# Maximum entropy principle approach to a non-isothermal
Maxwell-Stefan diffusion model

Benjamin Anwasia  Universidade do Minho, Centro de Matemática, Campus de
Gualtar, 4710-057 Braga, Portugal  New York University Abu Dhabi, Abu Dhabi,
United Arab Emirates<EMAIL_ADDRESS>Srboljub Simić  Department of
Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg
Dositeja Obradovića 3, 21000 Novi Sad, Serbia<EMAIL_ADDRESS>
###### Abstract

In this study we apply the maximum entropy principle to derive the properly
scaled velocity distribution function of Boltzmann equations for mixtures,
which leads to a non-isothermal Maxwell-Stefan diffusion model. We also
analyze the entropy balance law and derive the kinetic entropy production from
the scaled distribution function.

###### keywords:

diffusion , Maxwell-Stefan model , maximum entropy principle

###### MSC:

[2010] 35Q20 , 82C40 , 76R50

mytitlenotemytitlenotefootnotetext: Authors acknowledge financial support
(B.A.) from the COST Action, CA18232 MAT-DYN-NET, supported by the “European
Cooperation in Science and Technology” through the “Short Term Scientific
Mission” at the Department of Mathematics and Informatics, University of Novi
Sad, Serbia, and (S.S.) from the Ministry of Education, Science and
Technological Development of the Republic of Serbia (Grant No.
451-03-9/2021-14/200125).

##  1 Introduction

Diffusion is a process of motion of one specie relative to another. If the
motion occurs relative to a background medium, the process is usually
described by the Fick’s law or some of its generalizations. When cross-
diffusion is more prominent, like in gaseous mixtures, the Maxwell-Stefan
diffusion model is more appropriate [1]. It turned out that cross-diffusion
models of Maxwell-Stefan kind can be used as a starting point for diverse
applications, ranging from engineering [2] to medicine [3], and they also
inspired new approaches to diffusion phenomena [4].

Along with applications, mathematical analysis of Maxwell-Stefan equations
emerged as equally important but relatively new subject [5, 6]. It covers the
aspects of existence and uniqueness of solutions [7, 8], large-time
asymptotics [9] and relaxation limits [10]. Our concern, however, will be the
formal derivation of Maxwell-Stefan diffusion equations and analysis of their
dissipative character.

From the macroscopic point of view, Maxwell-Stefan diffusion model consists of
mass balance equations for the constituents, and a kind of momentum balance
relations adjoined with a closure relation due to their linear dependence. It
can be derived either in the continuum framework [11, 12], or starting from
the Boltzmann equations (BEs) for mixtures [13, 14, 15, 16]. In the latter
approach, crucial step in derivation of Maxwell-Stefan equations is the
asymptotic limit enabled through appropriate scaling of the governing
equations.

In this study we shall take the kinetic approach to Maxwell-Stefan equations.
It is based upon _formal_ diffusive scaling of the BEs, and crucial
_assumption_ that velocity distribution functions have the following form:

$f^{\alpha}_{i}(t,{{\boldsymbol{x}}},{\boldsymbol{v}})=\rho_{i}^{\alpha}\left(\frac{m_{i}}{2\pi
k_{\mathrm{B}}T}\right)^{3/2}\exp\left(-\frac{m_{i}|{\boldsymbol{v}}-\alpha{\boldsymbol{u}}^{\alpha}_{i}|^{2}}{2k_{\mathrm{B}}T}\right),\quad
i=1,\ldots,S,$ (1)

for some small parameter $\alpha\ll 1$, where
${\boldsymbol{v}}\\!\in\\!\mathbb{R}^{3}$ are the particle velocities,
${\boldsymbol{u}}^{\alpha}_{i}\\!\in\\!\mathbb{R}^{3}$ are the macroscopic
species velocities, $m_{i}$ are the species atomic masses, $k_{\mathrm{B}}$ is
the Boltzmann constant, $\rho_{i}^{\alpha}$ are the species mass densities and
$T$ is the mixture temperature.

Instead of assuming, our aim is to _derive_ the scaled velocity distribution
function (1) appropriate for description of diffusion processes. To that end,
we shall apply the maximum entropy principle (MEP) in the scaled
(dimensionless) form (Theorem 1). Moreover, we shall put our analysis in a
broader context of non-isothermal processes. We shall also derive the entropy
balance law in the scaled form and compute the entropy production (Theorem 2)
for the non-isothermal Maxwell-Stefan model, starting from the scaled velocity
distribution function. Computation of the velocity distribution function from
MEP will be _exact_. Therefore, we shall derive the moment equations (Maxwell-
Stefan model) and the entropy balance law for the mixture of monatomic
Maxwellian particles, and in such a way retain the exact character of the
results.

##  2 The Boltzmann equation for inert mixtures

We consider a mixture of monatomic ideal inert gases: a gaseous mixture of
non-reactive species consisted of particles endowed with translational degrees
of freedom only. For each species ${\cal A}_{i}$ with $i=1,2,3\dots,S$, let
$f_{i}:=f_{i}(t,{{\boldsymbol{x}}},{\boldsymbol{v}})$ be its distribution
function, representing the densities of particles of species ${\cal A}_{i}$
which at time $t\in\mathbb{R}^{+}$ are located at position
${{\boldsymbol{x}}}\in\mathbb{R}^{3}$ and have velocity
${\boldsymbol{v}}\in\mathbb{R}^{3}$. In the absence of external forces, the
evolution of the non-reactive mixture can be described by the following system
of BEs:

$\displaystyle\frac{\partial f_{i}}{\partial
t}+{\boldsymbol{v}}\cdot\frac{\partial f_{i}}{\partial{\bf{x}}}$
$\displaystyle=\underbrace{\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\Big{[}f_{i}({\boldsymbol{v}}^{\prime})f_{i}({\boldsymbol{v}}_{*}^{\prime})-f_{i}({\boldsymbol{v}})f_{i}({\boldsymbol{v}}_{*})\Big{]}\,B_{i}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma})\,d\boldsymbol{\sigma}\,d{\boldsymbol{v}}_{*}}_{Q_{i}^{m}(f_{i},f_{i})}$
(2) $\displaystyle+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{S}\underbrace{\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\Big{[}f_{i}({\boldsymbol{v}}^{\prime})f_{j}({\boldsymbol{v}}_{*}^{\prime})-f_{i}({\boldsymbol{v}})f_{j}({\boldsymbol{v}}_{*})\Big{]}\,B_{ij}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma})\,d\boldsymbol{\sigma}\,d{\boldsymbol{v}}_{*}}_{Q^{b}_{ij}(f_{i},f_{j})},$

for $i=1,2,3,\dots,S,$ where the first and second terms on the right-hand side
describe collisions between particles of the same species and between
particles of different species, respectively. They are called the mono-species
and bi-species collision operators with ${\boldsymbol{v}}^{\prime}$,
${\boldsymbol{v}}^{\prime}_{*}$ defined as follows:

${\boldsymbol{v}}^{\prime}=\frac{1}{m_{i}+m_{j}}\big{(}m_{i}{\boldsymbol{v}}+m_{j}{\boldsymbol{v}}_{*}+m_{j}|{\boldsymbol{v}}-{\boldsymbol{v}}_{*}|\boldsymbol{\sigma}),\qquad{\boldsymbol{v}}_{*}^{\prime}=\frac{1}{m_{i}+m_{j}}\big{(}m_{i}{\boldsymbol{v}}+m_{j}{\boldsymbol{v}}_{*}-m_{i}|{\boldsymbol{v}}-{\boldsymbol{v}}_{*}|\boldsymbol{\sigma}),$
(3)

where $\boldsymbol{\sigma}\in\mathbb{S}^{2}$ is an arbitrary unit vector. The
formulas given in equation (3) represent all the possible solutions
$({\boldsymbol{v}}^{\prime},{\boldsymbol{v}}^{\prime}_{*})\in\mathbb{R}^{3}\times\mathbb{R}^{3}$
of the system of equations composed of the momentum and kinetic energy
conservation law, which are respectively defined as follows:

$m_{i}{\boldsymbol{v}}+m_{j}{\boldsymbol{v}}_{*}=m_{i}{\boldsymbol{v}}^{\prime}+m_{j}{\boldsymbol{v}}^{\prime}_{*},\qquad\frac{1}{2}m_{i}|{\boldsymbol{v}}|^{2}+\frac{1}{2}m_{j}|{\boldsymbol{v}}_{*}|^{2}=\frac{1}{2}m_{i}|{\boldsymbol{v}}^{\prime}|^{2}+\frac{1}{2}m_{j}|{\boldsymbol{v}}^{\prime}_{*}|^{2}.$
(4)

###  2.1 Scaled system of Boltzmann equations

To write the system (2) of BEs in dimensionless form, we first choose
macroscopic scales for the independent variables $t,{{\boldsymbol{x}}}$ and
${\boldsymbol{v}}$ associated with the distribution function.To this end, let
$\tau,L$ and $T_{0}$ respectively be the macroscopic time, space and reference
temperature scales. These macroscopic scales define two macroscopic velocity
scales: ${u}_{0}=\frac{L}{\tau}$ and
${c}_{0}=\sqrt{\frac{5k_{B}T_{0}}{3m_{0}}}$, where ${u}_{0}$ is the speed at
which some macroscopic portion of the gas is transported over a distance $L$
in time $\tau$, while ${c}_{0}$ is the speed of sound in a monatomic gas with
$m_{0}$ the average atomic mass of the mixture. Using the macroscopic scales:
$\tau,L$ and ${c}_{0}$, one can define the dimensionless time, space,
velocity, distribution function and cross-sections:

$\displaystyle\displaystyle\widehat{t}=\frac{t}{\tau},\quad\widehat{{\boldsymbol{x}}}=\frac{{\boldsymbol{x}}}{L},\qquad\widehat{\boldsymbol{v}}=\frac{{\boldsymbol{v}}}{c_{0}},\quad{\widehat{f}_{i}}(\widehat{t},\widehat{{\boldsymbol{x}}},\widehat{\boldsymbol{v}})=\frac{L^{3}c_{0}^{3}}{N}f_{i}(t,{{\boldsymbol{x}}},{\boldsymbol{v}}),\quad\widehat{B}_{i}(\widehat{\boldsymbol{v}},\widehat{\boldsymbol{v}}_{*},\widehat{\boldsymbol{\sigma}})=\frac{1}{c_{0}4\pi
r^{2}}B_{i}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma}),$ (5)

respectively, and rewrite the system (2) of BEs in dimensionless form in which
only two parameters appear—Mach number $\mathrm{Ma}$ which multiplies
$\partial\widehat{f_{i}}/\partial\widehat{t}$, and a reciprocal of the Knudsen
number $\mathrm{Kn}$ which multiplies the scaled collision operators:

$\mathrm{Ma}=\frac{u_{0}}{c_{0}},\quad\mathrm{Kn}=\frac{\text{mean free
path}}{\text{macroscopic length scale}}=\frac{L^{3}}{N\times 4\pi
r^{2}}\times\frac{1}{L}.$

In the sequel we shall assume that $\mathrm{Ma}=\mathrm{Kn}=\alpha\ll 1$,
which amounts to diffusive scaling of the BEs (2). Removing the hats for
simplicity and adding superscript $\alpha$ to the distribution function so as
to differentiate the dimensionless system of BEs from equation (2), we obtain:

$\alpha\frac{\partial f_{i}^{\alpha}}{\partial
t}+{\boldsymbol{v}}\cdot\frac{\partial
f_{i}^{\alpha}}{\partial{{\boldsymbol{x}}}}=\frac{1}{\alpha}{Q}_{i}^{m}(f^{\alpha}_{i},f^{\alpha}_{i})+\frac{1}{\alpha}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{S}{Q}^{b}_{ij}(f^{\alpha}_{i},f^{\alpha}_{j}).$ (6)

###  2.2 Moment Equations

Multiplying the scaled system of BEs (6) by an arbitrary function of the
velocity variable $\psi_{i}({\boldsymbol{v}})$, and then integrate the
resulting equation over all values of the velocity ${\boldsymbol{v}}$ gives
the moment equations:

$\alpha\frac{\partial}{\partial
t}\int_{\mathbb{R}^{3}}\\!\\!\psi_{i}({\boldsymbol{v}})f_{i}^{\alpha}\,d{\boldsymbol{v}}+\frac{\partial}{\partial{{\boldsymbol{x}}}}\int_{\mathbb{R}^{3}}\\!\\!{\boldsymbol{v}}\psi_{i}({\boldsymbol{v}})f_{i}^{\alpha}\,d{\boldsymbol{v}}\\!=\\!\frac{1}{\alpha}\int_{\mathbb{R}^{3}}\\!\\!\psi_{i}({\boldsymbol{v}}){Q}_{i}^{m}(f^{\alpha}_{i},f^{\alpha}_{i})\,d{\boldsymbol{v}}+\frac{1}{\alpha}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}\int_{\mathbb{R}^{3}}\\!\\!\psi_{i}({\boldsymbol{v}}){Q}^{b}_{ij}(f^{\alpha}_{i},f^{\alpha}_{j})\,d{\boldsymbol{v}},$
(7)

where the integrals on the right-hand side are the weak formulation of the
mono-species and bi-species collision operators respectively defined as
follows:

$\displaystyle\int_{\mathbb{R}^{3}}\psi_{i}({\boldsymbol{v}})Q_{i}^{m}(f^{\alpha}_{i},f^{\alpha}_{i})\,d{\boldsymbol{v}}=-\frac{1}{4}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\Big{[}f_{i}^{\alpha}({\boldsymbol{v}}^{\prime})f_{i}^{\alpha}({\boldsymbol{v}}_{*}^{\prime})-f_{i}^{\alpha}({\boldsymbol{v}})f_{i}^{\alpha}({\boldsymbol{v}}_{*})\Big{]}$
$\displaystyle\hskip
173.56198pt\times\big{[}\psi_{i}({\boldsymbol{v}}^{\prime})+\psi_{i}({\boldsymbol{v}}^{\prime}_{*})-\psi_{i}({\boldsymbol{v}})-\psi_{i}({\boldsymbol{v}}_{*})\big{]}B_{i}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma})\,d\boldsymbol{\sigma}\,d{\boldsymbol{v}}_{*}\,d{\boldsymbol{v}},$
(8)
$\displaystyle\int_{\mathbb{R}^{3}}\psi_{i}({\boldsymbol{v}})Q^{b}_{ij}(f^{\alpha}_{i},f^{\alpha}_{j})\,d{\boldsymbol{v}}\\!=\\!\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}f_{i}^{\alpha}({\boldsymbol{v}})f_{j}^{\alpha}({\boldsymbol{v}}_{*})\Big{[}\psi_{i}({\boldsymbol{v}}^{\prime})-\psi_{i}({\boldsymbol{v}})\Big{]}B_{ij}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma})\,d\boldsymbol{\sigma}\,d{\boldsymbol{v}}_{*}d{\boldsymbol{v}}.$
(9)

##  3 Maximum entropy principle

The maximum entropy principle (MEP) determines the approximate velocity
distribution function, compatible with selected moments, through maximization
of the kinetic entropy [17, 18, 19, 20, 21]. Our aim is to apply MEP and
_derive_ the properly scaled velocity distribution function which leads to a
non-isothermal Maxwell-Stefan model. To that end we introduce the following

###### Definition 1

Kinetic entropy $H(t,{{\boldsymbol{x}}})$ is defined as:

$H(t,{{\boldsymbol{x}}}):=\sum_{i=1}^{S}H_{i}(t,{{\boldsymbol{x}}}),\quad
H_{i}(t,{{\boldsymbol{x}}}):=-k_{\mathrm{B}}\int_{\mathbb{R}^{3}}f_{i}\log(b_{i}f_{i})d{\boldsymbol{v}},$
(10)

where $b_{i}$ is the dimensional constant which makes the argument of the
$\log$ function dimensionless. Moments of the distribution functions,
representing partial mass, momentum and energy densities, are defined as:

$\begin{split}\rho_{i}(t,{{\boldsymbol{x}}}):=\int_{\mathbb{R}^{3}}m_{i}f_{i}d{\boldsymbol{v}},&\quad\rho_{i}(t,{{\boldsymbol{x}}}){\boldsymbol{u}}_{i}(t,{{\boldsymbol{x}}}):=\int_{\mathbb{R}^{3}}m_{i}{\boldsymbol{v}}f_{i}d{\boldsymbol{v}},\\\
\rho_{i}(t,{{\boldsymbol{x}}})|{\boldsymbol{u}}_{i}(t,{{\boldsymbol{x}}})|^{2}+3&\rho_{i}(t,{{\boldsymbol{x}}})\frac{k_{B}}{m_{i}}T(t,{{\boldsymbol{x}}}):=\int_{\mathbb{R}^{3}}m_{i}|{\boldsymbol{v}}|^{2}f_{i}d{\boldsymbol{v}},\end{split}$
(11)

where $\rho_{i}$ is the mass density and ${\boldsymbol{u}}_{i}$ is the mean
velocity of the specie $\mathcal{A}_{i}$, while $T$ is the common kinetic
temperature.

It will be assumed that all the species have the same kinetic temperature $T$.
This simplifying assumption is appropriate for non-isothermal diffusion
problems, see [22].

In the sequel we shall use the dimensionless macroscopic variables:

$\widehat{\rho}_{i}=\frac{L^{3}}{m_{i}N}\rho_{i},\quad\widehat{\boldsymbol{u}}_{i}=\frac{\boldsymbol{u}_{i}}{u_{0}},\quad\widehat{T}=\frac{T}{T_{0}},\quad\widehat{H}_{i}=\frac{L^{3}}{k_{\mathrm{B}}N}H_{i},\quad\widehat{b}_{i}=\frac{N}{L^{3}c_{0}^{3}}b_{i}.$
(12)

and drop the hat notation for convenience. The variational problem (MEP) may
now be formulated in dimensionless form as follows.

#### Maximum entropy principle

Find the velocity distribution functions
$f_{i}^{\alpha}(t,{{\boldsymbol{x}}},{\boldsymbol{v}})$ which maximize the
kinetic entropy:

$H^{\alpha}(t,{{\boldsymbol{x}}})=-\int_{\mathbb{R}^{3}}\sum_{i=1}^{n}f_{i}^{\alpha}\log(b_{i}f_{i}^{\alpha})d{\boldsymbol{v}}\to\mathrm{max},$
(13)

subject to the following constraints:

$\begin{split}\rho_{i}^{\alpha}(t,{{\boldsymbol{x}}})=\int_{\mathbb{R}^{3}}f_{i}^{\alpha}d{\boldsymbol{v}},&\quad\alpha\rho_{i}^{\alpha}(t,{{\boldsymbol{x}}}){\boldsymbol{u}}_{i}^{\alpha}(t,{{\boldsymbol{x}}})=\int_{\mathbb{R}^{3}}{\boldsymbol{v}}f_{i}^{\alpha}d{\boldsymbol{v}},\\\
\alpha^{2}\rho_{i}^{\alpha}(t,{{\boldsymbol{x}}})|{\boldsymbol{u}}_{i}^{\alpha}(t,{{\boldsymbol{x}}})|^{2}+&3\rho_{i}^{\alpha}(t,{{\boldsymbol{x}}})\frac{3}{5}\frac{m_{0}}{m_{i}}T(t,{{\boldsymbol{x}}})=\int_{\mathbb{R}^{3}}|{\boldsymbol{v}}|^{2}f_{i}^{\alpha}d{\boldsymbol{v}}.\end{split}$
(14)

We would like to note that small parameter $\alpha$ naturally appears in
dimensionless constraints (14) due to scaling introduced in previous Section.

###### Theorem 1

The velocity distribution functions which solve the variational problem
(13)-(14) have the following form:

$f_{i}^{\alpha}(t,{{\boldsymbol{x}}},{\boldsymbol{v}})=\left(\frac{5}{3m_{0}}\right)^{3/2}\rho_{i}^{\alpha}(t,{{\boldsymbol{x}}})\left(\frac{m_{i}}{2\pi
T(t,{{\boldsymbol{x}}})}\right)^{3/2}\exp\left(-\frac{5}{3m_{0}}\frac{m_{i}|{\boldsymbol{v}}-\alpha{\boldsymbol{u}}_{i}^{\alpha}(t,{{\boldsymbol{x}}})|^{2}}{2T(t,{{\boldsymbol{x}}})}\right).$
(15)

###### Proof 1

Proof of the Theorem 1 relies on the method of multipliers. Starting from
(13)-(14) we define extended functional:

$\mathcal{H}^{\alpha}(t,{{\boldsymbol{x}}}):=\int_{\mathbb{R}^{3}}\sum_{i=1}^{n}\left(f_{i}^{\alpha}\log(b_{i}f_{i}^{\alpha})+\lambda_{i}^{(0)}f_{i}^{\alpha}+\boldsymbol{\lambda}_{i}^{(1)}\cdot{\boldsymbol{v}}f_{i}^{\alpha}+\lambda_{i}^{(2)}|{\boldsymbol{v}}|^{2}f_{i}^{\alpha}\right)d{\boldsymbol{v}}\equiv\int_{\mathbb{R}^{3}}\mathcal{L}({\boldsymbol{v}},f_{i}^{\alpha},\nabla_{{\boldsymbol{v}}}f_{i}^{\alpha})d{\boldsymbol{v}},$

where $\lambda_{i}^{(0)}(t,{{\boldsymbol{x}}})\in\mathbb{R}$,
$\boldsymbol{\lambda}_{i}^{(1)}(t,{{\boldsymbol{x}}})\in\mathbb{R}^{3}$ and
$\lambda_{i}^{(2)}(t,{{\boldsymbol{x}}})\in\mathbb{R}$ are the unknown
multipliers. Since the density function $\mathcal{L}$ does not depend on
$\nabla_{{\boldsymbol{v}}}f_{i}^{\alpha}$, necessary condition for extremum is
reduced to $\partial\mathcal{L}/\partial f_{i}^{\alpha}=0$, $i=1,\ldots,S$,
which yields $S$ uncoupled equations that can be solved for $f_{i}^{\alpha}$:

$f_{i}^{\alpha}=\frac{1}{b_{i}}\exp\left[-\left(1+\lambda_{i}^{(0)}+\boldsymbol{\lambda}_{i}^{(1)}\cdot{\boldsymbol{v}}+\lambda_{i}^{(2)}|{\boldsymbol{v}}|^{2}\right)\right].$

Plugging the last expression into the constraints (14), after rather
straightforward computation the following relations which determine the
multipliers are obtained:

$\lambda_{i}^{(2)}=\frac{5}{3}\frac{m_{i}}{m_{0}}\frac{1}{2T},\quad\frac{\boldsymbol{\lambda}_{i}^{(1)}}{2\lambda_{i}^{(2)}}=-\alpha{\boldsymbol{u}}_{i}^{\alpha},\quad\frac{1}{b_{i}}\exp\left(-1-\lambda_{i}^{(0)}\right)\exp\left(\frac{|\boldsymbol{\lambda}_{i}^{(1)}|^{2}}{4\lambda_{i}^{(2)}}\right)\left(\frac{\pi}{\lambda_{i}^{(2)}}\right)^{3/2}=\rho_{i}^{\alpha}.$

The proof is completed by returning these expressions into the maximizer.

##  4 Non-isothermal Maxwell-Stefan diffusion equations

The Maxwell-Stefan diffusion equations for the mixture of Maxwell molecules
were derived in [13], and in the non-isothermal setting in [22], while in [16]
the case of reactive mixture of polyatomic gases was considered. Since our
interest is on inert mixtures, we shall give the non-isothermal Maxwell-Stefan
diffusion equations here for the completeness. They can be obtained from the
moment equations (7) by using the scaled distribution function (15) obtained
by MEP. The mass/concentrantion and momentum balance laws for the species are
obtained taking $\psi_{i}(\boldsymbol{v})=1$ and
$\psi_{i}(\boldsymbol{v})=\boldsymbol{v}$, respectively. The energy
conservation law for the mixture is obtained taking
$\psi_{i}(\boldsymbol{v})=|\boldsymbol{v}|^{2}/2$ and summing up all the
energy balance laws for the species. The resulting equations read:

$\displaystyle\frac{\partial\rho_{i}^{\alpha}}{\partial
t}+\frac{\partial}{\partial{{\boldsymbol{x}}}}(\rho_{i}^{\alpha}{\boldsymbol{u}}_{i}^{\alpha})=0,$
(16) $\displaystyle\alpha^{2}\left[\frac{\partial}{\partial
t}(\rho_{i}^{\alpha}{\boldsymbol{u}}_{i}^{\alpha})+\frac{\partial}{\partial{{\boldsymbol{x}}}}\left(\rho_{i}^{\alpha}{\boldsymbol{u}}_{i}^{\alpha}\otimes{\boldsymbol{u}}_{i}^{\alpha}\right)\right]+\frac{3m_{0}}{5m_{i}}\frac{\partial}{\partial{{\boldsymbol{x}}}}(\rho_{i}^{\alpha}T)=2\pi\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}||b_{ij}||_{L^{1}}\frac{m_{j}}{m_{i}+m_{j}}\rho_{i}^{\alpha}\rho_{j}^{\alpha}({\boldsymbol{u}}^{\alpha}_{j}-{\boldsymbol{u}}^{\alpha}_{i})$
$\displaystyle\frac{\partial}{\partial
t}\sum_{i=1}^{S}\left(\alpha^{2}\frac{1}{2}\rho_{i}^{\alpha}|{\boldsymbol{u}}_{i}^{\alpha}|^{2}+\frac{3m_{0}}{5m_{i}}\frac{3}{2}\rho_{i}^{\alpha}T\right)+\frac{\partial}{\partial{{\boldsymbol{x}}}}\sum_{i=1}^{S}\left[\left(\alpha^{2}\frac{1}{2}\rho_{i}^{\alpha}|{\boldsymbol{u}}_{i}^{\alpha}|^{2}+\frac{3m_{0}}{5m_{i}}\frac{5}{2}\rho_{i}^{\alpha}T\right){\boldsymbol{u}}_{i}^{\alpha}\right]=0.$

Taking the limit as $\alpha\to 0^{+}$, one obtains the non-isothermal Maxwell-
Stefan diffusion equations:

$\displaystyle\frac{\partial\rho_{i}}{\partial
t}+\frac{\partial}{\partial{{\boldsymbol{x}}}}(\rho_{i}{\boldsymbol{u}}_{i})=0,$
(17)
$\displaystyle\frac{3m_{0}}{5m_{i}}\frac{\partial}{\partial{{\boldsymbol{x}}}}(\rho_{i}T)=2\pi\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}||b_{ij}||_{L^{1}}\frac{m_{j}}{m_{i}+m_{j}}\rho_{i}\rho_{j}({\boldsymbol{u}}_{j}-{\boldsymbol{u}}_{i})$
$\displaystyle\frac{3}{2}\frac{\partial}{\partial
t}\sum_{i=1}^{S}\left(\rho_{i}T\right)+\frac{5}{2}\frac{\partial}{\partial{{\boldsymbol{x}}}}\sum_{i=1}^{S}\left(\rho_{i}{\boldsymbol{u}}_{i}T\right)=0,$

where

$\rho_{i}=\lim_{\alpha\rightarrow
0^{+}}\rho_{i}^{\alpha}\quad\text{and}\quad{\boldsymbol{u}}_{i}=\lim_{\alpha\rightarrow
0^{+}}{\boldsymbol{u}}_{i}^{\alpha}.$

Equations (16) and (17) are up to scaling completely equivalent to the
equations derived in [13, 22].

##  5 Entropy balance law

To derive the macroscopic entropy balance law from the BEs (2), we first have
to define the entropy flux and the entropy production.

###### Definition 2

Entropy flux $\boldsymbol{\Phi}(t,{\boldsymbol{x}})$ and entropy production
$D(t,{\boldsymbol{x}})$ are defined as:

$\displaystyle\boldsymbol{\Phi}(t,{\boldsymbol{x}}):=\sum_{i=1}^{S}\boldsymbol{\Phi}_{i}(t,{\boldsymbol{x}}),\quad\boldsymbol{\Phi}_{i}(t,{\boldsymbol{x}}):=-k_{\mathrm{B}}\int_{\mathbb{R}^{3}}\boldsymbol{v}f_{i}\log(b_{i}f_{i})d\boldsymbol{v},$
$\displaystyle
D(t,{\boldsymbol{x}}):=\sum_{i=1}^{S}D_{i}(t,{\boldsymbol{x}}),\quad
D_{i}(t,{\boldsymbol{x}}):=-k_{\mathrm{B}}\int_{\mathbb{R}^{3}}{Q}_{i}^{m}(f_{i},f_{i})\log(b_{i}f_{i})d\boldsymbol{v}-k_{\mathrm{B}}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}\int_{\mathbb{R}^{3}}{Q}^{b}_{ij}(f_{i},f_{j})\log(b_{i}f_{i})d\boldsymbol{v}.$

By introducing dimensionless quantities:

$\hat{\boldsymbol{\Phi}}_{i}=\frac{L^{3}}{k_{\mathrm{B}}Nc_{0}}\boldsymbol{\Phi}_{i},\quad\hat{D}_{i}=\frac{L^{6}}{k_{\mathrm{B}}N^{2}4\pi
r^{2}c_{0}}D_{i},$

and dropping hat notation for convenience, one may determine the dimensionless
partial entropy fluxes and partial entropy productions:

$\begin{split}\boldsymbol{\Phi}_{i}^{\alpha}(t,{\boldsymbol{x}})&=-\int_{\mathbb{R}^{3}}\boldsymbol{v}f_{i}^{\alpha}\log(b_{i}f_{i}^{\alpha})d\boldsymbol{v},\\\
D_{i}^{\alpha}(t,{\boldsymbol{x}})&=-\int_{\mathbb{R}^{3}}{Q}_{i}^{m}(f^{\alpha}_{i},f^{\alpha}_{i})\log(b_{i}f_{i}^{\alpha})d\boldsymbol{v}-\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}\int_{\mathbb{R}^{3}}Q^{b}_{ij}(f^{\alpha}_{i},f^{\alpha}_{j})\log(b_{i}f_{i}^{\alpha})d\boldsymbol{v}.\end{split}$
(18)

Setting $\psi_{i}(\boldsymbol{v})=-\log(b_{i}f_{i}^{\alpha})$ in (7), one
obtains the following scaled entropy balance law:

$\alpha\frac{\partial H^{\alpha}}{\partial
t}+\frac{\partial\boldsymbol{\Phi}^{\alpha}}{\partial{\boldsymbol{x}}}=\frac{1}{\alpha}D^{\alpha}.$
(19)

It is a matter of straightforward computation to show that dimensionless
entropy densities and fluxes read:

$H_{i}^{\alpha}(t,{\boldsymbol{x}})=\left\\{\log\left[b_{i}\left(\frac{5}{3m_{0}}\right)^{\frac{3}{2}}\rho_{i}^{\alpha}\left(\frac{m_{i}}{2\pi
T}\right)^{\frac{3}{2}}\right]-\frac{3}{2}\right\\}\rho_{i}^{\alpha},\quad\boldsymbol{\Phi}_{i}^{\alpha}(t,{\boldsymbol{x}})=\alpha
H_{i}^{\alpha}(t,{\boldsymbol{x}}){\boldsymbol{u}}_{i}^{\alpha}$ (20)

However, computation of the entropy production is summarized in the following
Theorem.

###### Theorem 2

The dimensionless partial entropy productions read:

$D_{i}^{\alpha}=\alpha^{2}\frac{10\pi}{3Tm_{0}}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}||b_{ij}||_{L^{1}}\frac{m_{i}m_{j}\rho_{i}^{\alpha}\rho_{j}^{\alpha}}{m_{i}+m_{j}}\left[\frac{m_{i}{\boldsymbol{u}}_{i}^{\alpha}+m_{j}{\boldsymbol{u}}_{j}^{\alpha}}{m_{i}+m_{j}}\cdot({\boldsymbol{u}}_{j}^{\alpha}-{\boldsymbol{u}}_{i}^{\alpha})-{\boldsymbol{u}}_{i}^{\alpha}\cdot({\boldsymbol{u}}_{j}^{\alpha}-{\boldsymbol{u}}_{i}^{\alpha})\right],$
(21)

while the dimensionless total entropy production have the form:

$D^{\alpha}=\sum_{i=1}^{S}D_{i}^{\alpha}=\alpha^{2}\frac{5\pi}{3Tm_{0}}\sum_{i=1}^{S}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}\left[||b_{ij}||_{L^{1}}\frac{m_{i}m_{j}}{m_{i}+m_{j}}\rho_{i}^{\alpha}\rho_{j}^{\alpha}\left|{\boldsymbol{u}}_{j}^{\alpha}-{\boldsymbol{u}}_{i}^{\alpha}\right|^{2}\right].$
(22)

###### Proof 2

To obtain the partial entropy, we need to evaluate the integrals given in
equation (18) and we will focus on the case Maxwell molecules. This means that
the collision cross-sections depend on the
${\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma}$ through the
deviation angle $\theta\in[0,\pi]$ between
${\boldsymbol{v}}-{\boldsymbol{v}}_{*}$ and $\boldsymbol{\sigma}$ only.
Specifically,

$B_{ij}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma})=b_{ij}\bigg{(}\frac{{\boldsymbol{v}}-{\boldsymbol{v}}_{*}}{|{\boldsymbol{v}}-{\boldsymbol{v}}_{*}|}\cdot\boldsymbol{\sigma}\bigg{)}=b_{ij}(\cos\theta).$

Moreover, we assume that $b_{ij}$ is an even function and is such that
$b_{ij}\in L^{1}(-1,1)$ following Grad’s angular cut-off assumption.
Therefore,

$\int_{\mathbb{S}^{2}}B_{ij}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma})\,d\boldsymbol{\sigma}=2\pi||b_{ij}||_{L^{1}}\quad\text{and}\quad\int_{\mathbb{S}^{2}}\boldsymbol{\sigma}B_{ij}({\boldsymbol{v}},{\boldsymbol{v}}_{*},\boldsymbol{\sigma})\,d\boldsymbol{\sigma}=0,$
(23)

see [13]. Furthermore, using the definition of the species distribution
functions (15) in $\log(b_{i}f^{\alpha}_{i})$, one can rewrite equation (18)
as follows:

$\displaystyle D_{i}^{\alpha}$
$\displaystyle=\frac{5m_{i}}{6Tm_{0}}k_{B}\underbrace{\int_{\mathbb{R}^{3}}|{\boldsymbol{v}}|^{2}{Q}_{i}^{m}(f^{\alpha}_{i},f^{\alpha}_{i})\,d{\boldsymbol{v}}}_{\cal
A}-\alpha\frac{10m_{i}k_{B}}{6Tm_{0}}{\boldsymbol{u}}_{i}^{\alpha}\cdot\underbrace{\int_{\mathbb{R}^{3}}{\boldsymbol{v}}{Q}_{i}^{m}(f^{\alpha}_{i},f^{\alpha}_{i})\,d{\boldsymbol{v}}}_{\cal
B}$ $\displaystyle+\frac{5m_{i}}{6Tm_{0}}k_{B}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}\underbrace{\int_{\mathbb{R}^{3}}|{\boldsymbol{v}}|^{2}{Q}^{b}_{ij}(f^{\alpha}_{i},f^{\alpha}_{j})\,d{\boldsymbol{v}}}_{\cal
E}-\alpha\frac{10m_{i}k_{B}}{6Tm_{0}}{\boldsymbol{u}}_{i}^{\alpha}\cdot\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}\underbrace{\int_{\mathbb{R}^{3}}{\boldsymbol{v}}{Q}^{b}_{ij}(f^{\alpha}_{i},f^{\alpha}_{j})\,d{\boldsymbol{v}}}_{\cal
F}$ (24)
$\displaystyle\\!+\\!k_{B}\Bigg{\\{}\frac{{\alpha}^{2}5m_{i}}{6Tm_{0}}|{\boldsymbol{u}}_{i}^{\alpha}|^{2}-\log\Bigg{[}b_{i}\bigg{(}\frac{5}{3m_{0}}\bigg{)}^{\\!\frac{3}{2}}\rho_{i}^{\alpha}\bigg{(}\frac{m_{i}}{2\pi
T}\bigg{)}^{\\!\frac{3}{2}}\Bigg{]}\Bigg{\\}}\Bigg{(}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{S}\underbrace{\int_{\mathbb{R}^{3}}\\!\\!{Q}^{b}_{ij}(f^{\alpha}_{i},f^{\alpha}_{j})d{\boldsymbol{v}}}_{\cal
G}+\underbrace{\int_{\mathbb{R}^{3}}\\!\\!{Q}_{i}^{m}(f^{\alpha}_{i},f^{\alpha}_{i})d{\boldsymbol{v}}}_{\cal
H}\Bigg{)}.$

The integrals ${\cal A}$, ${\cal B}$, ${\cal G}$ and ${\cal H}$ all vanish,
that is

${\cal A}=0\quad{\cal B}=0,\quad{\cal G}=0\quad\text{and}\quad{\cal H}=0.$
(25)

To show that the integrals ${\cal H}$ and ${\cal G}$ both vanish, we use the
weak formulation of the mono-species and bi-species collision operators given
in equations (8) and (9), respectively, with $\psi({\boldsymbol{v}})=1$. To
show that the integrals ${\cal A}$ and ${\cal B}$ both vanish, we use the weak
formulation of the mono-species collision operator (8) with
$\psi({\boldsymbol{v}})=|{\boldsymbol{v}}|^{2}$ and
$\psi({\boldsymbol{v}})={\boldsymbol{v}}$, respectively, together with the
conservation laws of kinetic energy and linear momentum for mono-species
collisions: equations (4)2 and (4)1 with $m_{j}$ replaced by $m_{i}$.
Furthermore, using the weak form of the bi-species collision operator (9) with
$\psi({\boldsymbol{v}})=|{\boldsymbol{v}}|^{2}$ and
$\psi({\boldsymbol{v}})={\boldsymbol{v}}$, respectively, together with
equations (3) and (23), one obtains that the values of the integrals ${\cal
E}$ and ${\cal F}$ are as given below:

${\cal
E}=\alpha^{2}4\pi||b_{ij}||_{L^{1}}\frac{m_{j}\rho_{i}^{\alpha}\rho_{j}^{\alpha}}{(m_{i}+m_{j})^{2}}(m_{i}{\boldsymbol{u}}_{i}^{\alpha}+m_{j}{\boldsymbol{u}}_{j}^{\alpha})\cdot({\boldsymbol{u}}_{j}^{\alpha}-{\boldsymbol{u}}_{i}^{\alpha}),\quad{\cal
F}=\alpha\frac{m_{j}}{m_{i}+m_{j}}2\pi||b_{ij}||_{L^{1}}\rho_{i}^{\alpha}\rho_{j}^{\alpha}\big{(}{\boldsymbol{u}}_{j}^{\alpha}-{\boldsymbol{u}}_{i}^{\alpha}\big{)}.$
(26)

Substituting equations (25) and (26) into (24) gives (21). Summing equation
(21) over all species gives (22).

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