{ji}(v_{*},v,\sigma)\implies
b_{ij}\left(\frac{v-v_{*}}{|v-v_{*}|}\cdot\sigma\right)=b_{ji}\left(\frac{v_{*}-v}{|v-v_{*}|}\cdot\sigma\right),$

and that, by parity, $b_{ij}(\cos\theta)=b_{ji}(\cos\theta)$.

For $\ell\in\\{1,2,3\\}$, we denote by $w_{(\ell)}$ the $\ell$-th component of
any vector $w\in{\mathbb{R}}^{3}$.

If we introduce the polar variable $\varphi\in[0,2\pi]$, we can find the
relationships between the Euclidean coordinates of $\sigma$ and the spherical
ones, namely

$\sigma_{(1)}=\sin\theta\cos\varphi,\quad\sigma_{(2)}=\sin\theta\sin\varphi,\quad\sigma_{(3)}=\cos\theta.$

The kinetic collision kernel $\Phi(|v-v_{*}|)$ is assumed to be analytic in
the following sense: there exists a family
$\\{a_{n}\\}_{n\in\mathbb{N}^{*}}\subset{\mathbb{R}}$ such that $\Phi$ can be
written as a uniformly converging even power series:

(13)
$\displaystyle\Phi(|v-v_{*}|)=\sum_{n\in\mathbb{N}^{*}}a_{n}|v-v_{*}|^{2n}.$

###  4.2. Scaled equation

In order to arrive at the diffusive limit, we introduce a scaling parameter
$0<\varepsilon\ll 1$ which represents the mean free path. We denote the
corresponding unknown distribution functions as $(f_{i}^{\varepsilon})_{1\leq
i\leq I}$. Each distribution function $f_{i}^{\varepsilon}$ solves the
following scaled version of (8):

(14)
$\varepsilon\,\partial_{t}f_{i}^{\varepsilon}+v\cdot\nabla_{x}f_{i}^{\varepsilon}=\frac{1}{\varepsilon}\sum_{j=1}^{I}Q_{ij}(f_{i}^{\varepsilon},f_{j}^{\varepsilon}),\quad\mbox{on
}{\mathbb{R}}_{+}\times{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}.$

Finally, we define the corresponding concentrations
$(c_{i}^{\varepsilon})_{1\leq i\leq I}$ as the zero-th order moment of the
distribution functions $f_{i}^{\varepsilon}(t,x,v)$:

$c_{i}^{\varepsilon}(t,x)=\int_{{\mathbb{R}}^{3}}f_{i}^{\varepsilon}(t,x,v)\,\mathrm{d}v,\quad\mbox{
for }(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{3};$

this relationship is a fundamental link between the kinetic equations and the
Maxwell-Stefan description.

###  4.3. Ansatz

As we are interested in pure diffusion dynamics, we suppose that the initial
data $f^{in}(x,v)$ for the multi-species Boltzmann equations (14) are such
that

$\displaystyle\int_{{\mathbb{R}}^{3}}f_{i}^{\mbox{\tiny in}}(x,v)\,{\rm
d}v=c_{i}^{\mbox{\tiny
in}}(x),\qquad\int_{{\mathbb{R}}^{3}}vf_{i}^{\mbox{\tiny in}}(x,v)\,{\rm
d}v=\mathcal{O}(\varepsilon),$

where

$c_{i}^{\mbox{\tiny in}}\,:\,{\mathbb{R}}^{3}\to{\mathbb{R}}_{+}$

are $\varepsilon$-independent. We moreover suppose that

$\sum_{i=1}^{I}c_{i}^{\mbox{\tiny in}}=1\quad\mbox{on }{\mathbb{R}}^{3},$

which of course implies that each $c_{i}^{\mbox{\tiny in}}$ lies in $[0,1]$.
As in [9], we assume that the evolution following (14) keeps the distribution
functions $f^{\varepsilon}_{i}(t,x,v)$ in the local Maxwellian state, with a
homogeneous temperature $T$. We hence suppose that there exist

$c_{i}^{\varepsilon}\,:\,{\mathbb{R}}_{+}\times{\mathbb{R}}^{3}\to{\mathbb{R}}_{+},\qquad
u_{i}^{\varepsilon}\,:\,{\mathbb{R}}_{+}\times{\mathbb{R}}^{3}\to{\mathbb{R}}^{3},\qquad
1\leq i\leq I,$

such that

(15)
$f^{\varepsilon}_{i}(t,x,v)={c^{\varepsilon}_{i}(t,x)}\,\left(\frac{m_{i}}{2\pi
k\,T}\right)^{3/2}e^{-{m_{i}}|v-\varepsilon
u^{\varepsilon}_{i}(t,x)|^{2}/2kT},\quad\mbox{ for
}(t,x,v)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}.$

The starting point of our analysis is the following result, proved in [9]:

######  Proposition 1.

Under the assumption (15) on the distribution functions
$f^{\varepsilon}_{i}(t,x,v)$, we have the following mass balance equations for
all $1\leq i\leq I$:

(16)
$\partial_{t}c_{i}^{\varepsilon}+\nabla_{x}\cdot(c_{i}^{\varepsilon}u_{i}^{\varepsilon})=0\qquad\mbox{
on }{\mathbb{R}}_{+}\times{\mathbb{R}}^{3}.$

We further have the following momentum balance for all $1\leq i\leq I$:

(17)
$\varepsilon^{2}\left[\partial_{t}\left(c_{i}^{\varepsilon}u_{i}^{\varepsilon}\right)+\nabla_{x}\cdot\left(c_{i}^{\varepsilon}u_{i}^{\varepsilon}\otimes
u_{i}^{\varepsilon}\right)\right]+\frac{kT}{m_{i}}\nabla_{x}c_{i}^{\varepsilon}=\Theta^{\varepsilon}_{i}\quad\mbox{
on }{\mathbb{R}}_{+}\times{\mathbb{R}}^{3},$

where the $\ell$-th component of $\Theta^{\varepsilon}_{i}$ is given by

(18)
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}=\frac{1}{\varepsilon}\sum_{j\neq
i}\frac{m_{j}}{m_{i}+m_{j}}\int_{{\mathbb{R}}^{6}}\int_{{\mathbb{S}}^{2}}B_{ij}(v,v_{*},\sigma)f_{i}^{\varepsilon}(v)f_{j}^{\varepsilon}(v_{*})\,\left({v_{*}}_{(\ell)}-v_{(\ell)}+|v-v_{*}|\sigma_{(\ell)}\right)\,\mathrm{d}\sigma\,\mathrm{d}v_{*}\,\mathrm{d}v.$

Remark. In Equation (17), we have stressed the actual order in $\varepsilon$
of various terms. In particular, as shown in the next lemma, the quantities
$\Theta^{\varepsilon}_{i}$ are of order $\mathcal{O}(1)$. This is of the same
order as $\nabla_{x}c^{\varepsilon}_{i}$ in (17). On the other hand, the first
two terms on the left hand side of (17) are of order
$\mathcal{O}(\varepsilon^{2})$. This clarifies how the diffusive scaling acts
on various macroscopic quantities associated with the solutions to the
Boltzmann system (14).

Our next task is to analyze the right hand side $\Theta^{\varepsilon}_{i}$ of
the momentum balance (17). Observe that $\Theta^{\varepsilon}_{i}$ depends on
the independent variables $(t,x)$.

######  Lemma 1.

The $\Theta^{\varepsilon}_{i}(t,x)$ term in the momentum balance (17) can be
asymptotically approximated as follows:

(19)
$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}(t,x)=\sum_{j\not=i}\Delta_{ij}\left(c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{j})^{\varepsilon}_{(\ell)}-c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{i}^{\varepsilon})_{(\ell)}\right)+\mathcal{O}(\varepsilon),$

where $\Delta_{ij}$ are given by

(20) $\begin{array}[]{cl}\displaystyle\Delta_{ij}=&\displaystyle
a_{0}\frac{2\pi m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})}+\,a_{1}\frac{10\pi
kT\|b_{ij}\|_{L^{1}}}{m_{i}}+\sum_{n\geq
2}a_{n}\frac{m_{j}(m_{i}m_{j})^{3/2}\|b_{ij}\|_{L^{1}}}{4\pi^{2}(m_{i}+m_{j})(k\,T)^{3}}\int_{{\mathbb{R}}^{6}}\Big{\\{}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\times\\\\[5.69046pt]
&\displaystyle\prod_{1\leq r\leq
3}\Big{(}\sum_{\alpha+\beta=2n_{r}}\frac{(2n_{r})!}{\alpha!\beta!}\left([{v}_{(r)}]^{\alpha}[-v_{*(r)}]^{\beta}\right)\Big{)}\Big{)}e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\Big{\\}}\,\mathrm{d}v_{*}\,\mathrm{d}v.\end{array}$

Proof

For readers’ convenience, let us rewrite the expression (18) for
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$:

$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}=\frac{1}{\varepsilon}\sum_{j\neq
i}\frac{m_{j}}{m_{i}+m_{j}}\int_{{\mathbb{R}}^{6}}\int_{{\mathbb{S}}^{2}}B_{ij}(v,v_{*},\sigma)f_{i}^{\varepsilon}(v)f_{j}^{\varepsilon}(v_{*})\,\left({v_{*}}_{(\ell)}-v_{(\ell)}+|v-v_{*}|\sigma_{(\ell)}\right)\,\mathrm{d}\sigma\,\mathrm{d}v_{*}\,\mathrm{d}v.$

The term containing $\sigma_{(\ell)}$ in (18) vanishes. Indeed, both terms for
$\ell=1$ or $2$ are zero because

$\int_{0}^{2\pi}\sin\varphi\,\mathrm{d}\varphi=\int_{0}^{2\pi}\cos\varphi\,\mathrm{d}\varphi=0,$

and for $\ell=3$, because $b_{ij}$ is even, one has

$\int_{{\mathbb{S}}^{2}}b_{ij}\left(\frac{v-v_{*}}{|v-v_{*}|}\cdot\sigma\right)\sigma_{(3)}\,\mathrm{d}\sigma=2\pi\int_{0}^{\pi}\sin\theta\cos\theta\,b_{ij}(\cos\theta)\,\mathrm{d}\theta=2\pi\int_{-1}^{1}\eta\,b_{ij}(\eta)\,\mathrm{d}\eta=0.$

Gathering the remaining part of the expression for
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$, we have:

(21)
$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}=\frac{1}{\varepsilon}\sum_{j\neq
i}\frac{2\pi
m_{j}\|b_{ij}\|_{L^{1}}}{m_{i}+m_{j}}\int_{{\mathbb{R}}^{6}}\Phi(|v-v_{*}|)f_{i}^{\varepsilon}(v)f_{j}^{\varepsilon}(v_{*})\,\left({v_{*}}_{(\ell)}-v_{(\ell)}\right)\,\mathrm{d}v_{*}\,\mathrm{d}v.$

We now substitute the ansatz (15) for $f_{i}^{\varepsilon}(v)$ and
$f_{j}^{\varepsilon}(v_{*})$ in (21). By writing the power series expansion
(13) for the kinetic collision kernel we obtain:

$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$
$\displaystyle=\sum_{j\not=i}c_{i}^{\varepsilon}c_{j}^{\varepsilon}\left\\{\left[a_{0}\frac{2\pi
m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})}+\,a_{1}\frac{10\pi
kT\|b_{ij}\|_{L^{1}}}{m_{i}}\right]\left[(u_{j})^{\varepsilon}_{(\ell)}-(u_{i})^{\varepsilon}_{(\ell)}\right]\right.$
$\displaystyle+\sum_{n\geq 2}a_{n}$
$\displaystyle\left.\frac{m_{j}(m_{i}m_{j})^{3/2}\|b_{ij}\|_{L^{1}}}{4\pi^{2}\varepsilon(m_{i}+m_{j})(k\,T)^{3}}\int_{{\mathbb{R}}^{6}}|v-v_{*}|^{2n}\left({v_{*}}_{(\ell)}-v_{(\ell)}\right)e^{-{m_{i}}|v-\varepsilon
u^{\varepsilon}_{i}|^{2}/2kT}e^{-{m_{j}}|v_{*}-\varepsilon
u^{\varepsilon}_{j}|^{2}/2kT}\,\mathrm{d}v_{*}\,\mathrm{d}v\right\\}.$

The computations that lead to the coefficients of $a_{0}$ can be found in [9].
The coefficients of the other terms $a_{i}$ are evaluated by performing the
change of variables: $(v,v_{*})\mapsto(v+\varepsilon
u^{\varepsilon}_{i},v_{*}+\varepsilon u^{\varepsilon}_{j})$ and then computing
various Gaussian integrals (whose explicit expressions can be found in the
Appendix).

Making a change of variables: $(v,v_{*})\mapsto(v+\varepsilon
u^{\varepsilon}_{i},v_{*}+\varepsilon u^{\varepsilon}_{j})$ in the integral
term of the expression written above yields:

$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$
$\displaystyle=\sum_{j\not=i}c_{i}^{\varepsilon}c_{j}^{\varepsilon}\left\\{\left[a_{0}\frac{2\pi
m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})}+\,a_{1}\frac{10\pi
kT\|b_{ij}\|_{L^{1}}}{m_{i}}\right]\left[(u_{j})^{\varepsilon}_{(\ell)}-(u_{i})^{\varepsilon}_{(\ell)}\right]\right.$
$\displaystyle+\sum_{n\geq
2}a_{n}\frac{m_{j}(m_{i}m_{j})^{3/2}\|b_{ij}\|_{L^{1}}}{4\pi^{2}\varepsilon(m_{i}+m_{j})(k\,T)^{3}}\left.\\!\\!\int_{{\mathbb{R}}^{6}}\\!\\!|v+\varepsilon
u^{\varepsilon}_{i}-v_{*}-\varepsilon
u^{\varepsilon}_{j}|^{2n}\\!\left({v_{*}}_{(\ell)}+\varepsilon(u_{j})^{\varepsilon}_{(\ell)}-v_{(\ell)}-\varepsilon(u_{i}^{\varepsilon})_{(\ell)}\right)e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\,\mathrm{d}v_{*}\,\mathrm{d}v\right\\}\\!.$

We employ the multinomial theorem:

$(b_{1}+b_{2}+\cdots+b_{k})^{n}=\sum_{\begin{subarray}{c}j_{1},\ j_{2},\dots,\
j_{k}\\\ 0\leq j_{i}\leq n\text{ for each }i\\\ \text{ and
}j_{1}+\ldots+j_{k}=n\end{subarray}}\binom{n}{j_{1},j_{2},\dots,j_{k}}b_{1}^{\,\,j_{1}}b_{2}^{\,\,j_{2}}\cdots
b_{k}^{\,\,j_{k}}$

where the multinomial coefficients are

$\displaystyle\binom{n}{j_{1},j_{2},\dots,j_{k}}=\dfrac{n!}{j_{1}!j_{2}!\dots
j_{k}!}.$

This yields

$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}=\sum_{j\not=i}c_{i}^{\varepsilon}c_{j}^{\varepsilon}\left\\{\left[a_{0}\frac{2\pi
m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})}+\,a_{1}\frac{10\pi
kT\|b_{ij}\|_{L^{1}}}{m_{i}}\right]\left[(u_{j})^{\varepsilon}_{(\ell)}-(u_{i})^{\varepsilon}_{(\ell)}\right]\right.$
$\displaystyle+\sum_{n\geq
2}a_{n}\frac{m_{j}(m_{i}m_{j})^{3/2}\|b_{ij}\|_{L^{1}}}{4\pi^{2}\varepsilon(m_{i}+m_{j})(k\,T)^{3}}\int_{{\mathbb{R}}^{6}}\Big{\\{}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\prod_{1\leq
r\leq
3}\left({v}_{(r)}+\varepsilon(u_{i}^{\varepsilon})_{(r)}-v_{*(r)}-\varepsilon(u_{j})^{\varepsilon}_{(r)}\right)^{2n_{r}}\Big{)}\times$
$\displaystyle\left({v_{*}}_{(\ell)}+\varepsilon(u_{j})^{\varepsilon}_{(\ell)}-v_{(\ell)}-\varepsilon(u_{i}^{\varepsilon})_{(\ell)}\right)e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\Big{\\}}\,\mathrm{d}v_{*}\,\mathrm{d}v\bigg{\\}}.$

Another application of the multinomial theorem in the previous expression
yields:

$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}=$
$\displaystyle\sum_{j\not=i}c_{i}^{\varepsilon}c_{j}^{\varepsilon}\left\\{\left[a_{0}\frac{2\pi
m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})}+\,a_{1}\frac{10\pi
kT\|b_{ij}\|_{L^{1}}}{m_{i}}\right]\left[(u_{j})^{\varepsilon}_{(\ell)}-(u_{i})^{\varepsilon}_{(\ell)}\right]\right.$
$\displaystyle+\sum_{n\geq 2}a_{n}$
$\displaystyle\frac{m_{j}(m_{i}m_{j})^{3/2}\|b_{ij}\|_{L^{1}}}{4\pi^{2}\varepsilon(m_{i}+m_{j})(k\,T)^{3}}\int_{{\mathbb{R}}^{6}}\Big{\\{}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\times$
$\displaystyle\prod_{1\leq r\leq
3}\Big{(}\sum_{\alpha+\beta+\gamma+\lambda=2n_{r}}\frac{(2n_{r})!}{\alpha!\beta!\gamma!\lambda!}\left([{v}_{(r)}]^{\alpha}[-v_{*(r)}]^{\beta}[\varepsilon(u_{i}^{\varepsilon})_{(r)}]^{\gamma}[-\varepsilon(u_{j})^{\varepsilon}_{(r)}]^{\lambda}\right)\Big{)}\Big{)}\times$
$\displaystyle\left({v_{*}}_{(\ell)}+\varepsilon(u_{j})^{\varepsilon}_{(\ell)}-v_{(\ell)}-\varepsilon(u_{i}^{\varepsilon})_{(\ell)}\right)e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\Big{\\}}\,\mathrm{d}v_{*}\,\mathrm{d}v\bigg{\\}}.$

The terms of $\mathcal{O}(\varepsilon^{-1})$ in
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$ are the following:

$\displaystyle\sum_{j\not=i}c_{i}^{\varepsilon}c_{j}^{\varepsilon}\bigg{\\{}\sum_{n\geq
2}a_{n}\frac{m_{j}(m_{i}m_{j})^{3/2}\|b_{ij}\|_{L^{1}}}{4\pi^{2}(m_{i}+m_{j})(k\,T)^{3}}\int_{{\mathbb{R}}^{6}}\Big{\\{}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\times$
$\displaystyle\prod_{1\leq r\leq
3}\left(\sum_{\alpha+\beta=2n_{r}}\frac{(2n_{r})!}{\alpha!\beta!}\left([{v}_{(r)}]^{\alpha}[-v_{*(r)}]^{\beta}\right)\right)\Big{)}\left({v_{*}}_{(\ell)}-v_{(\ell)}\right)e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\Big{\\}}\,\mathrm{d}v_{*}\,\mathrm{d}v\bigg{\\}}.$

Observe that all the terms in the above sum vanish since the integrands are
odd with respect to the variables $v$ or $v_{*}$. Hence there is no
contribution of the terms of $\mathcal{O}(\varepsilon^{-1})$ to
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$. Now, we move on to consider
the terms of order $\mathcal{O}(1)$ in
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$, which have the form:

$\displaystyle\sum_{j\not=i}c_{i}^{\varepsilon}c_{j}^{\varepsilon}\bigg{\\{}\Big{(}a_{0}$
$\displaystyle\frac{2\pi
m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})}+\,a_{1}\frac{10\pi
kT\|b_{ij}\|_{L^{1}}}{m_{i}}\Big{)}\left((u_{j})^{\varepsilon}_{(\ell)}-(u_{i}^{\varepsilon})_{(\ell)}\right)$
$\displaystyle+\sum_{n\geq 2}a_{n}$
$\displaystyle\frac{m_{j}(m_{i}m_{j})^{3/2}\|b_{ij}\|_{L^{1}}}{4\pi^{2}(m_{i}+m_{j})(k\,T)^{3}}\int_{{\mathbb{R}}^{6}}\Big{\\{}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\prod_{1\leq
r\leq
3}\Big{(}\sum_{\alpha+\beta=2n_{r}}\frac{(2n_{r})!}{\alpha!\beta!}\times$
$\displaystyle\left([{v}_{(r)}]^{\alpha}[-v_{*(r)}]^{\beta}\right)\Big{)}\Big{)}\left((u_{j})^{\varepsilon}_{(\ell)}-(u_{i}^{\varepsilon})_{(\ell)}\right)e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\Big{\\}}\,\mathrm{d}v_{*}\,\mathrm{d}v\bigg{\\}}.$

Hence, we indeed have the asymptotic behaviour of
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$ as in (19) with the
coefficients $\Delta_{ij}$ given by (20).

$\square$

Remark. The kinetic collision kernel for the three dimensional hard spheres,
i.e. $\Phi(|v-v_{*}|)=|v-v_{*}|$ is not an analytic function of $v-v_{*}$.
Hence our approach cannot be directly applied to this case. However, one could
approximate the hard sphere kernel by an analytic expression of the type (13)
and then perform the computations on the approximate series. This would yield
an approximation on the binary diffusion coefficients for the hard sphere
case.

###  4.4. Limiting behavior of the system

Now, we are equipped to state the main result of this article. Putting
together the results of Proposition 1 and Lemma 1, we have indeed proved at
the formal level the following theorem.

######  Theorem 2.

The local Maxwellian states $(\ref{ansatz})$ are solution of the initial value
problem for the system of scaled Boltzmann equations $(\ref{eqbo})$ if
$(c_{i}^{\varepsilon},u_{i}^{\varepsilon})$ solves

(22)
$\displaystyle\partial_{t}c_{i}^{\varepsilon}+\nabla_{x}\cdot\left(c_{i}^{\varepsilon}u_{i}^{\varepsilon}\right)=0,$
(23)
$\displaystyle\nabla_{x}c_{i}^{\varepsilon}=\sum_{j\not=i}\tilde{\Delta}_{ij}\left(c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{j})^{\varepsilon}_{(\ell)}-c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{i}^{\varepsilon})_{(\ell)}\right)+\mathcal{O}(\varepsilon),$

with the coefficients $\tilde{\Delta}_{ij}$ given by

(24) $\begin{array}[]{cl}\displaystyle\tilde{\Delta}_{ij}=&\displaystyle
a_{0}\frac{2\pi
m_{i}m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})kT}+\,a_{1}10\pi\|b_{ij}\|_{L^{1}}\\\\[14.22636pt]
&\displaystyle+\sum_{n\geq
2}a_{n}\,\frac{2\,\pi\|b_{ij}\|_{L^{1}}}{k\,T}\frac{(m_{i}m_{j})}{(m_{i}+m_{j})}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\sum_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta,\rho,\eta\in
2{\mathbb{N}}^{*}\\\ \alpha+\beta=2n_{1}\\\ \gamma+\delta=2n_{2}\\\
\rho+\eta=2n_{3}\end{subarray}}\frac{(2n_{1})!}{\alpha!\beta!}\frac{(2n_{2})!}{\gamma!\delta!}\frac{(2n_{3})!}{\rho!\eta!}\times\\\\[8.5359pt]
&\mathcal{E}(\alpha,\beta,\gamma,\delta,\rho,\eta)\left(\frac{kT}{m_{i}}\right)^{(\alpha+\gamma+\rho)/2}\left(\frac{kT}{m_{j}}\right)^{(\beta+\delta+\eta)/2}\Big{)},\end{array}$

where

(25)
$\displaystyle\mathcal{E}(\alpha,\beta,\gamma,\delta,\rho,\eta):=\left((\alpha-1)(\alpha-3)\cdots
1\right)\left((\beta-1)(\beta-3)\cdots
1\right)\left((\gamma-1)(\gamma-3)\cdots 1\right)\times$
$\displaystyle\left((\delta-1)(\delta-3)\cdots
1\right)\left((\rho-1)(\rho-3)\cdots 1\right)\left((\eta-1)(\eta-3)\cdots
1\right).$

Proof

From (17) and (19), we have:

$\displaystyle\nabla_{x}c^{\varepsilon}_{i}=\sum_{j\not=i}\frac{m_{i}}{kT}\Delta_{ij}\left(c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{j})^{\varepsilon}_{(\ell)}-c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{i}^{\varepsilon})_{(\ell)}\right)+\mathcal{O}(\varepsilon).$

As the coefficients $\Delta_{ij}$ involve Gaussian integrals, by using the
expressions from the Appendix, we first compute the integral terms in (20):

$\displaystyle\int_{{\mathbb{R}}^{6}}\Big{\\{}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\times\prod_{1\leq
r\leq
3}\Big{(}\sum_{\alpha+\beta=2n_{r}}\frac{(2n_{r})!}{\alpha!\beta!}\left([{v}_{(r)}]^{\alpha}[-v_{*(r)}]^{\beta}\right)\Big{)}\Big{)}\times$
$\displaystyle
e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\Big{\\}}\,\mathrm{d}v_{*}\,\mathrm{d}v$
$\displaystyle=\int_{{\mathbb{R}}^{6}}\\!\\!\Big{\\{}\Big{(}\\!\\!\sum_{n_{1}+n_{2}+n_{3}=n}\\!\\!\frac{n!}{n_{1}!n_{2}!n_{3}!}\\!\sum_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta,\rho,\eta\in
2{\mathbb{N}}^{*}\\\ \alpha+\beta=2n_{1}\\\ \gamma+\delta=2n_{2}\\\
\rho+\eta=2n_{3}\end{subarray}}\\!\\!\frac{(2n_{1})!}{\alpha!\beta!}\frac{(2n_{2})!}{\gamma!\delta!}\frac{(2n_{3})!}{\rho!\eta!}[{v}_{(1)}]^{\alpha}[v_{*(1)}]^{\beta}[{v}_{(2)}]^{\gamma}[v_{*(2)}]^{\delta}[{v}_{(3)}]^{\rho}[v_{*(3)}]^{\eta}\Big{)}\times$
$\displaystyle
e^{-{m_{i}}|v|^{2}/2kT}e^{-{m_{j}}|v_{*}|^{2}/2kT}\Big{\\}}\,\mathrm{d}v_{*}\,\mathrm{d}v$

Proceeding as in the Appendix (see (27)), the above expression can be computed
and equals:

$\displaystyle\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\sum_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta,\rho,\eta\in
2{\mathbb{N}}^{*}\\\ \alpha+\beta=2n_{1}\\\ \gamma+\delta=2n_{2}\\\
\rho+\eta=2n_{3}\end{subarray}}\frac{(2n_{1})!}{\alpha!\beta!}\frac{(2n_{2})!}{\gamma!\delta!}\frac{(2n_{3})!}{\rho!\eta!}\mathcal{E}(\alpha,\beta,\gamma,\delta,\rho,\eta)\times$
$\displaystyle\left(\frac{kT}{m_{i}}\right)^{(\alpha+\gamma+\rho)/2}\left(\frac{kT}{m_{j}}\right)^{(\beta+\delta+\eta)/2}\left(\frac{2\pi
kT}{m_{i}}\right)^{3/2}\left(\frac{2\pi kT}{m_{j}}\right)^{3/2},$

where $\mathcal{E}$ is given by (25). Substituting the above expression for
the integrals in $\Delta_{ij}$ would yield an explicit expression for
$\tilde{\Delta}_{ij}=m_{i}\Delta_{ij}/kT$:

$\displaystyle\tilde{\Delta}_{ij}=$ $\displaystyle\displaystyle
a_{0}\frac{2\pi
m_{i}m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})kT}+\,a_{1}10\pi\|b_{ij}\|_{L^{1}}$
$\displaystyle\displaystyle+\sum_{n\geq
2}a_{n}\,\frac{\|b_{ij}\|_{L^{1}}}{4\pi^{2}(m_{i}+m_{j})}\frac{(m_{i}m_{j})^{5/2}}{(kT)^{4}}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\sum_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta,\rho,\eta\in
2{\mathbb{N}}^{*}\\\ \alpha+\beta=2n_{1}\\\ \gamma+\delta=2n_{2}\\\
\rho+\eta=2n_{3}\end{subarray}}\frac{(2n_{1})!}{\alpha!\beta!}\frac{(2n_{2})!}{\gamma!\delta!}\frac{(2n_{3})!}{\rho!\eta!}\times$
$\displaystyle\mathcal{E}(\alpha,\beta,\gamma,\delta,\rho,\eta)\left(\frac{kT}{m_{i}}\right)^{(\alpha+\gamma+\rho)/2}\left(\frac{kT}{m_{j}}\right)^{(\beta+\delta+\eta)/2}\left(\frac{2\pi
kT}{m_{i}}\right)^{3/2}\left(\frac{2\pi kT}{m_{j}}\right)^{3/2}\Big{)}$
$\displaystyle=$ $\displaystyle\displaystyle a_{0}\frac{2\pi
m_{i}m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})kT}+\,a_{1}10\pi\|b_{ij}\|_{L^{1}}$
$\displaystyle\displaystyle+\sum_{n\geq
2}a_{n}\,\frac{2\,\pi\|b_{ij}\|_{L^{1}}}{k\,T}\frac{(m_{i}m_{j})}{(m_{i}+m_{j})}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\sum_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta,\rho,\eta\in
2{\mathbb{N}}^{*}\\\ \alpha+\beta=2n_{1}\\\ \gamma+\delta=2n_{2}\\\
\rho+\eta=2n_{3}\end{subarray}}\frac{(2n_{1})!}{\alpha!\beta!}\frac{(2n_{2})!}{\gamma!\delta!}\frac{(2n_{3})!}{\rho!\eta!}\times$
$\displaystyle\mathcal{E}(\alpha,\beta,\gamma,\delta,\rho,\eta)\left(\frac{kT}{m_{i}}\right)^{(\alpha+\gamma+\rho)/2}\left(\frac{kT}{m_{j}}\right)^{(\beta+\delta+\eta)/2}\Big{)}.$

$\square$

Note that the coefficients $\tilde{\Delta}_{ij}$ are symmetric with respect to
each pair of species since $b_{ij}=b_{ji}$ and that the structure is much more
intricate than the corresponding binary diffusion coefficients computed with
Maxwellian cross section in [9].

Remark. Observe that the first term in the expression for
$\tilde{\Delta}_{ij}$ is nothing but the expression obtained in [9]. Note also
that $\tilde{\Delta}_{ij}$ in $(\ref{eq:thm:Delta-ij})$ is given in terms of
the reduced masses of the species and the temperature $T$ except for the term
involving $a_{1}$ in the analytic expression $(\ref{eq:Phi-power})$. In
particular, if the kinetic collision kernel has the form
$\Phi(|v-v_{*}|)=|v-v_{*}|^{2}$, then the preceding computations in the paper
yield the following expression for
$\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}$:

$\displaystyle\left(\Theta^{\varepsilon}_{i}\right)_{(\ell)}=\sum_{j\not=i}\frac{2\pi
m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})}\left(\frac{5kT}{m_{i}}+\frac{5kT}{m_{j}}\right)\left(c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{j})^{\varepsilon}_{(\ell)}-c^{\varepsilon}_{i}c^{\varepsilon}_{j}(u_{i}^{\varepsilon})_{(\ell)}\right)+\mathcal{O}(\varepsilon),$

thus giving the following expression for $\tilde{\Delta}_{ij}$:

$\displaystyle\tilde{\Delta}_{ij}=a_{1}10\pi\|b_{ij}\|_{L^{1}}.$

The rest of the terms in the expression (24), however, do depend on the
temperature $T$ and the dependence is non-trivial.

In the following, let us set

$F_{i}^{\varepsilon}(t,x)=\frac{1}{\varepsilon}\int_{{\mathbb{R}}^{3}}v\,f_{i}^{\varepsilon}(t,x,v)\,\mathrm{d}v=c_{i}^{\varepsilon}(t,x)u_{i}^{\varepsilon}(t,x),\quad\mbox{
for }(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{3},$

and denote, for any $t\geq 0$ and $x\in{\mathbb{R}}^{3}$,

$c_{i}(t,x)=\lim_{\varepsilon\to 0^{+}}c_{i}^{\varepsilon}(t,x),\qquad
F_{i}(t,x)=\lim_{\varepsilon\to 0^{+}}F_{i}^{\varepsilon}(t,x).$

In the limit, Equations (22)–(23) give a system of equations, which has the
following form for the density-flux set of unknown $(c_{i},F_{i})$:

$\displaystyle\displaystyle\partial_{t}c_{i}+\nabla_{x}\cdot
F_{i}=0,\phantom{\int}$ (26)
$\displaystyle\displaystyle\nabla_{x}c_{i}=-\sum_{j\neq
i}\tilde{\Delta}_{ij}({c_{j}F_{i}-c_{i}F_{j}}).$

An argument similar to the ones in [9], which consists in writing the
approximation of the conservation of the kinetic energy at the leading order
in $\varepsilon$ for the Boltzmann system, allows to deduce that

$c=\sum c_{i}=\sum c_{i}^{\mbox{\tiny in}}=1.$

We can hence obtain, in the limit, the Maxwell-Stefan system in the case of
analytic cross-section and Grad’s cutoff assumption:

$\left\\{\begin{array}[]{l}\displaystyle\partial_{t}c_{i}+\nabla_{x}\cdot
F_{i}=0\quad\mbox{on }{\mathbb{R}}_{+}\times{\mathbb{R}}^{3},\\\ \\\
\displaystyle-c\,\nabla_{x}n_{i}=\frac{1}{c}\sum_{j\neq
i}\frac{c_{j}F_{i}-c_{i}F_{j}}{\textrm{\DH}_{ij}}\quad\mbox{on
}{\mathbb{R}}_{+}^{*}\times{\mathbb{R}}^{3},\end{array}\right.$

where the binary diffusion coefficient have the form

$\displaystyle\textrm{\DH}_{ij}=\frac{1}{c}\displaystyle\Bigg{\\{}a_{0}\frac{2\pi
m_{i}m_{j}\|b_{ij}\|_{L^{1}}}{(m_{i}+m_{j})kT}+\,a_{1}10\pi\|b_{ij}\|_{L^{1}}+\sum_{n\geq
2}a_{n}\,\frac{2\,\pi\|b_{ij}\|_{L^{1}}}{k\,T}\frac{(m_{i}m_{j})}{(m_{i}+m_{j})}\Big{(}\sum_{n_{1}+n_{2}+n_{3}=n}\frac{n!}{n_{1}!n_{2}!n_{3}!}\times$
$\displaystyle\sum_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta,\rho,\eta\in
2{\mathbb{N}}^{*}\\\ \alpha+\beta=2n_{1}\\\ \gamma+\delta=2n_{2}\\\
\rho+\eta=2n_{3}\end{subarray}}\frac{(2n_{1})!}{\alpha!\beta!}\frac{(2n_{2})!}{\gamma!\delta!}\frac{(2n_{3})!}{\rho!\eta!}\mathcal{E}(\alpha,\beta,\gamma,\delta,\rho,\eta)\left(\frac{kT}{m_{i}}\right)^{(\alpha+\gamma+\rho)/2}\left(\frac{kT}{m_{j}}\right)^{(\beta+\delta+\eta)/2}\Big{)}\Bigg{\\}}^{-1}.$

##  5\. Appendix

The objective of this appendix is the explicit computation of the Gaussian
integrals appearing in the expression for ${\Delta}_{ij}$ given by (20)
leading to the expression of $\tilde{\Delta}_{ij}$ given by (24). We have the
following normalization:

$\displaystyle\int_{{\mathbb{R}}^{3}}\left(\frac{m_{i}}{2\pi
kT}\right)^{3/2}e^{-m_{i}|v|^{2}/2kT}\,{\rm d}v=1.$

We observe that

$\displaystyle\left(\frac{m_{i}}{2\pi
kT}\right)^{3/2}\int_{{\mathbb{R}}^{3}}[v_{(1)}]^{\alpha}\,[v_{(2)}]^{\beta}\,[v_{(3)}]^{\gamma}\,e^{-m_{i}|v|^{2}/2kT}\,{\rm
d}v_{(1)}\,{\rm d}v_{(2)}\,{\rm d}v_{(3)}$
$\displaystyle=-\left(\frac{m_{i}}{2\pi
kT}\right)^{3/2}\int_{{\mathbb{R}}^{3}}\left(\frac{kT}{m_{i}}\right)[v_{(1)}]^{\alpha-1}\,\partial_{v_{(1)}}e^{-m_{i}|v|^{2}/2kT}\,[v_{(2)}]^{\beta}\,[v_{(3)}]^{\gamma}\,\,{\rm
d}v_{(1)}\,{\rm d}v_{(2)}\,{\rm d}v_{(3)}.$

Performing integration by parts with respect to $v_{(1)}$ variable we arrive
at

$\displaystyle\left(\frac{m_{i}}{2\pi
kT}\right)^{3/2}\int_{{\mathbb{R}}^{3}}\left(\frac{kT}{m_{i}}\right)(\alpha-1)[v_{(1)}]^{\alpha-2}\,e^{-m_{i}|v|^{2}/2kT}\,[v_{(2)}]^{\beta}\,[v_{(3)}]^{\gamma}\,\,{\rm
d}v_{(1)}\,{\rm d}v_{(2)}\,{\rm d}v_{(3)}$
$\displaystyle=-\left(\frac{m_{i}}{2\pi
kT}\right)^{3/2}\int_{{\mathbb{R}}^{3}}\left(\frac{kT}{m_{i}}\right)^{2}(\alpha-1)[v_{(1)}]^{\alpha-3}\,\partial_{v_{(1)}}e^{-m_{i}|v|^{2}/2kT}\,[v_{(2)}]^{\beta}\,[v_{(3)}]^{\gamma}\,\,{\rm
d}v_{(1)}\,{\rm d}v_{(2)}\,{\rm d}v_{(3)}.$

Performing an integration by parts in the $v_{(1)}$ variable again yields:

$\displaystyle\left(\frac{m_{i}}{2\pi
kT}\right)^{3/2}\int_{{\mathbb{R}}^{3}}\left(\frac{kT}{m_{i}}\right)^{2}(\alpha-1)(\alpha-3)[v_{(1)}]^{\alpha-4}\,e^{-m_{i}|v|^{2}/2kT}\,[v_{(2)}]^{\beta}\,[v_{(3)}]^{\gamma}\,\,{\rm
d}v_{(1)}\,{\rm d}v_{(2)}\,{\rm d}v_{(3)}$

Continuing in a similar fashion with respect to all the variables, i.e.
$v_{(1)},v_{(2)},v_{(3)}$, we arrive at the following expression:

(27) $\displaystyle\left(\frac{m_{i}}{2\pi
kT}\right)^{3/2}\int_{{\mathbb{R}}^{3}}[v_{(1)}]^{\alpha}\,[v_{(2)}]^{\beta}\,[v_{(3)}]^{\gamma}\,e^{-m_{i}|v|^{2}/2kT}\,{\rm
d}v_{(1)}\,{\rm d}v_{(2)}\,{\rm d}v_{(3)}$
$\displaystyle=\left((\alpha-1)(\alpha-3)\cdots
1\right)\left((\beta-1)(\beta-3)\cdots
1\right)\left((\gamma-1)(\gamma-3)\cdots
1\right)\left(\frac{kT}{m_{i}}\right)^{(\alpha+\beta+\gamma)/2}.$

Acknowledgments: The authors thank the referees for their useful comments and
remarks. The authors appreciate the fruitful suggestions of Claude Bardos
during the preparation of this article. This work was partially funded by the
projects Kimega (ANR-14-ACHN-0030-01) and Kibord (ANR-13-BS01-0004). H.H.
acknowledges the support of the ERC grant MATKIT. F.S. would like to thank the
Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and
hospitality during the programme Mathematical Modelling and Analysis of
Complex Fluids and Active Media in Evolving Domains. This work was partially
supported by EPSRC grant no EP/K032208/1.

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