11institutetext: Thomas Borsoni 22institutetext: ENS Paris-Saclay, Université
Paris-Saclay, Gif-sur-Yvette, France
Università di Parma, Parma, Italia
22email<EMAIL_ADDRESS>33institutetext: Marzia Bisi
44institutetext: Università di Parma, Parma, Italia
44email<EMAIL_ADDRESS>55institutetext: Maria Groppi 66institutetext:
Università di Parma, Parma, Italia
66email<EMAIL_ADDRESS>
# A general framework for the kinetic modelling of polyatomic gases

Thomas Borsoni  Marzia Bisi  Maria Groppi

(Received: date / Accepted: date)

###### Abstract

A general framework for the kinetic modelling of non-relativistic polyatomic
gases is proposed, where each particle is characterized both by its velocity
and by its internal state, and the Boltzmann collision operator involves
suitably weighted integrals over the space of internal energies. The
description of the internal structure of a molecule is kept highly general,
and this allows classical and semi-classical models, such as the monoatomic
gas description, the continuous internal energy structure, and the description
with discrete internal energy levels, to fit our framework. We prove the
H-Theorem for the proposed kinetic equation of Boltzmann type in this general
setting, and characterize the equilibrium Maxwellian distribution and the
thermodynamic number of degrees of freedom. Euler equations are derived, as
zero-order approximation in a suitable asymptotic expansion. In addition,
within this general framework it is possible to build up new models, highly
desirable for physical applications, where rotation and vibration are
precisely described. Examples of models for the Hydrogen Fluoride gas are
presented.

## Acknowledgements

This study was initiated while T.B. was visiting the Department of
Mathematical, Physical and Computer Sciences, University of Parma, with the
financial support of the Ecole Normale Superieure Paris-Saclay. The authors
thank the support by the University of Parma, by the Italian National Group of
Mathematical Physics (GNFM-INdAM), and by the Italian National Research
Project ”Multiscale phenomena in Continuum Mechanics: singular limits, off-
equilibrium and transitions” (Prin 2017YBKNCE).

###### Contents

  1. 1 Introduction
  2. 2 General setting and collision rules
  3. 3 Boltzmann model
    1. 3.1 Boltzmann collision operator
    2. 3.2 Note about degeneracy
    3. 3.3 Weak formulation
    4. 3.4 Collision invariants
  4. 4 H Theorem and collision equilibria
  5. 5 Euler limit
  6. 6 Equipartition Theorem
  7. 7 Models within the framework
    1. 7.1 Existing models
    2. 7.2 Combination of the continuous and discrete models
    3. 7.3 Building a model within the framework
  8. 8 Reduction to one real variable and comparison with Borgnakke-Larsen
  9. 9 Conclusions

##  1 Introduction

The purpose of this paper is to propose a consistent general Boltzmann–type
model for a polyatomic gas, able to include the kinetic models already
exploited in the pertinent literature, but also to give rise to more accurate
descriptions of internal energies of polyatomic molecules. The construction of
a reliable mathematical model for polyatomic gaseous particles is highly
desirable in view of physical applications, since the main constituents of the
atmosphere are polyatomic. A review of possible problems involving monoatomic
and polyatomic species, possibly undergoing also chemical reactions, may be
found in Nagnibeda-Kustova ; Zhdanov . For this reason, in last decades the
investigation of polyatomic particles has gained interest not only in the
frame of kinetic theory, but also in Extended Thermodynamics Ruggeri-Sugiyama-
new  and in the derivation of accurate schemes for fluid-dynamics equations
Xu-book2013 ; Toro-book2014 .

In kinetic theory, non–translational degrees of freedom of polyatomic
particles are usually described by means of an internal energy variable, that
may be assumed discrete or continuous. The basic features of the model with a
finite set of discrete energy levels may be found in groppi1999kinetic ;
Giovangigli ; the gas is considered as a sort of mixture of monatomic
components, each one characterized by its energy level, and particles interact
by binary collisions preserving total energy, but with possible exchange of
energy between its kinetic and internal (excitation) forms, allowing thus
particle transitions from one energy component to another. The kinetic model
based on a continuous internal energy parameter $I$ has been proposed by
Borgnakke and Larsen in borgnakke1975statistical  and then extensively
investigated in desvillettes1997modele ; desvillettes2005kinetic ; the gas
distribution function turns out to depend also on this energy variable, and
macroscopic fields and Boltzmann operators involve integrations in $\text{d}I$
with an associated integration weight $\varphi(I)$. This measure
$\varphi(I)\,\text{d}I$ is a parameter of the model, and different options for
it allow to reproduce any desired number of internal degrees of freedom. With
the commonly adopted choice $\varphi(I)=I^{\alpha}$, this number of internal
degrees of freedom is independent of the temperature of the gas (describing
thus polytropic gases). On the other hand, the set of discrete internal energy
levels proposed in wang1951transport ; groppi1999kinetic  allows to obtain a
temperature-dependent number of degrees of freedom Bisi-Spiga-RicMat2017 ,
with a specific heat at constant volume which resembles the physical laws of
statistical mechanics Landau . Both formulations with discrete and continuous
energy turn out to be well suited also in presence of simple chemical
reactions (see for instance groppi1999kinetic ; desvillettes2005kinetic ) and
discrete energy levels are also well adapted to problems involving complex
chemistry since real molecules actually have discrete energy levels
Giovangigli .

Since Boltzmann equations are quite complicated to deal with, simpler kinetic
models have been proposed, mainly of BGK or ES-BGK type, both for gases or
mixtures with discrete energies GroSpiPoF2004 ; BGS-PhysRevE ; Bisi-Travaglini
and for the continuous energy description Struchtrup ; Andries-etal-2000 ;
Brull-Schneider2009 ; Bisi-Monaco-Soares . Hydrodynamic limits of kinetic
equations have been performed, at Euler or Navier–Stokes accuracy, in order to
build up consistent fluid–dynamic equations for the main macroscopic fields,
focusing the attention on the transport coefficients, especially on the bulk
viscosity or, equivalently, on the so-called dynamical pressure, a correction
to the classical scalar pressure due to the polyatomic structure of molecules
Milana-2014 ; Ruggeri-PhysRevE-2017 ; Bisi-Ruggeri-Spiga . Such kinetic or
hydrodynamic equations have been applied to simple physical problems, as the
existence of shock-wave profiles Kosuge-Aoki-2018 ; Kosuge-Kuo-Aoki ;
Taniguchi-etal-2016 ; Pavic-Madjarevic-Simic , the Poiseuille and thermal
creep flows Funagane-etal ; LeiWu-etal , and suitable boundary conditions have
been derived in order to allow the investigation of flow problems with solid
boundaries Hattori-etal .

The main drawback of these kinetic models is the fact that they wish to
describe all internal features of polyatomic molecules by means of a single
internal energy parameter (discrete or continuous). In order to improve the
applicability of Boltzmann equations to real problems, more accurate models
would be needed, able to separate the vibrational degrees of freedom from the
rotational ones. More precisely, since the gap between two subsequent discrete
levels is much lower for rotational energy than for vibrational energy
Herzberg , a reasonable way of modeling could approximate the rotational part
by means of a continuous variable, keeping the vibrational part discrete. This
is the main motivation for the present work; indeed, we would like to merge
the existing discrete and continuous energy models in a general (abstract)
framework, in order to combine their strength and obtain a full description of
both rotation and vibration. Models with discrete vibrational degrees of
freedom and continuous rotational states have long been used in the literature
in order to analyse vibrational state-to-state non-equilibrium phenomena
Nagnibeda-Kustova ; Kal20 . Some attempts to distinguish vibrational from
rotational energy have been performed also at kinetic level: in the discrete
energy formulation, this aim could be achieved by choosing discrete energies
depending on several discrete indices Giovangigli , but also other models have
been proposed, of BGK or Fokker-Planck–type Mathiaud-Mieussens ; Dauvois-etal
or in the frame of Extended Thermodynamics Ruggeri-vibrot  (involving two
continuous energy variables). The idea of introducing two different
temperatures (translational and internal) in the kinetic model for a
polyatomic gas, investigating then their influence on volume viscosity and
other transport coefficients, has been developed in Nagnibeda-Kustova ; BG11 ;
Aoki-Bisi-etal-PRE . Moreover, a proper rovibrational collision model has been
built up in order to study the internal energy excitation and dissociation
processes behind a strong shockwave in a nitrogen flow PMMJ14 .

We propose here a general kinetic framework for the description of a single
polyatomic gas, in which each particle is characterized, in addition to its
position and velocity, also by a suitable internal state $\zeta$. The space of
all admissible internal states $\mathcal{E}$ is endowed with a measure $\mu$.
In other words, instead of considering a continuous or a discrete variable to
describe the internal state of a given molecule, we use a single parameter
$\zeta$, and integration over the continuous variable or summation over the
discrete one are replaced by a single integration over $\zeta$ against the
given proper measure $\mu$. Moreover, we clearly distinguish the state of a
molecule $\zeta$ and the energy function $\varepsilon(\zeta)$, which
associates to each state a suitable energy value.

The continuous and the discrete energy models may be included into this
framework by choosing $\mathcal{E}=\mathbb{R}_{+}$ and
$\mathcal{E}=\mathbb{N}$, respectively. We prove that, even keeping the space
$\mathcal{E}$, the measure $\mu$ and the energy function $\varepsilon(\zeta)$
generic (not explicit), the collision rule determining post-collision
velocities may be defined, and the corresponding Boltzmann equation fulfills
the expected consistency properties. More precisely, the Boltzmann H theorem
holds in the generic framework, and collision equilibria turn out to be
provided by Maxwellian distributions depending on the internal energy function
$\varepsilon(\zeta)$. The technical steps of the proofs rely essentially on
the conservation of momentum and of total energy, therefore there is no need
of fixing from the beginning the microscopic energy structure of particles. We
remark that our framework describes isotropic gases, where polarization
effects due to external fields acting on (non ionized) polyatomic gases may be
neglected MBKK90 .

In more detail, the content of the paper is organized as follows. In section
2, we introduce the framework and its assumptions, and we make the collision
rules explicit. Our method is inspired by desvillettes1997modele ;
desvillettes2005kinetic ; bisi2005kinetic , with some necessary modifications,
since the procedure in such references is specifically designed for a
continuous or discrete internal variable. Section 3 focuses then on the
Boltzmann collision operator and on some of its main properties, as the weak
formulation and the characterisation of collision invariants. In section 4 we
prove the validity of the Boltzmann H-theorem, and we investigate the
equilibrium Maxwellian distribution and the thermodynamic number of internal
degrees of freedom. In section 5, the hydrodynamic Euler equations
corresponding to our general model are derived. Sections 6 and 7 are devoted
to modelling: we explain how to recover the monoatomic, continuous and
discrete formulations, and we give insights on how to build consistent new
models within our framework, including a description able to separate rotation
and vibrations of polyatomic particles; examples of possible models are
provided and commented on, with reference to the pertinent gas–dynamics
literature. In Section 8, we show how is possible to reduce models fitting the
general setting to a one-real-variable description with a suitable measure,
similar to the continuous model with integration weight. We address the
problem of computing this measure from the general setting, and show that the
combination of a continuous model with weight and a discrete model reduces to
a continuous model with a different weight that can be made explicit, allowing
an easier treatment for numerical applications. Finally, section 9 contains
some concluding remarks and perspectives.

##  2 General setting and collision rules

We consider a gas composed of one species of molecules, which mass we denote
by $m\in\mathbb{R}_{+}^{*}$. Each molecule is microscopically described by its
velocity, denoted by $v$, and its internal state, denoted by $\zeta$. The
latter is usually related to rotation or vibration that occurs inside the
molecule. We assume that there exists a set of all possible internal states
$\mathcal{E}$, and that this set is a measure space. All molecules, being of
the same species, are related to the same space $\mathcal{E}$. Secondly, we
assume that to each internal state $\zeta\in\mathcal{E}$ corresponds an
energy, that is, there exists an internal energy function
$\varepsilon:\mathcal{E}\to\mathbb{R}$. The energy of a molecule with velocity
$v$ and internal state $\zeta$ is then equal to
$\displaystyle\frac{m}{2}|v|^{2}+\varepsilon(\zeta)$, the sum of its kinetic
and internal parts.

###### Definition 2.1

We call space of internal states a non-empty measure space
$(\mathcal{E},\mathcal{A},\mu)$, with $\mu(\mathcal{E})\neq 0$.

To clarify here, $\mathcal{E}$ is the space of internal states, $\mathcal{A}$
is a $\sigma$-algebra on $\mathcal{E}$, and $\mu$ is a (non-negative) measure
on $(\mathcal{E},\mathcal{A})$. In the following definition,
$\text{Bor}(\mathbb{R})$ is the set of Borelians of $\mathbb{R}$.

###### Definition 2.2

We call internal energy function a
$(\mathcal{A},\text{Bor}(\mathbb{R}))$-measurable function
$\varepsilon:\mathcal{E}\to\mathbb{R}.$

In our framework, we only require two assumptions on
$(\mathcal{E},\mathcal{A},\mu)$ and $\varepsilon$, which both physically make
sense. The first is to assume that $\varepsilon$ admits an infimum.

###### Definition and assumption 2.3

_We denote by $\varepsilon^{0}$ the essential infimum of $\varepsilon$ in
$\bar{\mathbb{R}}$ on $\mathcal{E}$ under the measure $\mu$, that is_

$\varepsilon^{0}:=\emph{inf
ess}_{\mu}\\{\varepsilon\\}=\emph{sup}\,\Big{\\{}R\in\mathbb{R},\;\emph{s.t.
}\mu(\varepsilon<R)=0\Big{\\}},$ (2.1)

_and assume that $\varepsilon^{0}\in\mathbb{R}$_.

###### Definition 2.4

We define the grounded internal energy function $\bar{\varepsilon}$ by

$\bar{\varepsilon}:=\varepsilon-\varepsilon^{0}.$ (2.2)

Note that by definition of $\varepsilon^{0}$, $\text{inf
ess}_{\mu}\\{\bar{\varepsilon}\\}=0$.

###### Remark 2.1

The reader shall remark that in this paper, the value of $\varepsilon^{0}$ has
no importance. Indeed, $\varepsilon^{0}$ represents the fundamental energy of
configuration of the molecule, and it plays a role only when chemical
reactions are involved.

The second assumption is the well-definedness of the partition function.

###### Definition and assumption 2.5

_We define the partition function $Z$, for all $\beta>0$, as_

$Z(\beta):=\int_{\mathcal{E}}e^{-\beta\bar{\varepsilon}(\zeta)}\,\emph{d}\mu(\zeta),$
(2.3)

_and assume that for all $\beta>0$, $Z(\beta)<\infty$._

The choice of $\bar{\varepsilon}$ instead of simply $\varepsilon$ in the
definition of $Z$ is explained in Remark 4.2. Assumptions 2.3 and 2.5 imply
that $\forall R\in\mathbb{R}$, $\mu(\varepsilon<R)<\infty$, which itself
implies that $\mu$ is $\sigma$-finite.

###### Proposition 2.1

$Z$ is $\mathcal{C}^{\infty}$ on $\mathbb{R}_{+}^{*}$ and for all
$k\in\mathbb{N}$ and $\beta>0$ and, denoting its derivatives by $\displaystyle
Z^{(k)}(\beta)=\frac{{\rm d}^{k}Z}{{\rm d}\beta^{k}}$, we have

$Z^{(k)}(\beta)=\int_{\mathcal{E}}(-\bar{\varepsilon}(\zeta))^{k}\,e^{-\beta\bar{\varepsilon}(\zeta)}\,\emph{d}\mu(\zeta).$
(2.4)

###### Proof

Remarking that for all $x\in\mathbb{R}_{+}$ and $k\in\mathbb{N}$,
$e^{\frac{x}{2}}\geq\frac{\left(\frac{x}{2}\right)^{k}}{k!}$, we get the
inequality $x^{k}\,e^{-x}\leq 2^{k}\,k!\,e^{-\frac{x}{2}}$. Since
$\bar{\varepsilon}\geq 0$ $\mu$-a.e., we get for all $k\in\mathbb{N}$ and
$\beta>0$,
$\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)^{k}\,e^{-\beta\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)<+\infty$
and the result comes by dominated convergence. $\square$

We consider a binary collision and define the collision rule. The states
(velocity and internal state) of the pre-collision molecules are denoted by
$(v,\zeta)$ and $(v_{*},\zeta_{*})$, and the states of the post-collision
molecules are denoted by $(v^{\prime},\zeta^{\prime})$ and
$(v^{\prime}_{*},\zeta^{\prime}_{*})$. In each collision momentum and total
energy are conserved

$\begin{cases}\hskip 83.5ptmv+mv_{*}=mv^{\prime}+mv^{\prime}_{*}\\\
\displaystyle\frac{m}{2}|v|^{2}+\varepsilon(\zeta)+\frac{m}{2}|v_{*}|^{2}+\varepsilon(\zeta_{*})=\frac{m}{2}|v^{\prime}|^{2}+\varepsilon(\zeta^{\prime})+\frac{m}{2}|v^{\prime}_{*}|^{2}+\varepsilon(\zeta^{\prime}_{*}).\end{cases}$
(2.5)

The parallelogram identity yields

$\displaystyle|v-v_{*}|^{2}$
$\displaystyle=2\left(|v|^{2}+|v_{*}|^{2}\right)-|v+v_{*}|^{2}$
$\displaystyle=|v^{\prime}-v^{\prime}_{*}|^{2}+\frac{4}{m}(\varepsilon(\zeta^{\prime})+\varepsilon(\zeta^{\prime}_{*})-\varepsilon(\zeta)-\varepsilon(\zeta_{*})).$

We set

$\Delta\left(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\right):=\frac{1}{4}|v-v_{*}|^{2}+\frac{1}{m}(\varepsilon(\zeta)+\varepsilon(\zeta_{*})-\varepsilon(\zeta^{\prime})-\varepsilon(\zeta^{\prime}_{*})),$
(2.6)

we then have

$\frac{1}{4}|v^{\prime}-v^{\prime}_{*}|^{2}=\Delta\left(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\right).$
(2.7)

We deduce that the collision can occur only _when $\Delta$ is non-negative_,
and in this case, by a classical argument, there exists
$\omega\in\mathbb{S}^{2}$ (two-dimensional unit sphere) such that

$\begin{cases}v^{\prime}&=\displaystyle\frac{v+v_{*}}{2}+\sqrt{\Delta}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
v^{\prime}_{*}&=\displaystyle\frac{v+v_{*}}{2}-\sqrt{\Delta}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\end{cases}$
(2.8)

where $T_{\omega}$ is the symmetry with respect to
$(\mathbb{R}\omega)^{\perp}$,

$\forall\,V\in\mathbb{S}^{2},\quad T_{\omega}\left[V\right]:=V-2(\omega\cdot
V)\,\omega\;\in\mathbb{S}^{2}.$

It is not required to extend formula (2.8) to the set
$\big{\\{}(v,v)\;\text{s.t. }v\in\mathbb{R}^{3}\big{\\}}$, since it is
negligible.

From the previous computation, it is clear that not any collision is possible.
If, before collision, two molecules are in low energy internal states with
close velocities, they cannot reach too much high energy internal states after
collision. For this reason, we fix at first the pre- and post-collision
internal states, and then we consider the space of pre-collision velocity
pairs that make the collision possible. In the following definition,
$E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ is the set of pre-
collision velocity pairs $(v,v_{*})$ so that a collision
$(v,\zeta),(v_{*},\zeta_{*})\rightarrow(\cdot,\zeta^{\prime}),(\cdot,\zeta^{\prime}_{*})$
is allowed.

###### Definition 2.6

For all $\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$, we
define

$E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]:=\Big{\\{}(v,v_{*})\in\mathbb{R}^{3}\times\mathbb{R}^{3}\text{
s.t.
}\Delta\left(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\right)\geq
0\Big{\\}},$ (2.9)

where $\Delta$ is defined by (2.6).

It can be proved that for all
$\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$, the
interior of $E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ is non-
empty, unbounded and arcwise connected.

Now we can define the transformation linking the pre and post-collision
velocities.

###### Definition 2.7

We define for all
$\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$ and
$\omega\in\mathbb{S}^{2}$,

$S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]:\begin{cases}E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]&\to\hskip
100.0ptE[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}]\\\
\displaystyle\quad(v,v_{*})&\mapsto\quad\displaystyle\left(\frac{v+v_{*}}{2}+\sqrt{\Delta}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right],\frac{v+v_{*}}{2}-\sqrt{\Delta}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\right).\end{cases}$
(2.10)

###### Lemma 2.1

$S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ is well-
defined and a bijection, with

$\left(S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]\right)^{-1}=S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}].$

###### Proof

Let $\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$,
$\omega\in\mathbb{S}^{2}$. First, we show that the transformation is well-
defined. We fix $(v,v_{*})\in
E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$. We have that
$\Delta(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})\geq 0$. We
define
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.
Then

$\displaystyle\Delta(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*})$
$\displaystyle=\frac{1}{4}|v^{\prime}-v^{\prime}_{*}|^{2}+\frac{1}{m}(\varepsilon(\zeta^{\prime})+\varepsilon(\zeta^{\prime}_{*})-\varepsilon(\zeta)-\varepsilon(\zeta_{*}))$
$\displaystyle=\frac{1}{4}\left(2\sqrt{\Delta(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})}\right)^{2}+\frac{1}{m}(\varepsilon(\zeta)+\varepsilon(\zeta_{*})-\varepsilon(\zeta)-\varepsilon(\zeta_{*}))$
$\displaystyle=\Delta(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})+\frac{1}{m}(\varepsilon(\zeta^{\prime})+\varepsilon(\zeta^{\prime}_{*})-\varepsilon(\zeta)-\varepsilon(\zeta_{*}))$
$\displaystyle=\frac{1}{4}|v-v_{*}|^{2}\geq 0.$

Thus $(v^{\prime},v^{\prime}_{*})\in
E[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}]$. The application is
well-defined.

Let us now prove that the transformation
$S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ is a
bijection. We fix $(v^{\prime},v^{\prime}_{*})\in
E[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}]$ and define
$(v,v_{*})=S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}](v^{\prime},v^{\prime}_{*})$.
First, we just proved (with inverted notations) that $(v,v_{*})\in
E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ and
$\displaystyle\Delta(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})=\frac{1}{4}|v^{\prime}-v^{\prime}_{*}|^{2}$.
Now we also remark that

$v+v_{*}=v^{\prime}+v^{\prime}_{*},$

and

$T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]=T_{\omega}\left[T_{\omega}\left[\frac{v^{\prime}-v^{\prime}_{*}}{|v^{\prime}-v^{\prime}_{*}|}\right]\right]=\frac{v^{\prime}-v^{\prime}_{*}}{|v^{\prime}-v^{\prime}_{*}|},$

because $T_{\omega}$ is a symmetry. Since
$\displaystyle\Delta(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})=\frac{1}{4}|v^{\prime}-v^{\prime}_{*}|^{2}$,
we deduce that

$\displaystyle\frac{v+v_{*}}{2}+\sqrt{\Delta(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]=\frac{v^{\prime}+v^{\prime}_{*}}{2}+\frac{1}{2}|v^{\prime}-v^{\prime}_{*}|\frac{v^{\prime}-v^{\prime}_{*}}{|v^{\prime}-v^{\prime}_{*}|}=v^{\prime}.$

Analogously,

$\frac{v+v_{*}}{2}-\sqrt{\Delta(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]=\frac{v^{\prime}+v^{\prime}_{*}}{2}-\frac{1}{2}|v^{\prime}-v^{\prime}_{*}|\frac{v^{\prime}-v^{\prime}_{*}}{|v^{\prime}-v^{\prime}_{*}|}=v^{\prime}_{*},$

so that
$S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})=(v^{\prime},v^{\prime}_{*})$,
which ends the proof.

$\square$

###### Lemma 2.2

The Jacobian of the transformation
$S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ is given by
the formula

$J\left[S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]\right](v,v_{*})=\frac{|v^{\prime}-v^{\prime}_{*}|}{|v-v_{*}|},$
(2.11)

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.

###### Proof

This proof is based on that one of Lemma 1 in desvillettes2005kinetic . We
decompose the transformation in a chain of elementary changes of variable. For
the sake of clarity, we fix
$\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$ and
$\omega\in\mathbb{S}^{2}$, and denote the transformation
$S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ by simply
writing

$(v,v_{*})\mapsto(v^{\prime},v^{\prime}_{*}).$

First, we pass to the center-of-mass reference frame. The transformations
$A_{1}:(v,v_{*})\mapsto(g,G)$ where $g=v-v_{*}$, $G=(v+v_{*})/2$, and
$B_{1}:(g^{\prime},G^{\prime})\mapsto(v^{\prime},v^{\prime}_{*})$ have both
Jacobian equal to 1. Since $G=G^{\prime}$, we are led to study the
transformation $C_{1}:g\mapsto g^{\prime}.$ Now, we pass to spherical
coordinates for $g$ and $g^{\prime}$. We perform the transformations
$A_{2}:g\mapsto(|g|,g/|g|)$ and
$B_{2}:(|g^{\prime}|,g^{\prime}/|g^{\prime}|)\mapsto g^{\prime}$, and finally
we study the transformation
$C_{2}:(|g|,g/|g|)\mapsto(|g^{\prime}|,g^{\prime}/|g^{\prime}|)$ taking into
account the determinant $|g^{\prime}|^{2}/|g|^{2}$ of the Jacobian coming from
$A_{2}$ and $B_{2}$. Since $T_{\omega}$ is a symmetry, we have

$\displaystyle\hskip
13.5ptJ\left[\left(|g|,\frac{g}{|g|}\right)\mapsto\left(2\sqrt{\frac{1}{4}|g|^{2}+\frac{1}{m}(\varepsilon(\zeta)+\varepsilon(\zeta_{*})-\varepsilon(\zeta^{\prime})-\varepsilon(\zeta^{\prime}_{*}))},T_{\omega}\left[\frac{g}{|g|}\right]\right)\right]$
$\displaystyle=J\left[|g|\mapsto
2\sqrt{\frac{1}{4}|g|^{2}+\frac{1}{m}(\varepsilon(\zeta)+\varepsilon(\zeta_{*})-\varepsilon(\zeta^{\prime})-\varepsilon(\zeta^{\prime}_{*}))}\right]$
$\displaystyle=\frac{\frac{1}{2}|g|}{\sqrt{\frac{1}{4}|g|^{2}+\frac{1}{m}(\varepsilon(\zeta)+\varepsilon(\zeta_{*})-\varepsilon(\zeta^{\prime})-\varepsilon(\zeta^{\prime}_{*}))}}=\frac{|g|}{|g^{\prime}|}.$

Multiplying by $|g^{\prime}|^{2}/|g|^{2}$, we finally obtain

$J\left[S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]\right](v,v_{*})=\frac{|g^{\prime}|}{|g|}=\frac{|v^{\prime}-v^{\prime}_{*}|}{|v-v_{*}|},$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.
$\square$

##  3 Boltzmann model

We consider a distribution function $f$ that represents the density of
molecules of the studied gas. This density depends on 4 parameters:

  * •

2 macroscopic parameters: the time $t\in\mathbb{R}_{+}$ and the position in
space $x$, typically $x\in\mathbb{R}^{3}$ or $\mathbb{T}^{3}$

  * •

2 microscopic parameters: the velocity $v\in\mathbb{R}^{3}$ and the internal
state $\zeta\in\mathcal{E}$

As a distribution function, $f$ is non-negative, so that we study
$f(t,x,v,\zeta)\in\mathbb{R}_{+}$. Moreover, we assume that for all $(t,x)$,
$f(t,x,\cdot,\cdot)\in
L^{1}(\mathbb{R}^{3}\times\mathcal{E},\,\text{d}v\,\text{d}\mu(\zeta))$. The
evolution of this distribution function is governed by the Boltzmann equation

$\partial_{t}f(t,x,v,\zeta)+v\cdot\nabla_{x}f(t,x,v,\zeta)=\mathcal{B}(f,f)(t,x,v,\zeta),$
(3.1)

where $\mathcal{B}$ is the Boltzmann collision operator we define in a
following subsection.

If $\phi$ defined on $\mathbb{R}^{3}\times\mathcal{E}$ is a microscopic
extensive quantity (test function, often called molecular property), then the
associated macroscopic quantity is

$\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\phi(v,\zeta)\,f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta).$

Typical choice for the molecular properties are $\phi=m$, $\phi=mv$ and
$\displaystyle\phi=\frac{m}{2}|v|^{2}+\varepsilon(\zeta)$, giving the mass
density $\rho$, velocity $u$ and total energy density $e$, respectively,
namely

$\displaystyle\rho(t,x):=\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle
u(t,x):=\frac{1}{\rho(t,x)}\,\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,v\,f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle
e(t,x):=\frac{1}{\rho(t,x)}\,\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\left(\frac{m}{2}\,|v|^{2}+\varepsilon(\zeta)\right)\,f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta).$

We can also define the temperature by

$T(t,x):=\Theta^{-1}\left(\frac{1}{\rho(t,x)}\,\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\left(\frac{m}{2}\,|v-u|^{2}+\bar{\varepsilon}(\zeta)\right)\,f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\right),$
(3.2)

where we recall that by definition (equation (2.2))
$\bar{\varepsilon}=\varepsilon-\varepsilon^{0}$, while
$\Theta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a suitable function
related to the integral of the specific heat at constant volume (the precise
definition will be given later, in equation (4.8) in Section 4).

###  3.1 Boltzmann collision operator

The Boltzmann operator, representing the effect of collisions on the change of
velocities and internal states of each molecule, is composed by a loss and a
gain term

$\mathcal{B}=\mathcal{B}_{+}-\mathcal{B}_{-}.$

We focus on the microscopic aspect and put temporarily aside the variables $t$
and $x$. When two molecules collide, not all possible post-collision states
are equiprobable. This idea is translated in the concept of the collision (or
scattering) kernel $b$, which encapsulates the information on the interaction
potential (for example a hard sphere or Lennard-Jones one). As well known, the
collision kernel is related to the differential cross section $\sigma$ through
the formula $b=|v-v_{*}|\sigma$.

In this paper, for a purpose of generality, following the line of
groppi2004two  we make as few assumptions as possible on the collision kernel
$b$.

###### Definition 3.1

The collision kernel $b$ is a measurable function

$b:\begin{cases}\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathcal{E}\times\mathcal{E}\times\mathcal{E}\times\mathcal{E}\times\mathbb{S}^{2}\to\mathbb{R}_{+}\\\
(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\mapsto
b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega).\end{cases}$
(3.3)

###### Assumption 3.2

_Symmetry. For a.e. $v,v_{*}\in\mathbb{R}^{3}$, for $\mu$-a.e.
$\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$ and a.e.
$\omega\in\mathbb{S}^{2}$,_

$b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)=b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime}_{*},\zeta^{\prime},\omega)=b(v_{*},v,\zeta_{*},\zeta,\zeta^{\prime}_{*},\zeta^{\prime},\omega).$
(3.4)

###### Assumption 3.3

_Micro-reversibility. For a.e. $v,v_{*}\in\mathbb{R}^{3}$, for $\mu$-a.e.
$\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$ and a.e.
$\omega\in\mathbb{S}^{2}$,_

$|v-v_{*}|\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)=|v^{\prime}-v^{\prime}_{*}|\,b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega).$
(3.5)

_where_
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.

###### Definition and assumption 3.4

_We denote by
$\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$
the subset of $\mathbb{S}^{2}$ such that for all $\omega\in\mathbb{S}^{2}$,_

$\omega\in\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]\iff
b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)>0.$

_We assume that for a.e. $v,v_{*}\in\mathbb{R}^{3}$ and $\mu$-a.e.
$\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$,_

$\displaystyle(v,v_{*})\in\,$ $\displaystyle
E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]\iff\quad\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]\emph{
has positive measure},$
$\displaystyle\mathbb{S}^{2}\setminus\mathbb{S}^{2}[v,v_{*},\zeta,\zeta,\zeta,\zeta]\emph{
has zero measure}.$ (3.6)

This assumption 3.4 has the following meaning: if a collision is possible,
then there exists a non-negligible set of angles $\omega$ for which the kernel
is positive; if, on the other hand, a collision is not allowed, then the
corresponding kernel is zero. Finally, if the collision is elastic, then the
set of angles $\omega$ for which the kernel is positive is the whole sphere
$\mathbb{S}^{2}$ (minus a negligible set). We thus allow, for inelastic
collisions, the kernel to be zero for a non-negligible set of angles. This is
an important feature, already present for example in groppi2004two  for the
case of reacting spheres.

Note that the symmetry and micro-reversibility assumptions are not in
contradiction with the positivity assumption, due to the fact the the function
$\mathbf{1}_{\Delta\geq 0}$ also has symmetry and reversibility properties.
Moreover, setting
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$,
the symmetry and micro-reversibility assumptions imply

$\mathbb{S}^{2}[v_{*},v,\zeta_{*},\zeta,\zeta^{\prime}_{*},\zeta^{\prime}]=\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]=\mathbb{S}^{2}[v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}].$

We are now ready to define the Boltzmann collision operator. The loss term of
the collision operator expresses the fact that a collision involving a
molecule with state $(v,\zeta)$ causes a change of the state. Thus for all
states $(v,\zeta)\in\mathbb{R}^{3}\times\mathcal{E}$, and recalling that $b$
is null for not admissible collisions, the loss term reads as

$\mathcal{B}_{-}(f,f)(v,\zeta):=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}f(v,\zeta)f(v_{*},\zeta_{*})\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$ (3.7)

The gain term of the collision operator expresses the production of the state
$(v,\zeta)$ by collisions. Thus the gain term is obtained by considering _all
collisions_ for which $(v,\zeta)$ is a post-collision state. We have already
proved in Lemma 2.1 that if
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$
then
$(v,v_{*})=S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}](v^{\prime},v^{\prime}_{*})$,
therefore the gain term will be symmetric to the loss one as expected. In more
detail, denoting by $\chi$ the Dirac mass on $\mathbb{R}^{3}$, the gain term
is defined by

$\displaystyle\mathcal{B}_{+}(f,f)(v,\zeta):=\iiint_{\mathcal{E}^{3}}\iint_{E[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}]}\int_{\mathbb{S}^{2}}\,\mathcal{\chi}(\tilde{v}=v)\,f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega)$
$\displaystyle\text{d}\omega\,\text{d}v^{\prime}\,\text{d}v^{\prime}_{*}\,\text{d}\mu^{\otimes
3}(\zeta^{\prime},\zeta^{\prime}_{*},\zeta_{*}),$

where just for convenience we define
$(\tilde{v},\tilde{v}_{*})=S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}](v^{\prime},v^{\prime}_{*})$,
in order to be able to perform the change of variables
$(\tilde{v},\tilde{v}_{*})=S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}](v^{\prime},v^{\prime}_{*})$
(recalling that
$S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}]$ is a
$\mathcal{C}^{1}$-diffeomorphism on
$E[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}]$, except on a negligible
set) leading to

$\displaystyle\mathcal{B}_{+}(f,f)(v,\zeta)=\iiint_{\mathcal{E}^{3}}\iint_{E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]}\int_{\mathbb{S}^{2}}\,\mathcal{\chi}(\tilde{v}=v)\,f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,J\left[(S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}])^{-1}\right]\,b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega)$
$\displaystyle\text{d}\omega\,\text{d}\tilde{v}\,\text{d}\tilde{v}_{*}\,\text{d}\mu^{\otimes
3}(\zeta^{\prime},\zeta^{\prime}_{*},\zeta_{*})$
$\displaystyle=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,J\left[(S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}])^{-1}\right]\,b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega)\,\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta^{\prime},\zeta^{\prime}_{*},\zeta_{*}),$

where in last line we went back to the original notation, changing
$\tilde{v}_{*}$ to $v_{*}$. The Jacobian of
$(S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}])^{-1}=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$
is given by equation (2.11), and it provides

$\displaystyle\mathcal{B}_{+}(f,f)(v,\zeta)=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,\frac{|v^{\prime}-v^{\prime}_{*}|}{|v-v_{*}|}\,b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega)\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta^{\prime},\zeta^{\prime}_{*},\zeta_{*}).$

Using now the micro-reversibility condition on $b$, Assumption 3.3, we finally
get

$\mathcal{B}_{+}(f,f)(v,\zeta)=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}),$ (3.8)

where we recall that
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.
Since $f$ and $b$ are non-negative measurable functions, the gain and loss
terms are always defined. In order to define the full collision term, we
impose the following assumption on $f$.

###### Assumption 3.5

_For a.e. $v\in\mathbb{R}^{3}$ and for $\mu$-a.e. $\zeta\in\mathcal{E}$,_

$\mathcal{B}_{-}(f,f)(v,\zeta)+\mathcal{B}_{+}(f,f)(v,\zeta)<\infty.$

Writing $b(\cdot)\equiv
b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)$ to
lighten the notations, the full Boltzmann collision operator writes, for all
$v\in\mathbb{R}^{3}$ and $\zeta\in\mathcal{E}$,

$\mathcal{B}(f,f)(v,\zeta)=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*})\Big{)}\;b(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}),$ (3.9)

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.

###### Remark 3.1

Alternatively, one can define the Boltzmann collision operator using
transition probabilities instead of $b$, see Giovangigli .

###  3.2 Note about degeneracy

The concept of degeneracy appears when the internal description is done
through the direct consideration of energy levels. For instance, in a quantum
description, different proper states can sometimes have the same energy. The
number of proper states sharing the same energy level is called the
_degeneracy_ of this energy level. When degeneracy is considered in kinetic
theory, it usually appears in the micro-reversibility condition of the
collision kernel (see for example Section 4.2.4 in Giovangigli
giovangigli2012multicomponent , equations (4.2.7) and (4.2.8)), and ultimately
in the formulation of the Boltzmann integral. We show in this subsection that,
without loss of generality, it is possible to assume no degeneracy in the
micro-reversibility condition and take this into account through the measure
$\mu$. Note also that if the internal description is done through the proper
states instead of directly through the energy levels, there is no degeneracy
to consider.

The degeneracy is a measurable positive function on the space of internal
states

$a:\mathcal{E}\to\mathbb{R}_{+}^{*}.$

The micro-reversibility condition taking degeneracy into account writes (see
Section 4.2.4 in Giovangigli giovangigli2012multicomponent , equation (4.2.8))

$|v-v_{*}|\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,a(\zeta)a(\zeta_{*})=|v^{\prime}-v^{\prime}_{*}|\,b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega)\,a(\zeta^{\prime})a(\zeta^{\prime}_{*}).$
(3.10)

Classical symmetry condition for cross-sections should be valid only if the
molecular states are non-degenerate FK72 . However, Waldmann Wa58  has shown
that semi-classical Boltzmann modelling is indeed admissible for all
molecules, even with degeneracy properties, provided the quantum mechanical
cross-sections are replaced by suitably degeneracy averaged cross-sections
Wa58 ; EG94 . Symmetry property (3.10) may be obtained owing to the invariance
of the Hamiltonian under the combined operation of space inversion and time
reversal Wa58 ; MS64 . We remark that in the classical model with a continuous
internal energy variable $I$ borgnakke1975statistical ; desvillettes1997modele
; desvillettes2005kinetic , the weight function $\varphi(I)$ in the Boltzmann
kernel plays the role of the degeneracy function $a(\cdot)$.

With the same reasoning as in the previous subsection 3.1, the collision term
writes in this case

$\mathcal{B}(f,f)(v,\zeta)=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\left(\frac{a(\zeta)a(\zeta_{*})}{a(\zeta^{\prime})a(\zeta^{\prime}_{*})}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*})\right)\;b(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

Let us now define

$\tilde{b}(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)=\frac{b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)}{a(\zeta^{\prime})a(\zeta^{\prime}_{*})},\quad\quad\text{d}\tilde{\mu}(\zeta)=a(\zeta)\,\text{d}\mu(\zeta),\quad\quad\tilde{f}(v,\zeta)=\frac{f(v,\zeta)}{a(\zeta)}.$

Then $\tilde{b}$ verifies the assumptions of our framework. Now let us set

$\tilde{\mathcal{B}}(\tilde{f},\tilde{f})(v,\zeta)=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\Big{(}\tilde{f}(v^{\prime},\zeta^{\prime})\tilde{f}(v^{\prime}_{*},\zeta^{\prime}_{*})-\tilde{f}(v,\zeta)\tilde{f}(v_{*},\zeta_{*})\Big{)}\;\tilde{b}(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\text{d}\tilde{\mu}^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

Then

$\displaystyle\tilde{\mathcal{B}}(\tilde{f},\tilde{f})(v,\zeta)$
$\displaystyle=\frac{1}{a(\zeta)}\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\left(\frac{f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})}{a(\zeta^{\prime})a(\zeta^{\prime}_{*})}-\frac{f(v,\zeta)f(v_{*},\zeta_{*})}{a(\zeta)a(\zeta_{*})}\right)\;b(\cdot)\,a(\zeta)a(\zeta_{*})\;\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})$
$\displaystyle=\frac{1}{a(\zeta)}\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\left(\frac{a(\zeta)a(\zeta_{*})}{a(\zeta^{\prime})a(\zeta^{\prime}_{*})}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*})\right)\;b(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})$
$\displaystyle=\frac{1}{a(\zeta)}\mathcal{B}(f,f)(v,\zeta).$

It follows that

$\tilde{\mathcal{B}}(\tilde{f},\tilde{f})(v,\zeta)\text{d}\tilde{\mu}(\zeta)=\frac{1}{a(\zeta)}\mathcal{B}(f,f)(v,\zeta)a(\zeta)\text{d}\mu(\zeta)=\mathcal{B}(f,f)(v,\zeta)\text{d}\mu(\zeta)$

and

$\tilde{f}(v,\zeta)\text{d}\tilde{\mu}(\zeta)=\frac{1}{a(\zeta)}f(v,\zeta)a(\zeta)\text{d}\mu(\zeta)=f(v,\zeta)\text{d}\mu(\zeta).$

Thus it is equivalent to study the model $(\mathcal{E},\mathcal{A},\mu)$ with
the kernel $b$ and degeneracy $a$, and the model
$(\mathcal{E},\mathcal{A},\tilde{\mu})$ with the kernel $\tilde{b}$ and no
degeneracy. In our framework, degeneracy can be simply included in the measure
on the space of internal states, and does not appear in the micro-
reversibility condition.

###  3.3 Weak formulation

In this section, we study the weak formulation of the collision integral that
will later enable us to deduce the conservation laws and collision invariants.

###### Proposition 3.1

Let $\psi:\mathbb{R}^{3}\times\mathcal{E}\to\mathbb{R}$ be a measurable
function such that

$\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{-}(f,f)(v,\zeta)\left|\psi(v,\zeta)\right|\,\emph{d}v\,\emph{d}\mu(\zeta)+\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{+}(f,f)(v,\zeta)\left|\psi(v,\zeta)\right|\,\emph{d}v\,\emph{d}\mu(\zeta)<\infty.$

Then

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\psi(v,\zeta)\,\emph{d}v\,\emph{d}\mu(\zeta)=-\frac{1}{4}\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\Big{(}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*})\Big{)}$
$\displaystyle\times\Big{[}\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})-\psi(v,\zeta)-\psi(v_{*},\zeta_{*})\Big{]}b(\cdot)\,\emph{d}\omega\,\emph{d}v\,\emph{d}v_{*}\,\emph{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.

###### Proof

First we have, thanks to the symmetry conditions on $b$ and Fubini’s Theorem,

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{-}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v,\zeta)f(v_{*},\zeta_{*})\,\psi(v,\zeta)b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})$
$\displaystyle=\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v_{*},\zeta_{*})f(v,\zeta)\,\psi(v_{*},\zeta_{*})b(v_{*},v,\zeta_{*},\zeta,\zeta^{\prime}_{*},\zeta^{\prime},\omega)\,\text{d}\omega\,\text{d}v_{*}\,\text{d}v\,\text{d}\mu^{\otimes
4}(\zeta_{*},\zeta,\zeta^{\prime}_{*},\zeta^{\prime})$
$\displaystyle=\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v,\zeta)f(v_{*},\zeta_{*})\,\psi(v_{*},\zeta_{*})b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}),$

thus

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{-}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\frac{1}{2}\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v,\zeta)f(v_{*},\zeta_{*})\,[\psi(v,\zeta)+\psi(v_{*},\zeta_{*})]\,b(\cdot)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

Now by renaming all the variables, we get

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{-}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\frac{1}{2}\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,[\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})]b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega)\,\text{d}\omega\,\text{d}v^{\prime}\text{d}v^{\prime}_{*}\,\text{d}\mu^{\otimes
4}(\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}).$

We now perform the change of variables
$(v,v_{*})=S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}](v^{\prime},v^{\prime}_{*})$,
use the formula for the Jacobian (2.11) and the micro-reversibility condition
and obtain

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{-}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\frac{1}{2}\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,[\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})]\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.
We deduce that

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{-}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\hskip
260.0pt$
$\displaystyle=\frac{1}{4}\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,[\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})]+f(v,\zeta)f(v_{*},\zeta_{*})\,[\psi(v,\zeta)+\psi(v_{*},\zeta_{*})]\Big{)}$
$\displaystyle
b(\cdot)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.

On the other hand, since

$\mathcal{B}_{+}(f,f)(v,\zeta)=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}),$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$,
we have

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{+}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,\psi(v,\zeta)\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

Applying the change of variables
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$
using the formula for the Jacobian, equation (2.11), together with the micro-
reversibility condition on $b$, Assumption 3.3, we obtain

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{+}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,\psi(v,\zeta)\,b(v^{\prime},v^{\prime}_{*},\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*},\omega)\,\text{d}\omega\,\text{d}v^{\prime}\,\text{d}v^{\prime}_{*}\,\text{d}\mu^{\otimes
4}(\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*})$

where $v$ in the last integral is defined by
$(v,v_{*})=S_{\omega}[\zeta^{\prime},\zeta^{\prime}_{*},\zeta,\zeta_{*}](v^{\prime},v^{\prime}_{*})$.
By simply renaming the variables (exchange of prime and non-prime), we get

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{+}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,f(v,\zeta)f(v_{*},\zeta_{*})\,\psi(v^{\prime},\zeta^{\prime})\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})$

where $v^{\prime}$ is defined by
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.
By using the same arguments as previously, we get

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{+}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\hskip
260.0pt$
$\displaystyle=\frac{1}{4}\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})\,[\psi(v,\zeta)+\psi(v_{*},\zeta_{*})]+f(v,\zeta)f(v_{*},\zeta_{*})\,[\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})]\Big{)}$
$\displaystyle
b(\cdot)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.
Putting the previous results together yields

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\psi(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=-\frac{1}{4}\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\Big{(}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*})\Big{)}$
$\displaystyle\times\Big{[}\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})-\psi(v,\zeta)-\psi(v_{*},\zeta_{*})\Big{]}b(\cdot)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

$\square$

###  3.4 Collision invariants

###### Definition 3.6

$\psi$ is a _collision invariant_ if and only if $\text{for $\mu$-a.e.
}\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E},\text{ a.e.
}(v,v_{*})\in E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ and a.e.
$\omega\in\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$,

$\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})=\psi(v,\zeta)+\psi(v_{*},\zeta_{*}),$
(3.11)

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.

First, we immediately deduce from the conservation laws (2.5) that if there
exists $\alpha\in\mathbb{R},\;\beta\in\mathbb{R}^{3}$ and
$\gamma\in\mathbb{R}$ such that for a.e. $v\in\mathbb{R}^{3}$ and $\mu$-a.e.
$\zeta\in\mathcal{E}$,

$\psi(v,\zeta)=\alpha+\beta\cdot
v+\gamma\left(\frac{m}{2}\,|v|^{2}+\varepsilon(\zeta)\right),$

then $\psi$ is a collision invariant. In this subsection, we prove that all
collision invariants verifying a certain integrability assumption have this
form.

###### Lemma 3.1

For all $\zeta\in\mathcal{E}$ and $\omega\in\mathbb{S}^{2}$,

$S_{\omega}[\zeta,\zeta,\zeta,\zeta]\equiv\bar{S}_{\omega}:\begin{cases}\mathbb{R}^{3}\times\mathbb{R}^{3}\hskip
20.0pt\longrightarrow\hskip 50.0pt\mathbb{R}^{3}\times\mathbb{R}^{3}\\\
\displaystyle(v,v_{*})\mapsto\left(\frac{v+v_{*}}{2}+\frac{|v-v_{*}|}{2}T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right],\frac{v+v_{*}}{2}-\frac{|v-v_{*}|}{2}T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\right)\end{cases}$

###### Proof

For any $v,v_{*}\in\mathbb{R}^{3}$,

$\Delta(v,v_{*},\zeta,\zeta,\zeta,\zeta)=\frac{1}{4}|v-v_{*}|^{2}+\frac{1}{m}(\varepsilon(\zeta)+\varepsilon(\zeta)-\varepsilon(\zeta)-\varepsilon(\zeta))=\frac{1}{4}|v-v_{*}|^{2}\geq
0$

so that

$E[\zeta,\zeta,\zeta,\zeta]=\Big{\\{}(v,v_{*})\in\mathbb{R}^{3}\times\mathbb{R}^{3}\text{
s.t. }\Delta(v,v_{*},\zeta,\zeta,\zeta,\zeta)\geq
0\Big{\\}}=\mathbb{R}^{3}\times\mathbb{R}^{3}.$

Now let $(v,v_{*})\in\mathbb{R}^{3}\times\mathbb{R}^{3}$. We set
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta,\zeta,\zeta](v,v_{*})$.
Then

$\displaystyle v^{\prime}$
$\displaystyle=\frac{v+v_{*}}{2}+\sqrt{\Delta(\cdot)}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]$
$\displaystyle=\frac{v+v_{*}}{2}+\frac{|v-v_{*}|}{2}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right],$

and analogously

$v^{\prime}_{*}=\frac{v+v_{*}}{2}-\frac{|v-v_{*}|}{2}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right].$

Thus we indeed have

$S_{\omega}[\zeta,\zeta,\zeta,\zeta]:\begin{cases}\displaystyle\mathbb{R}^{3}\times\mathbb{R}^{3}\hskip
20.0pt\longrightarrow\hskip 50.0pt\mathbb{R}^{3}\times\mathbb{R}^{3}\\\
\displaystyle(v,v_{*})\mapsto\left(\frac{v+v_{*}}{2}+\frac{|v-v_{*}|}{2}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right],\frac{v+v_{*}}{2}-\frac{|v-v_{*}|}{2}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\right).\end{cases}$

$\square$

The following Theorem 3.1 is the well-known characterization of collision
invariants in the elastic case.

###### Theorem 3.1

Let $r_{0}>0$ and $\phi\in
L^{1}\left(\mathbb{R}^{3},\,e^{-|v|^{2}/r_{0}}\,\emph{d}v\right)$ such that
for a.e. $(v,v_{*})\in\mathbb{R}^{3}\times\mathbb{R}^{3}\text{ and
}\omega\in\mathbb{S}^{2}$,

$\phi(v^{\prime}(v,v_{*},\omega))+\phi(v^{\prime}_{*}(v,v_{*},\omega))=\phi(v)+\phi(v_{*}).$

with
$(v^{\prime}(v,v_{*},\omega),v^{\prime}_{*}(v,v_{*},\omega))=\bar{S}_{\omega}(v,v_{*})$.
Then there exists
$(\alpha,\beta,\gamma)\in\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{R}$ such
that for a.e. $v\in\mathbb{R}^{3}$,

$\phi(v)=\alpha+\beta\cdot v+\frac{\gamma}{2}|v|^{2}.$

###### Proof

For the proof, we refer the reader to Bouchut and Golse bouchut2000kinetic ,
Chapter 2. $\square$

The following Theorem 3.2 characterises collision invariants in the general
setting.

###### Theorem 3.2

Let $\psi$ be a collision invariant, such that for $\mu$-a.e.
$\zeta\in\mathcal{E}$ there exists $r_{\zeta}>0$ such that
$\psi(\cdot,\zeta)\in
L^{1}\left(\mathbb{R}^{3},\,e^{-|v|^{2}/r_{\zeta}}\,\emph{d}v\right)$. Then
there exists
$(\alpha,\beta,\gamma)\in\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{R}$ such
that for $\mu$-a.e. $\zeta\in\mathcal{E}$ and a.e. $v\in\mathbb{R}^{3}$,

$\psi(v,\zeta)=\alpha+\beta\cdot
v+\gamma\left(\frac{m}{2}\,|v|^{2}+\varepsilon(\zeta)\right).$

###### Proof

Let us denote by $\tilde{\mathcal{E}}$ the set such that
$\mu(\mathcal{E}\setminus\tilde{\mathcal{E}})=0$, and for all
$\zeta\in\tilde{\mathcal{E}}$, there exists $r_{\zeta}>0$ such that
$\psi(\cdot,\zeta)\in
L^{1}\left(\mathbb{R}^{3},\,e^{-|v|^{2}/r_{\zeta}}\text{d}v\right)$ and for
all $\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\tilde{\mathcal{E}}$, a.e.
$(v,v_{*})\in E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ and a.e.
$\omega\in\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$,

$\psi(v^{\prime},\zeta^{\prime})+\psi(v^{\prime}_{*},\zeta^{\prime}_{*})=\psi(v,\zeta)+\psi(v_{*},\zeta_{*}),$

where $(v^{\prime},v^{\prime}_{*})$ are defined by
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.

First, we consider the equality for
$\zeta=\zeta_{*}=\zeta^{\prime}=\zeta^{\prime}_{*}\in\tilde{\mathcal{E}}$.
From Lemma 3.1, Assumption 3.4 (in our framework, all angles are allowed in
elastic collisions) and Theorem 3.1, there exists
$(\alpha_{\zeta},\beta_{\zeta},\gamma_{\zeta})\in\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{R}$
such that for a.e. $v\in\mathbb{R}^{3}$,

$\psi(v,\zeta)=\alpha_{\zeta}+\beta_{\zeta}\cdot
v+\gamma_{\zeta}\left(\frac{1}{2}|v|^{2}\right).$ (3.12)

Let us now consider again the equation (3.11), using formula (3.12),
$\zeta=\zeta_{*}\in\tilde{\mathcal{E}}$ and
$\zeta^{\prime}=\zeta^{\prime}_{*}\in\tilde{\mathcal{E}}$. It writes, for a.e.
$(v,v_{*})\in E[\zeta,\zeta,\zeta^{\prime},\zeta^{\prime}]$ and a.e.
$\omega\in\mathbb{S}^{2}[v,v_{*},\zeta,\zeta,\zeta^{\prime},\zeta^{\prime}]$,

$2\alpha_{\zeta}+\beta_{\zeta}\cdot(v+v_{*})+\gamma_{\zeta}\left(\frac{1}{2}|v|^{2}+\frac{1}{2}|v_{*}|^{2}\right)=2\alpha_{\zeta^{\prime}}+\beta_{\zeta^{\prime}}\cdot(v^{\prime}+v^{\prime}_{*})+\gamma_{\zeta^{\prime}}\left(\frac{1}{2}|v^{\prime}|^{2}+\frac{1}{2}|v^{\prime}_{*}|^{2}\right)$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta,\zeta^{\prime},\zeta^{\prime}](v,v_{*})$.
By assumption 3.4, at fixed $v,v_{*},\zeta,\zeta^{\prime}$, the set of angles
$\omega$ for which the above equation holds,
$\mathbb{S}^{2}[v,v_{*},\zeta,\zeta,\zeta^{\prime},\zeta^{\prime}]$, is not
negligible. Using the conservation equations (2.5), we thus get for a.e.
$v,v_{*}$ such that
$\displaystyle\frac{1}{4}|v-v_{*}|^{2}\geq\frac{2}{m}\,(\varepsilon(\zeta^{\prime})-\varepsilon(\zeta))$,

$2\left(\alpha_{\zeta}-\alpha_{\zeta^{\prime}}-\frac{\gamma_{\zeta^{\prime}}}{m}\left(\varepsilon(\zeta^{\prime})-\varepsilon(\zeta)\right)\right)+(\beta_{\zeta}-\beta_{\zeta^{\prime}})\cdot(v+v_{*})+(\gamma_{\zeta}-\gamma_{\zeta^{\prime}})\left(\frac{1}{2}|v|^{2}+\frac{1}{2}|v_{*}|^{2}\right)=0.$

A polynomial that vanishes on a set of infinite cardinal must have all its
coefficients equal to zero, hence

$\displaystyle\gamma_{\zeta}$ $\displaystyle=\gamma_{\zeta^{\prime}}$
$\displaystyle\beta_{\zeta}$ $\displaystyle=\beta_{\zeta^{\prime}}$
$\displaystyle\alpha_{\zeta}-\gamma_{\zeta^{\prime}}\frac{\varepsilon(\zeta)}{m}$
$\displaystyle=\alpha_{\zeta^{\prime}}-\gamma_{\zeta^{\prime}}\frac{\varepsilon(\zeta^{\prime})}{m}.$

We conclude that there exists
$(\alpha,\beta,\gamma)\in\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{R}$ such
that for a.e. $v\in\mathbb{R}^{3}$ and for all $\zeta\in\tilde{\mathcal{E}}$
(thus $\mu$-a.e. $\zeta\in\mathcal{E}$),

$\psi(v,\zeta)=\alpha+\beta\cdot
v+\gamma\left(\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\right).$

$\square$

##  4 H Theorem and collision equilibria

###### Theorem 4.1

H Theorem. Let $f:\mathbb{R}^{3}\times\mathcal{E}\to\mathbb{R}_{+}$ be a
measurable function such that for a.e. $v\in\mathbb{R}^{3}$ and $\mu$-a.e.
$\zeta\in\mathcal{E}$, $f(v,\zeta)>0$. We assume that

$\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{-}(f,f)(v,\zeta)\left|\emph{ln}(f)(v,\zeta)\right|\,\emph{d}v\,\emph{d}\mu(\zeta)+\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}_{+}(f,f)(v,\zeta)\left|\emph{ln}(f)(v,\zeta)\right|\,\emph{d}v\,\emph{d}\mu(\zeta)<\infty.$

Then

$\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\,\emph{ln}(f)(v,\zeta)\,\emph{d}v\,\emph{d}\mu(\zeta)\leq
0,$ (4.1)

and

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\,\emph{ln}$
$\displaystyle(f)(v,\zeta)\,\emph{d}v\,\emph{d}\mu(\zeta)=0$
$\displaystyle\Updownarrow$ $\displaystyle\mathcal{B}(f,f)(v,\zeta)=0\text{
for a.e. }$ $\displaystyle v\in\mathbb{R}^{3}\text{ and $\mu$-a.e.
}\zeta\in\mathcal{E}$ $\displaystyle\Updownarrow$
$\displaystyle\emph{ln}(f)\text{ is a }$ collision invariant.

###### Proof

This proof is highly inspired by a proof which can be found in
bouchut2000kinetic , Chapter 2. First we define, for $\mu$-a.e.
$\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E}$, a.e.
$(v,v_{*})\in E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ and a.e.
$\omega\in\mathbb{S}^{2}$,

$\mathcal{P}[f](v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega):=\frac{1}{4}(f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*}))\text{ln}\left(\frac{f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})}{f(v,\zeta)f(v_{*},\zeta_{*})}\right),$

where
$(v^{\prime},v^{\prime}_{*})=S_{\omega}[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}](v,v_{*})$.
Now by remarking that

$\forall x>0,\quad(x-1)\text{ln}(x)\geq 0,$

we have, since $f>0$ and considering $\displaystyle
x=\frac{f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})}{f(v,\zeta)f(v_{*},\zeta_{*})}>0$,

$\mathcal{P}[f](v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)=\frac{1}{4}\,f(v,\zeta)f(v_{*},\zeta_{*})\,(x-1)\text{ln}(x)\geq
0.$

Using Proposition 3.1 with $\psi=\text{ln}(f)$,

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\,\text{ln}(f)(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=-\iiiint_{\mathcal{E}^{4}}\iint_{(\mathbb{R}^{3})^{2}}\int_{\mathbb{S}^{2}}\mathcal{P}[f](v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,b(\cdot)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}).$

Now from $b\geq 0$ and $\mathcal{P}[f]\geq 0$,

$\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\,\text{ln}(f)(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\leq
0.$

Let us now focus on the case of equality. Since the integrand
$\mathcal{P}[f]\,b$ is non-negative, it must be equal to zero.

$\displaystyle\hskip
30.0pt\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\,\text{ln}(f)(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=0$
$\displaystyle\implies\iiiint_{\mathcal{E}^{4}}\iint_{E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]}\int_{\mathbb{S}^{2}}\mathcal{P}[f](v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,b(\cdot)\,\text{d}\omega\,\text{d}v\,\text{d}v_{*}\,\text{d}\mu^{\otimes
4}(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})=0$
$\displaystyle\implies\mathcal{P}[f](v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)\,b(v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)=0\quad\text{for
a.e. }v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega$
$\displaystyle\implies\mathcal{P}[f](v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)=0\quad\text{for
$\mu$-a.e.
}\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E},\text{ a.e.
}(v,v_{*})\in E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$
$\displaystyle\hskip 230.0pt\text{ and a.e.
}\omega\in\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}],$

where we recall that the set
$\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$ is
defined by Definition 3.4. Now since $f>0$ and

$\forall x>0,\quad(x-1)\text{ln}(x)=0\implies x=1,$

we finally obtain, writing
$\mathbb{S}^{2}[\cdot]\equiv\mathbb{S}^{2}[v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]$
to lighten the notations,

$\displaystyle\hskip
30.0pt\mathcal{P}[f](v,v_{*},\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*},\omega)=0\quad\text{for
$\mu$-a.e.
}\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E},\text{ a.e.
}(v,v_{*})\in E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}]\text{ and
a.e. }\omega\in\mathbb{S}^{2}[\cdot]$ $\displaystyle\implies
f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})=f(v,\zeta)f(v_{*},\zeta_{*})\;\,\text{for
$\mu$-a.e.
}\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E},\text{ a.e.
}(v,v_{*})\in E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}],\text{ a.e.
}\omega\in\mathbb{S}^{2}[\cdot]$ $\displaystyle\implies\text{ln}(f)\text{ is a
collision invariant}.$

Note also that

$\displaystyle\hskip 30.0pt\text{ln}(f)\text{ is a collision invariant}$
$\displaystyle\implies
f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})=f(v,\zeta)f(v_{*},\zeta_{*})\;\,\text{for
$\mu$-a.e.
}\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}\in\mathcal{E},\text{ a.e.
}(v,v_{*})\in E[\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*}],\text{ a.e.
}\omega\in\mathbb{S}^{2}[\cdot]$
$\displaystyle\implies\mathcal{B}(f,f)(v,\zeta)=0\quad\text{for a.e.
}v\in\mathbb{R}^{3}\text{ and $\mu$-a.e. }\zeta\in\mathcal{E}$
$\displaystyle\implies\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\mathcal{B}(f,f)(v,\zeta)\,\text{ln}(f)(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=0,$

which ends the proof. $\square$

Let us assume that there exists $f\in
L^{1}(\mathbb{R}^{3}\times\mathcal{E},\,\text{d}v\,\text{d}\mu(\zeta))$ such
that $f$ verifies assumptions of Theorem 4.1 and $\text{ln}(f)$ verifies the
hypotheses of Theorem 3.2. From Theorem 3.2,
$\exists\,(\alpha,\beta,\gamma)\in\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{R}$
such that for a.e. $v\in\mathbb{R}^{3}$ and $\mu$-a.e. $\zeta\in\mathcal{E}$,

$\text{ln}(f)(v,\zeta)=\alpha+\beta\cdot
v+\gamma\left(\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\right).$

Since $f\in
L^{1}(\mathbb{R}^{3}\times\mathcal{E},\,\text{d}v\,\text{d}\mu(\zeta))$, we
have $\gamma<0$. Let us now set

$T=-\frac{1}{k_{B}\gamma},\qquad u=-\frac{\beta}{m\gamma},\qquad
A=\text{exp}\left(\alpha-\frac{|\beta|^{2}}{2m\gamma}\right),$

where $k_{B}$ is the Boltzmann constant. Then

$f(v,\zeta)=A\,\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}-\frac{\varepsilon(\zeta)}{k_{B}T}\right).$

and it can be easily checked that $u$ is the average velocity of $f$, while
$T$ will provide its temperature (see below). Note that indeed $f\in
L^{1}(\mathbb{R}^{3}\times\mathcal{E},\,\text{d}v\,\text{d}\mu(\zeta))$ thanks
to Assumption 2.5. We define the density $\rho$

$\rho:=\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,f(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta).$

Then, almost everywhere,

$f=\mathcal{M}[\rho,u,T],$

where, for all $(v,\zeta)\in\mathbb{R}^{3}\times\mathcal{E}$,

$\mathcal{M}[\rho,u,T](v,\zeta):=\frac{\rho}{m}\;\frac{\displaystyle\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}-\frac{\varepsilon(\zeta)}{k_{B}T}\right)}{\displaystyle\left(\frac{2\pi
k_{B}T}{m}\right)^{3/2}\,\int_{\mathcal{E}}\text{exp}\left(-\frac{\varepsilon(\zeta_{*})}{k_{B}T}\right)\text{d}\mu(\zeta_{*})},$

which we can rewrite

$\mathcal{M}[\rho,u,T](v,\zeta)=\rho\;m^{1/2}\,(2\pi
k_{B}T)^{-3/2}\,Z\left(\frac{1}{k_{B}T}\right)^{-1}\;\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right),$
(4.2)

where $\bar{\varepsilon}$ and $Z$ are defined by equations (2.2) and (2.3).

First, note that by definition of $\rho$,

$\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=\rho>0.$

Also,

$\frac{1}{\rho}\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,v\,\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=m^{3/2}\,(2\pi
k_{B}T)^{-3/2}\;\int_{\mathbb{R}^{3}}v\,\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}\right)\,\text{d}v=u.$

The total energy density at collision equilibrium writes

$\displaystyle e^{eq}[u,T]$
$\displaystyle:=\frac{m}{\rho}\,\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\left(\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\right)\,\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=m^{3/2}(2\pi
k_{B}T)^{-3/2}\;\int_{\mathbb{R}^{3}}\frac{m}{2}|v|^{2}\,\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}\right)\,\text{d}v+\frac{\displaystyle\int_{\mathcal{E}}\varepsilon(\zeta)\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\displaystyle\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)},$

and is finite thanks to Assumption 2.5 and Proposition 2.1. The term
associated with the velocity is well-known, and a classical computation gives

$m^{3/2}(2\pi
k_{B}T)^{-3/2}\;\int_{\mathbb{R}^{3}}\frac{m}{2}|v|^{2}\,\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}\right)\,\text{d}v=\frac{m}{2}|u|^{2}+\frac{3}{2}k_{B}T.$

Remark that

$\frac{\displaystyle\int_{\mathcal{E}}\varepsilon(\zeta)\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\displaystyle\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}=\varepsilon^{0}+\frac{\displaystyle\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\displaystyle\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)},$

where we recall that $\varepsilon^{0}=\text{inf ess}_{\mu}\\{\varepsilon\\}$
and $\bar{\varepsilon}=\varepsilon-\varepsilon^{0}$, see equation (2.2).

###### Definition 4.1

We define the (thermodynamic) number of internal degrees of freedom $\delta$
by

$\delta(T):=\frac{2}{k_{B}T}\left(\frac{\displaystyle\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\displaystyle\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}\right).$
(4.3)

We notice that, since $\bar{\varepsilon}\geq 0$ $\mu$-a.e., $\delta\geq 0$.

The total energy density at collision equilibrium finally writes

$e^{eq}[u,T]=\varepsilon^{0}+\frac{m}{2}|u|^{2}+\frac{3+\delta(T)}{2}\,k_{B}T.$

$3+\delta(T)$ is the _total_ (thermodynamic) number of degrees of freedom at
temperature $T$, provided by the sum of the 3 translational degrees of freedom
and of the number of internal ones $\delta(T)$.

###### Definition 4.2

We define the specific heat at constant volume $c_{V}$, for all $T>0$,

$c_{V}(T):=\frac{3+D(T)}{2},$ (4.4)

where

$D(T):=\frac{\text{d}\big{(}T\delta(T)\big{)}}{\text{d}T}.$

###### Remark 4.1

We recall that, from Proposition 2.1, the partition function $Z$ defined by
equation (2.3) is $\mathcal{C}^{\infty}$. Remarking that

$T\delta(T)=-\frac{2}{k_{B}}\,\big{(}\text{ln}(Z)\big{)}^{\prime}\left(\frac{1}{k_{B}\,T}\right),$

then the function $T\mapsto T\delta(T)$ is $\mathcal{C}^{\infty}$ on
$\mathbb{R}_{+}^{*}$ and $D$ is thus well-defined.

In the following propositions we show various mathematical properties of the
function $\delta(T)$.

###### Proposition 4.1

If $\mu(\varepsilon=\varepsilon^{0})>0$, then

$\delta(T)\underset{T\to 0^{+}}{\longrightarrow}0\quad\text{and}\quad
D(T)\underset{T\to 0^{+}}{\longrightarrow}0.$

###### Proof

First, note that
$\mu(\varepsilon=\varepsilon^{0})>0\implies\mu(\bar{\varepsilon}=0)>0$. Since
$\mu$-a.e., $\bar{\varepsilon}\geq 0$, we then have for $k\in\mathbb{N}$

$\int_{\mathcal{E}}\,\left(\frac{\bar{\varepsilon}}{k_{B}T}\right)^{k}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)=\mathbf{1}_{k=0}\times\mu(\bar{\varepsilon}=0)+\int_{\bar{\varepsilon}>0}\,\left(\frac{\bar{\varepsilon}}{k_{B}T}\right)^{k}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta).$

Now remark that for all $y>0$, $e^{y/2}\geq 1+\frac{y}{2}+\frac{y^{2}}{8}$.
Thus

$ye^{-y}\leq 2e^{-y/2}\quad\text{and}\quad y^{2}e^{-y}\leq 8e^{-y/2}.$

Thus it comes

$\displaystyle\delta(T)$
$\displaystyle=2\,\frac{\displaystyle\int_{\mathcal{E}}\,\frac{\bar{\varepsilon}}{k_{B}T}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\displaystyle\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}\leq
2\,\frac{\displaystyle\int_{\bar{\varepsilon}>0}\,\frac{\bar{\varepsilon}}{k_{B}T}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\mu(\bar{\varepsilon}=0)}$
$\displaystyle\leq\frac{4}{\mu(\bar{\varepsilon}=0)}\,\int_{\bar{\varepsilon}>0}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{2k_{B}T}\right)\text{d}\mu(\zeta).$

We can prove that (see proof of Proposition 4.3) $0\leq D(T)\leq
2\,\frac{\int_{\mathcal{E}}\,\left(\frac{\bar{\varepsilon}}{k_{B}T}\right)^{2}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}$,
which implies

$0\leq D(T)\leq
2\,\frac{\displaystyle\int_{\bar{\varepsilon}>0}\,\left(\frac{\bar{\varepsilon}}{k_{B}T}\right)^{2}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\mu(\bar{\varepsilon}=0)}\leq\frac{16}{\mu(\bar{\varepsilon}=0)}\,\int_{\bar{\varepsilon}>0}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{2k_{B}T}\right)\text{d}\mu(\zeta).$

Now for all $\zeta\in\\{\bar{\varepsilon}>0\\}$,
$e^{-x\bar{\varepsilon}(\zeta)}\underset{x\to\infty}{\longrightarrow}0$. Since
$(\zeta\mapsto e^{-x\bar{\varepsilon}(\zeta)})_{x>0}$ is a non-decreasing
family of positive functions, we get by monotone convergence

$\int_{\bar{\varepsilon}>0}e^{-x\bar{\varepsilon}(\zeta)}\text{d}\mu(\zeta)\underset{x\to\infty}{\longrightarrow}0.$

It follows that
$\displaystyle\int_{\bar{\varepsilon}>0}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{2k_{B}T}\right)\text{d}\mu(\zeta)\underset{T\to
0^{+}}{\longrightarrow}0$, and thus, since $\mu(\bar{\varepsilon}=0)>0$,

$0\leq\delta(T)\leq\frac{4}{\mu(\bar{\varepsilon}=0)}\,\int_{\bar{\varepsilon}>0}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{2k_{B}T}\right)\text{d}\mu(\zeta)\underset{T\to
0^{+}}{\longrightarrow}0$

and

$0\leq
D(T)\leq\frac{16}{\mu(\bar{\varepsilon}=0)}\,\int_{\bar{\varepsilon}>0}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{2k_{B}T}\right)\text{d}\mu(\zeta)\underset{T\to
0^{+}}{\longrightarrow}0,$

so that finally

$\delta(T)\underset{T\to 0^{+}}{\longrightarrow}0\quad\text{and}\quad
D(T)\underset{T\to 0^{+}}{\longrightarrow}0.$

$\square$

###### Proposition 4.2

For all $T>0$,

$\delta(T)=\frac{1}{T}\int_{0}^{T}D(T^{\prime})\,\emph{d}T^{\prime}.$

###### Proof

We have for $T_{1},T_{2}>0$,
$\displaystyle\int_{T_{1}}^{T_{2}}D(T^{\prime})\,\text{d}T^{\prime}=T_{2}\delta(T_{2})-T_{1}\delta(T_{1})$.
We thus want to prove that $T\delta(T)\underset{T\to
0^{+}}{\longrightarrow}0$. We proceed by disjunction of case. First, if
$\mu(\varepsilon=\varepsilon^{0})>0$, then from Proposition 4.1,
$\delta(T)\underset{T\to 0^{+}}{\longrightarrow}0$, so we also have
$T\delta(T)\underset{T\to 0^{+}}{\longrightarrow}0$. If on the other hand
$\mu(\varepsilon=\varepsilon^{0})=0$, then since $\varepsilon^{0}=\text{inf
ess}_{\mu}\\{\varepsilon\\}$, we have $\mu(0<\bar{\varepsilon}\leq
r)=\mu(\varepsilon^{0}<\varepsilon\leq\varepsilon^{0}+r)>0$ for all $r>0$.
Then from Lemma 9.1 in Appendix, for all $r>0$,

$\displaystyle\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)$
$\displaystyle\underset{x\to\infty}{=}\int_{\bar{\varepsilon}\leq
r}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)+o\left(\int_{\bar{\varepsilon}\leq
r}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)\right),$
$\displaystyle\text{and}\quad\int_{\mathcal{E}}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)$
$\displaystyle\underset{x\to\infty}{=}\int_{\bar{\varepsilon}\leq
r}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)+o\left(\int_{\bar{\varepsilon}\leq
r}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)\right).$

Since $\mu(\\{0<\bar{\varepsilon}\leq r\\})>0$ and $\bar{\varepsilon}\geq 0$
$\mu$-a.e., $\int_{\bar{\varepsilon}\leq
r}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)>0$,
so that

$\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)\underset{x\to\infty}{\sim}\int_{\bar{\varepsilon}\leq
r}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta).$

Also, $\mu(\\{\bar{\varepsilon}\leq r\\})>0$ implies
$\int_{\bar{\varepsilon}\leq
r}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)>0$, so that

$\int_{\mathcal{E}}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)\underset{x\to\infty}{\sim}\int_{\bar{\varepsilon}\leq
r}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta).$

Hence

$\frac{\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}{\int_{\mathcal{E}}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}\underset{x\to\infty}{\sim}\frac{\int_{\bar{\varepsilon}\leq
r}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}{\int_{\bar{\varepsilon}\leq
r}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}.$

Thus there exists $x_{r}>0$ such that for all $x\geq x_{r}$,

$0\leq\frac{\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}{\int_{\mathcal{E}}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}\leq
2\,\frac{\int_{\bar{\varepsilon}\leq
r}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}{\int_{\bar{\varepsilon}\leq
r}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}\leq 2\,r.$

Since it is true for all $r>0$, we finally get

$\frac{\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}{\int_{\mathcal{E}}e^{-x\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)}\underset{x\to\infty}{\longrightarrow}0.$

It follows that in any case

$T\delta(T)=\frac{2}{k_{B}}\,\frac{\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)e^{-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}}\,\text{d}\mu(\zeta)}{\int_{\mathcal{E}}e^{-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}}\,\text{d}\mu(\zeta)}\underset{T\to
0^{+}}{\longrightarrow}0.$

$\square$

###### Remark 4.2

This Proposition 4.2 along with Remark 4.1 justify our choice to define the
partition function by
$\displaystyle\int_{\mathcal{E}}e^{-\beta\bar{\varepsilon}(\zeta)}\,\text{d}\mu(\zeta)$
(with $\bar{\varepsilon}$) instead of simply
$\displaystyle\int_{\mathcal{E}}e^{-\beta\varepsilon(\zeta)}\,\text{d}\mu(\zeta)$
(with $\varepsilon$). Indeed, if the second definition were taken, and
$\delta$ were still defined by
$-\frac{2}{k_{B}T}\,\big{(}\text{ln}(Z)\big{)}^{\prime}\left(\frac{1}{k_{B}\,T}\right)$,
then $\varepsilon^{0}\neq 0$ would imply $\displaystyle\delta(T)\underset{T\to
0}{\sim}\frac{2\,\varepsilon^{0}}{k_{B}T}$, which is non-physical. On the
other hand, choosing $\bar{\varepsilon}$ in the definition always yields
$T\delta(T)\underset{T\to 0}{\longrightarrow}0$, and thus the formula
$\displaystyle\delta(T)=\frac{1}{T}\int_{0}^{T}D(T^{\prime})\,\text{d}T^{\prime}$,
which is physically coherent.

As a consequence of Proposition 4.2, the total energy at the equilibrium may
be cast as

$e^{eq}[u,T]=\varepsilon^{0}+\frac{m}{2}|u|^{2}+\int_{0}^{T}c_{V}(T^{\prime})\,k_{B}\,\text{d}T^{\prime}.$

A probabilistic interpretation may also be provided for the functions
$\delta(T)$ and $D(T)$. Indeed, for $T>0$, we define the Gibbs (probability)
measure $\nu_{T}$ on $(\mathcal{E},\mathcal{A})$ by

$\frac{\text{d}\nu_{T}}{\text{d}\mu}(\zeta)=Z\left(\frac{1}{k_{B}T}\right)^{-1}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right),$
(4.5)

where $Z$ is the partition function defined by equation (2.3); then for all
$T>0$, $(\mathcal{E},\mathcal{A},\nu_{T})$ is a probability space. Since
$\bar{\varepsilon}:\mathcal{E}\to\mathbb{R}$ is
$(\mathcal{A},\text{Bor}(\mathbb{R}))$-measurable, it is a real random
variable on $(\mathcal{E},\mathcal{A},\nu_{T})$.

###### Proposition 4.3

For all $T>0$,

$\delta(T)=2\,\mathbb{E}_{\nu_{T}}\left[\frac{\bar{\varepsilon}}{k_{B}T}\right],$
(4.6)

and

$D(T)=2\,\emph{Var}_{\nu_{T}}\left[\frac{\bar{\varepsilon}}{k_{B}T}\right],$
(4.7)

where $\mathbb{E}_{\nu_{T}}$ and $\emph{Var}_{\nu_{T}}$ are respectively the
expectation and the variance under the probability $\nu_{T}$.

###### Proof

The following computations are possible thanks to Assumption 2.5. The first
part of the proposition comes from

$\frac{\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}=\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)\,Z\left(\frac{1}{k_{B}T}\right)^{-1}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)=\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)\,\text{d}\nu_{T}(\zeta)=\mathbb{E}_{\nu_{T}}[\bar{\varepsilon}].$

For equation (4.7), we remark that from Proposition 2.1,

$\mathbb{E}_{\nu_{T}}[\bar{\varepsilon}]=\frac{\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}=\frac{-Z^{\prime}\left(\frac{1}{k_{B}T}\right)}{Z\left(\frac{1}{k_{B}T}\right)},\quad\mathbb{E}_{\nu_{T}}\left[\bar{\varepsilon}^{2}\right]=\frac{\int_{\mathcal{E}}\bar{\varepsilon}(\zeta)^{2}\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}=\frac{Z^{\prime\prime}\left(\frac{1}{k_{B}T}\right)}{Z\left(\frac{1}{k_{B}T}\right)}.$

It follows that

$\displaystyle D(T)$
$\displaystyle=2\,\frac{\text{d}}{k_{B}\,\text{d}T}\left(\frac{-Z^{\prime}\left(\frac{1}{k_{B}T}\right)}{Z\left(\frac{1}{k_{B}T}\right)}\right)=\frac{2}{(k_{B}T)^{2}}\left(\frac{Z^{\prime\prime}\left(\frac{1}{k_{B}T}\right)}{Z\left(\frac{1}{k_{B}T}\right)}-\left(\frac{-Z^{\prime}\left(\frac{1}{k_{B}T}\right)}{Z\left(\frac{1}{k_{B}T}\right)}\right)^{2}\right)$
$\displaystyle=\frac{2}{(k_{B}T)^{2}}\left(\mathbb{E}_{\nu_{T}}\left[\bar{\varepsilon}^{2}\right]-(\mathbb{E}_{\nu_{T}}[\bar{\varepsilon}])^{2}\right)=\frac{2}{(k_{B}T)^{2}}\,\text{Var}_{\nu_{T}}\left[\bar{\varepsilon}\right]=2\,\text{Var}_{\nu_{T}}\left[\frac{\bar{\varepsilon}}{k_{B}T}\right].$

$\square$

###### Corollary 4.1

For all $T>0$, $D(T)\geq 0$.

###### Proof

It is an immediate consequence of Proposition 4.3. $\square$

###### Corollary 4.2

If there exists $R\in\mathbb{R}$ such that $\varepsilon\leq R$ $\mu$-a.e.,
then

$\delta(T)\underset{T\to\infty}{\longrightarrow}0,\quad\text{and}\quad
D(T)\underset{T\to\infty}{\longrightarrow}0.$

###### Proof

Necessarily $R\geq\varepsilon^{0}$, and from Proposition 4.3

$0\leq\delta(T)=2\,\mathbb{E}_{\nu_{T}}\left[\frac{\bar{\varepsilon}}{k_{B}T}\right]\leq\frac{2(R-\varepsilon^{0})}{k_{B}T}\underset{T\to\infty}{\longrightarrow}0,$

and

$0\leq
D(T)=2\,\text{Var}_{\nu_{T}}\left[\frac{\bar{\varepsilon}}{k_{B}T}\right]\leq
2\,\mathbb{E}_{\nu_{T}}\left[\left(\frac{\bar{\varepsilon}}{k_{B}T}\right)^{2}\right]\leq\frac{2(R-\varepsilon^{0})^{2}}{(k_{B}T)^{2}}\underset{T\to\infty}{\longrightarrow}0.$

$\square$

We define now the function $\Theta$, for all $T\in\mathbb{R}_{+}^{*}$, as

$\Theta(T):=\int_{0}^{T}c_{V}(T^{\prime})\,k_{B}\,\text{d}T^{\prime},$ (4.8)

such that

$e^{eq}[u,T]=\varepsilon^{0}+\frac{m}{2}|u|^{2}+\Theta(T).$

###### Proposition 4.4

$\Theta$ is continuous on $\mathbb{R}_{+}^{*}$, can be extended by continuity
to $\mathbb{R}_{+}$, setting $\Theta(0)=0$, increasing on $\mathbb{R}_{+}$,
with $\Theta(T)\underset{T\rightarrow\infty}{\longrightarrow}\infty$, and thus
a bijection from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$.

###### Proof

The result is immediate, since from Corollary 4.1 we have $\displaystyle
c_{V}\geq\frac{3}{2}$. $\square$

We then have

$T=\Theta^{-1}\left(\frac{m}{\rho}\,\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\left(\frac{m}{2}|v-u|^{2}+\bar{\varepsilon}(\zeta)\right)\,f(v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\right),$

which explains and rigorously justifies the definition of temperature stated
in equation (3.2).

##  5 Euler limit

Let us assume that there exists $f$ solution to the Boltzmann equation (3.1)
such that for all $t,x$, $f(t,x,\cdot,\cdot)$ is a Maxwellian, that is

$f(t,x,v,\zeta)=\mathcal{M}[\rho(t,x),u(t,x),T(t,x)](v,\zeta).$

Since

$\partial_{t}f(t,x,v,\zeta)+v\cdot\nabla_{x}f(t,x,v,\zeta)=\mathcal{B}(f,f)(t,x,v,\zeta)$

and

$\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\begin{pmatrix}m\\\ mv\\\
\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\end{pmatrix}\mathcal{B}(f,f)(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=0,$

we have

$\displaystyle\partial_{t}\left(\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\right)$
$\displaystyle+\text{div}_{x}\left(\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,vf(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\right)=0$
$\displaystyle\partial_{t}\left(\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,vf(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\right)$
$\displaystyle+\text{div}_{x}\left(\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,v\otimes
vf(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\right)=0$
$\displaystyle\partial_{t}\Bigg{(}\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}\left(\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\right)f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\Bigg{)}$
$\displaystyle+\text{div}_{x}\Bigg{(}\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}v\left(\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\right)f(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\Bigg{)}=0.$

Since $f(t,x,v,\zeta)=\mathcal{M}[\rho(t,x),u(t,x),T(t,x)](v,\zeta)$, we have
from the definitions of mass, momentum, and total energy

$\displaystyle\partial_{t}\rho$
$\displaystyle+\text{div}_{x}\left(\rho\,u\right)=0$
$\displaystyle\partial_{t}\left(\rho\,u\right)$
$\displaystyle+\text{div}_{x}\left(\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,v\otimes
v\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\right)=0$
$\displaystyle\partial_{t}\left(\rho\,\varepsilon^{0}+\frac{1}{2}\rho|u|^{2}+\frac{\rho}{m}\,\frac{3+\delta(T)}{2}\,k_{B}T\right)$
$\displaystyle+\text{div}_{x}\Bigg{(}\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}v\left(\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\right)\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)\Bigg{)}=0.$

We are left with two terms to compute. First, by a classical argument (see
bouchut2000kinetic , Chapter 2),

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}m\,v\otimes
v\,\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\rho\,m^{1/2}\,(2\pi
k_{B}T)^{-3/2}\,\int_{\mathbb{R}^{3}}m\,v\otimes
v\,\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}\right)\,\text{d}v$
$\displaystyle=\rho\,u\otimes u+\frac{\rho}{m}\,k_{B}T\,\mathbf{I_{d}},$

where $\mathbf{I_{d}}$ is the identity matrix. Also by a classical argument,

$\displaystyle\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}v\left(\frac{m}{2}|v|^{2}+\varepsilon(\zeta)\right)\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\rho\,m^{1/2}\,(2\pi
k_{B}T)^{-3/2}\,\int_{\mathbb{R}^{3}}v\frac{m}{2}|v|^{2}\,\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}\right)\,\text{d}v+\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}v\,\varepsilon(\zeta)\mathcal{M}[\rho,u,T](v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)$
$\displaystyle=\frac{1}{2}\rho\,|u|^{2}u+\frac{\rho}{m}\,k_{B}T\,u+\frac{\rho}{m}\,\frac{3}{2}k_{B}T\,u+\frac{\rho}{m}\,\left(\frac{\int_{\mathcal{E}}\varepsilon(\zeta)\,\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}{\int_{\mathcal{E}}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)}\right)u$
$\displaystyle=\frac{1}{2}\rho\,|u|^{2}u+\frac{\rho}{m}\,k_{B}T\,u+\frac{\rho}{m}\,\frac{3+\delta(T)}{2}k_{B}T\,u+\frac{\rho}{m}\,\varepsilon^{0}\,u.$

We set

$p(t,x):=\frac{\rho}{m}\,k_{B}T(t,x)\quad\text{and}\quad\theta(t,x):=\frac{3+\delta(T(t,x))}{2}k_{B}T(t,x).$

Remarking that
$\partial_{t}(\rho\,\varepsilon^{0})+\text{div}_{x}\left(\rho\,\varepsilon^{0}\,u\right)=\varepsilon^{0}\big{(}\partial_{t}\rho+\text{div}_{x}\left(\rho\,u\right)\big{)}=0$,
we obtain the compressible Euler set of equations, with $\rho$ the density,
$u$ the velocity, $p$ the pressure and $\theta$ the specific internal energy
density.

$\begin{cases}\displaystyle\hskip
62.4pt\partial_{t}\rho+\text{div}_{x}\left(\rho\,u\right)=0\\\
\displaystyle\hskip
45.6pt\partial_{t}\left(\rho\,u\right)+\text{div}_{x}\left(\rho\,u\otimes
u\right)+\nabla_{x}p=0\\\
\displaystyle\partial_{t}\left(\frac{1}{2}\rho|u|^{2}+\frac{\rho}{m}\,\theta\right)+\text{div}_{x}\left(\frac{1}{2}\rho|u|^{2}u+\frac{\rho}{m}\,\theta\,u+p\,u\right)=0.\end{cases}$
(5.1)

##  6 Equipartition Theorem

In this section we show how is possible to combine different measure spaces of
internal states into a unique kinetic model for the considered polyatomic gas.

###### Theorem 6.1

Equipartition Theorem. Let $L\in\mathbb{N}^{*}$. Consider
$(\mathcal{E}_{l},\mathcal{A}_{l},\mu_{l})_{1\leq l\leq L}$, and for all
$1\leq l\leq L$,
$\varepsilon_{l}:(\mathcal{E}_{l},\mathcal{A}_{l})\to(\mathbb{R},\emph{Bor}(\mathbb{R}))$
measurable such that Assumptions 2.3 and 2.5 hold for all
$(\mu_{l},\varepsilon_{l})$. Let us define

$\displaystyle\mathcal{E}=\mathcal{E}_{1}\times\dots\times\mathcal{E}_{L},\qquad\mathcal{A}=\mathcal{A}_{1}\otimes\dots\otimes\mathcal{A}_{L},\qquad\mu=\mu_{1}\otimes\dots\otimes\mu_{L}$

and

$\forall\zeta=(\zeta_{1},\dots,\zeta_{L})\in\mathcal{E},\qquad\varepsilon(\zeta)=\sum_{l=1}^{L}\varepsilon_{l}(\zeta_{l}).$

Then Assumptions 2.3 and 2.5 hold for $(\mu,\varepsilon)$, and for all $T>0$,
the Gibbs measure defined in (4.5) turns out to be

$\nu_{T}=\nu^{1}_{T}\otimes\dots\otimes\nu^{L}_{T}$

and moreover one has

$\delta(T)=\sum_{l=1}^{L}\delta_{l}(T),\qquad D(T)=\sum_{l=1}^{L}D_{l}(T).$

###### Proof

Let $T>0$. Let us first prove that
$\nu_{T}=\nu^{1}_{T}\otimes\dots\otimes\nu^{L}_{T}$. We first remark that
$\bar{\varepsilon}(\zeta)=\sum_{l=1}^{L}\bar{\varepsilon}_{l}(\zeta_{l})$. Let
$A=(A_{1},\dots,A_{L})\in\mathcal{A}_{1}\times\dots\times\mathcal{A}_{L}$.
Then

$\displaystyle\int_{A}\text{exp}\left(-\frac{\bar{\varepsilon}(\zeta)}{k_{B}T}\right)\text{d}\mu(\zeta)$
$\displaystyle=\int_{A_{1}}\dots\int_{A_{L}}\left(\text{exp}\left(-\frac{\sum_{l=1}^{L}\bar{\varepsilon}_{l}(\zeta_{l})}{k_{B}T}\right)\right)\,\text{d}\mu_{1}(\zeta_{1})\dots\text{d}\mu_{L}(\zeta_{L})$
$\displaystyle=\int_{A_{1}}\dots\int_{A_{L}}\left(\prod_{l=1}^{L}\text{exp}\left(-\frac{\bar{\varepsilon}_{l}(\zeta_{l})}{k_{B}T}\right)\right)\,\text{d}\mu_{1}(\zeta_{1})\dots\text{d}\mu_{L}(\zeta_{L})$
$\displaystyle=\prod_{l=1}^{L}\int_{A_{l}}\text{exp}\left(-\frac{\bar{\varepsilon}_{l}(\zeta_{l})}{k_{B}T}\right)\,\text{d}\mu_{l}(\zeta_{l}).$

Since this also holds for
$A=\mathcal{E}=\mathcal{E}_{1}\times\dots\times\mathcal{E}_{L}$, we then have

$\nu_{T}(A)=\prod_{l=1}^{L}\frac{\int_{A_{l}}\text{exp}\left(-\frac{\bar{\varepsilon}_{l}(\zeta_{l})}{k_{B}T}\right)\,\text{d}\mu_{l}(\zeta_{l})}{\int_{\mathcal{E}_{l}}\text{exp}\left(-\frac{\bar{\varepsilon}_{l}(\zeta_{l})}{k_{B}T}\right)\,\text{d}\mu_{l}(\zeta_{l})}=\prod_{l=1}^{L}\nu^{l}_{T}(A_{l}),$

that is $\nu_{T}=\nu^{1}_{T}\otimes\dots\otimes\nu^{L}_{T}$.

Each $\varepsilon_{l}$ (thus also $\bar{\varepsilon}_{l}$) can also be seen as
a real random variable on $(\mathcal{E},\mathcal{A},\nu_{T})$, by setting
$\varepsilon_{l}(\zeta)\equiv\varepsilon_{l}(\zeta_{l})$. It follows that

$\delta(T)=2\,\mathbb{E}_{\nu_{T}}\left[\frac{\bar{\varepsilon}}{k_{B}T}\right]=2\,\mathbb{E}_{\nu_{T}}\left[\sum_{l=1}^{L}\frac{\bar{\varepsilon}_{l}}{k_{B}T}\right]=2\sum_{l=1}^{L}\mathbb{E}_{\nu_{T}}\left[\frac{\bar{\varepsilon}_{l}}{k_{B}T}\right]=\sum_{l=1}^{L}\delta_{l}(T).$

Finally, since $\varepsilon_{1},\dots,\varepsilon_{L}$ are independent under
$\nu_{T}$ ($\nu_{T}$ is tensorized and they each depend on a different
variable),

$D(T)=2\,\text{Var}_{\nu_{T}}\left[\frac{\bar{\varepsilon}}{k_{B}T}\right]=2\,\text{Var}_{\nu_{T}}\left[\sum_{l=1}^{L}\frac{\bar{\varepsilon}_{l}}{k_{B}T}\right]=2\sum_{l=1}^{L}\text{Var}_{\nu_{T}}\left[\frac{\bar{\varepsilon}_{l}}{k_{B}T}\right]=\sum_{l=1}^{L}D_{l}(T).$

$\square$

We show now the results relevant to some particular options.

###### Proposition 6.1

Let $C_{1},C_{2}>0$, $\beta>0$ and $\alpha>-1$. Denoting by
$\emph{Bor}(\mathbb{R})$ the set Borelians of $\mathbb{R}$, we consider

$(\mathcal{E},\mathcal{A},\emph{d}\mu(\zeta))=\left(\mathbb{R},\emph{Bor}(\mathbb{R}),C_{1}|z|^{\alpha}\,\emph{d}z\right),\quad\varepsilon(z)=C_{2}|z|^{\beta}.$

Then for all $T>0$,

$\delta(T)=D(T)=\frac{2(\alpha+1)}{\beta}.$

###### Proof

Set $T>0$ and $\alpha>-1$. Performing the change of variables
$x=C_{2}|z|^{\beta}$, and then $y=x/(k_{B}T)$, we get

$\displaystyle\int_{\mathbb{R}}C_{2}|z|^{\beta}\,\text{exp}\left(-\frac{C_{2}|z|^{\beta}}{k_{B}T}\right)\,C_{1}|z|^{\alpha}\,\text{d}z$
$\displaystyle=\frac{2C_{1}C_{2}^{\frac{\alpha-1}{\beta}}}{\beta}\int_{0}^{\infty}x\,\text{exp}\left(-\frac{x}{k_{B}T}\right)\,x^{\frac{\alpha}{\beta}}\,x^{\frac{1-\beta}{\beta}}\text{d}x$
$\displaystyle=\frac{2C_{1}C_{2}^{\frac{\alpha-1}{\beta}}}{\beta}\int_{0}^{\infty}x^{\frac{\alpha+1}{\beta}}\,\text{exp}\left(-\frac{x}{k_{B}T}\right)\,\text{d}x$
$\displaystyle=\frac{2C_{1}C_{2}^{\frac{\alpha-1}{\beta}}}{\beta}(k_{B}T)^{\frac{\alpha+1}{\beta}+1}\int_{0}^{\infty}y^{\frac{\alpha+1}{\beta}}\,e^{-y}\,\text{d}y$
$\displaystyle=\frac{2C_{1}C_{2}^{\frac{\alpha-1}{\beta}}}{\beta}(k_{B}T)^{\frac{\alpha+1}{\beta}+1}\left(\frac{\alpha+1}{\beta}\right)\int_{0}^{\infty}y^{\frac{\alpha+1}{\beta}-1}\,e^{-y}\,\text{d}y.$

Analogously, performing the same changes of variables

$\displaystyle\int_{\mathbb{R}}\text{exp}\left(-\frac{C_{2}|z|^{\beta}}{k_{B}T}\right)\,C_{1}|z|^{\alpha}\,\text{d}z$
$\displaystyle=\frac{2C_{1}C_{2}^{\frac{\alpha-1}{\beta}}}{\beta}\int_{0}^{\infty}\text{exp}\left(-\frac{x}{k_{B}T}\right)\,x^{\frac{\alpha}{\beta}}\,x^{\frac{1-\beta}{\beta}}\text{d}x$
$\displaystyle=\frac{2C_{1}C_{2}^{\frac{\alpha-1}{\beta}}}{\beta}\int_{0}^{\infty}x^{\frac{\alpha+1}{\beta}-1}\,\text{exp}\left(-\frac{x}{k_{B}T}\right)\,\text{d}x$
$\displaystyle=\frac{2C_{1}C_{2}^{\frac{\alpha-1}{\beta}}}{\beta}(k_{B}T)^{\frac{\alpha+1}{\beta}}\int_{0}^{\infty}y^{\frac{\alpha+1}{\beta}-1}\,e^{-y}\,\text{d}y.$

We deduce that

$\delta(T)=\frac{2}{k_{B}T}\,\frac{\int_{\mathbb{R}}C_{2}|z|^{\beta}\,\text{exp}\left(-\frac{C_{2}|z|^{\beta}}{k_{B}T}\right)\,C_{1}|z|^{\alpha}\,\text{d}z}{\int_{\mathbb{R}}\text{exp}\left(-\frac{C_{2}|z|^{\beta}}{k_{B}T}\right)\,C_{1}|z|^{\alpha}\,\text{d}z}=\frac{2(\alpha+1)}{\beta}.$

Since $\delta$ does not depend on $T$, we have $\displaystyle
D(T)=\delta(T)=\frac{2(\alpha+1)}{\beta}$. $\square$

###### Corollary 6.1

We consider $a_{1},\dots,a_{d}>0$ and

$(\mathcal{E},\mathcal{A},\emph{d}\mu(\zeta))=\left(\mathbb{R}^{d},\emph{Bor}(\mathbb{R}^{d}),\emph{d}z\right),\qquad\varepsilon(z)=\sum_{l=1}^{d}a_{l}\,z_{l}^{2}.$

Then for all $T>0$,

$\delta(T)=D(T)=d.$

###### Proof

This Corollary is a consequence of Proposition 6.1 with $\alpha=0$, $\beta=2$,
and Proposition 6.1 with
$(\mathcal{E}_{l},\mathcal{A}_{l},\text{d}\mu_{l})=\left(\mathbb{R},\text{Bor}(\mathbb{R}),\text{d}z\right)$
and $\varepsilon_{l}(z)=a_{l}\,z^{2}$ for $1\leq l\leq d$. $\square$

##  7 Models within the framework

###  7.1 Existing models

In this subsection, we show that our framework encapsulates various existing
models. The parameters to choose in order to build the model are
$(\mathcal{E},\mathcal{A},\mu)$ and $\varepsilon$.

#### The monoatomic gas

The model for the monoatomic gas is the classical Boltzmann model, which
considers elastic collisions only. We refer the reader to bouchut2000kinetic .
To recover this model in our framework, simply set $\mathcal{E}=\\{0\\}$,
$\mathcal{A}=\mathcal{P}(\mathcal{E})$, $\mu(\\{0\\})=1$ and
$\varepsilon(0)=\varepsilon^{0}\in\mathbb{R}$. Then the state of a molecule is
described by $(v,\zeta)\in\mathbb{R}^{3}\times\\{0\\}$, thus simply by
$v\in\mathbb{R}^{3}$. The conservation laws become

$\begin{cases}\hskip 68.0ptmv+mv_{*}=mv^{\prime}+mv^{\prime}_{*}\\\
\displaystyle\frac{m}{2}|v|^{2}+\varepsilon^{0}+\frac{m}{2}|v_{*}|^{2}+\varepsilon^{0}=\frac{m}{2}|v^{\prime}|^{2}+\varepsilon^{0}+\frac{m}{2}|v^{\prime}_{*}|^{2}+\varepsilon^{0},\end{cases}$

which simplify into the conservation laws of an elastic collision

$\begin{cases}\hskip 20.5ptv+v_{*}=v^{\prime}+v^{\prime}_{*}\\\
\displaystyle|v|^{2}+|v_{*}|^{2}=|v^{\prime}|^{2}+|v^{\prime}_{*}|^{2}.\end{cases}$

From there, the classical post-collision velocities

$\begin{cases}v^{\prime}&=\displaystyle\frac{v+v_{*}}{2}+\frac{|v-v_{*}|}{2}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\\\
v^{\prime}_{*}&=\displaystyle\frac{v+v_{*}}{2}-\frac{|v-v_{*}|}{2}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\end{cases}$

are recovered. The number of internal degrees of freedom and heat capacity at
constant volume associated with this model are, as expected,

$\delta=0\quad\text{and}\quad c_{V}=\frac{3}{2},$

reproducing that fact that an atom has only the three translational degrees of
freedom.

#### The weighted model with one continuous variable

This model was originally proposed by Borgnakke et al. in
borgnakke1975statistical  and completed with the introduction of a weight
$\varphi$ of integration by Desvillettes et al. in desvillettes1997modele ;
desvillettes2005kinetic  in order to accurately describe polyatomic gases with
the introduction of a single parameter $I$ related to the internal energy of
the molecule. The state of a molecule is described by
$(v,I)\in\mathbb{R}^{3}\times\mathbb{R}_{+}$, and the associated total energy
is $\displaystyle\frac{m}{2}|v|^{2}+I$.

To recover this model in our framework, set $\mathcal{E}=\mathbb{R}_{+}$,
$\mathcal{A}$ the set of Borelians of $\mathbb{R}_{+}$,
$\text{d}\mu(I)=\varphi(I)\text{d}I$, and for $I\in\mathbb{R}_{+}$,
$\varepsilon(I)=I$. The conservation laws of a collision write, when the
collision is possible, as

$\begin{cases}\hskip 71.5ptmv+mv_{*}=mv^{\prime}+mv^{\prime}_{*}\\\
\displaystyle\frac{m}{2}|v|^{2}+I+\frac{m}{2}|v_{*}|^{2}+I_{*}=\frac{m}{2}|v^{\prime}|^{2}+I^{\prime}+\frac{m}{2}|v^{\prime}_{*}|^{2}+I^{\prime}_{*}.\end{cases}$

Since $I$ is a continuous parameter, in desvillettes2005kinetic , Desvillettes
et al. introduce $\displaystyle e=\frac{m}{4}|v-v_{*}|^{2}+I+I_{*}$,
$\displaystyle r=\frac{I^{\prime}}{I^{\prime}+I^{\prime}_{*}}$ and
$\displaystyle R=\frac{\frac{m}{4}|v^{\prime}-v^{\prime}_{*}|^{2}}{e}$. Using
these parameters, the post-collision velocities write in their formulation

$\begin{cases}v^{\prime}&=\displaystyle\frac{v+v_{*}}{2}+\sqrt{Re}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\\\
v^{\prime}_{*}&=\displaystyle\frac{v+v_{*}}{2}-\sqrt{Re}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\end{cases}$

Our formula for post-collision velocities is equivalent and writes, for
allowed collisions,

$\begin{cases}v^{\prime}&=\displaystyle\frac{v+v_{*}}{2}+\sqrt{\frac{1}{4}|v-v_{*}|^{2}+\frac{1}{m}(I+I_{*}-I^{\prime}-I^{\prime}_{*})}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\\\
v^{\prime}_{*}&=\displaystyle\frac{v+v_{*}}{2}-\sqrt{\frac{1}{4}|v-v_{*}|^{2}+\frac{1}{m}(I+I_{*}-I^{\prime}-I^{\prime}_{*})}\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]\end{cases}$

The main technical difference between the two approaches is that the authors
in desvillettes2005kinetic  fix all the pre–collision parameters (velocities,
internal energies, $r$ and $R$) and study then the transformation

$\tilde{S}_{\omega}:(v,v_{*},I,I_{*},r,R)\mapsto(v^{\prime},v^{\prime}_{*},I^{\prime},I^{\prime}_{*},r^{\prime},R^{\prime})$

defined on
$\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}_{+}\times\mathbb{R}_{+}\times[0,1]\times[0,1]$,
while, in our framework, we fix pre–collision velocities and all pre– and
post–interaction internal energies and we study the transformation

$S_{\omega}[I,I_{*},I^{\prime},I^{\prime}_{*}]:(v,v_{*})\mapsto(v^{\prime},v^{\prime}_{*})$

defined on $E[I,I_{*},I^{\prime},I^{\prime}_{*}]$.

We are able to prove that the two methods are equivalent. Indeed, we denote by
$\Phi$ the bijection

$\Phi:\begin{cases}\hskip 63.0pt\tilde{\Omega}\to\Omega\\\
(v,v_{*},I,I_{*},r,R)\mapsto(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*}),\end{cases}$

where
$\tilde{\Omega}=(\mathbb{R}^{3})^{2}\times(\mathbb{R}_{+})^{2}\times[0,1]^{2}$
and
$\Omega=\left\\{(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*})\in(\mathbb{R}^{3})^{2}\times(\mathbb{R}_{+})^{4},\;(v,v_{*})\in
E[I,I_{*},I^{\prime},I^{\prime}_{*}]\right\\}$. Its Jacobian can be computed
and is equal to

$J[\Phi](v,v_{*},I,I_{*},r,R)=(1-R)e^{2},$

where we recall that $\displaystyle
e=\frac{m}{4}|v-v_{*}|^{2}+I+I_{*}=\frac{m}{4}|v^{\prime}-v^{\prime}_{*}|^{2}+I^{\prime}+I^{\prime}_{*}=e^{\prime}$.
Then

$\displaystyle S_{\omega}[I,I_{*},I^{\prime},I^{\prime}_{*}](v,v_{*})$
$\displaystyle=(v^{\prime},v^{\prime}_{*})$
$\displaystyle\Phi\circ\tilde{S}_{\omega}\circ\Phi^{-1}(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*})$
$\displaystyle=(v^{\prime},v^{\prime}_{*},I^{\prime},I^{\prime}_{*},I,I_{*}),$

and we remark that (see Lemma 1. in desvillettes2005kinetic  and equation
(2.11))

$\displaystyle
J\left[\Phi\circ\tilde{S}_{\omega}\circ\Phi^{-1}\right](v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*})$
$\displaystyle=(1-R^{\prime})(e^{\prime})^{2}\,\frac{(1-R)|v^{\prime}-v^{\prime}_{*}|}{(1-R^{\prime})|v-v_{*}|}\,\frac{1}{(1-R)e^{2}}$
$\displaystyle=\frac{|v^{\prime}-v^{\prime}_{*}|}{|v-v_{*}|}=J\left[S_{\omega}[I,I_{*},I^{\prime},I^{\prime}_{*}]\right](v,v_{*}).$

The authors in desvillettes2005kinetic  consider a kernel
$B:\tilde{\Omega}\times\mathbb{S}^{2}\to\mathbb{R}_{+}$ with the micro-
reversibility conditions

$\displaystyle B(v,v_{*},I,I_{*},R,r,\omega)$
$\displaystyle=B(v_{*},v,I_{*},I,R,1-r,\omega)$ $\displaystyle
B(v,v_{*},I,I_{*},R,r,\omega)$
$\displaystyle=B(v^{\prime},v^{\prime}_{*},I^{\prime},I^{\prime}_{*},R^{\prime},r^{\prime},\omega).$

We define on $\Omega\times\mathbb{S}^{2}$

$b(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*},\omega)=\frac{1}{e^{2}}|v-v_{*}|^{-1}\,\frac{B(\cdot,\omega)\circ\Phi^{-1}(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*})}{\varphi(I)\varphi(I_{*})\varphi(I^{\prime})\varphi(I^{\prime}_{*})},$

and set $b$ to zero elsewhere. Then, since $e=e^{\prime}$,

$\displaystyle b(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*},\omega)$
$\displaystyle=b(v_{*},v,I_{*},I,I^{\prime}_{*},I^{\prime},\omega)$
$\displaystyle|v-v_{*}|\,b(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*},\omega)$
$\displaystyle=|v^{\prime}-v^{\prime}_{*}|\,b(v^{\prime},v^{\prime}_{*},I^{\prime},I^{\prime}_{*},I,I_{*},\omega),$

which are the symmetry and micro-reversibility conditions in our framework. It
follows that the difference of approach is purely technical, and does not lead
to theoretical discrepancies. The same distribution $f$ and macroscopic
quantities are recovered. Indeed, in this case, with $\zeta=I$,

$\displaystyle\mathcal{B}(f,f)(v,\zeta)=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*})\Big{)}\;b(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})$
$\displaystyle=\iiint_{(\mathbb{R}_{+})^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},I^{\prime})f(v^{\prime}_{*},I^{\prime}_{*})-f(v,I)f(v_{*},I_{*})\Big{)}\;b(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\varphi(I_{*})\varphi(I^{\prime})\varphi(I^{\prime}_{*})\,\text{d}I_{*}\,\text{d}I^{\prime}\,\text{d}I^{\prime}_{*}$
$\displaystyle=\int_{\mathbb{S}^{2}}\int_{\Omega[v,I]}\,\Big{(}f(v^{\prime},I^{\prime})f(v^{\prime}_{*},I^{\prime}_{*})-f(v,I)f(v_{*},I_{*})\Big{)}\;\frac{1}{e^{2}}|v-v_{*}|^{-1}\,B(\cdot,\omega)\circ\Phi^{-1}(\cdot)\,\frac{1}{\varphi(I)}\;\text{d}v_{*}\,\text{d}I_{*}\,\text{d}I^{\prime}\,\text{d}I^{\prime}_{*}\,\text{d}\omega$
$\displaystyle=\int_{\mathbb{S}^{2}}\int_{\tilde{\Omega}[v,I]}\,\Big{(}f(v^{\prime},I^{\prime})f(v^{\prime}_{*},I^{\prime}_{*})-f(v,I)f(v_{*},I_{*})\Big{)}\;|v-v_{*}|^{-1}\,B(\cdot,\omega)\frac{J[\Phi](\cdot)}{e^{2}}\,\frac{1}{\varphi(I)}\;\text{d}v_{*}\,\text{d}I_{*}\,\text{d}R\,\text{d}r\,\text{d}\omega\,$
$\displaystyle=\iint_{[0,1]^{2}}\int_{\mathbb{R}_{+}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},I^{\prime})f(v^{\prime}_{*},I^{\prime}_{*})-f(v,I)f(v_{*},I_{*})\Big{)}\;|v-v_{*}|^{-1}\,(1-R)\,B(\cdot,\omega)\,\frac{1}{\varphi(I)}\;\text{d}\omega\,\text{d}v_{*}\,\text{d}I_{*}\,\text{d}R\,\text{d}r\,,$

where we recall that $J[\Phi](v,v_{*},I,I_{*},r,R)=(1-R)e^{2}$ is the Jacobian
of $\Phi$. We denoted by $\Omega[v,I]$ the set of
$(v_{*},I_{*},I^{\prime},I^{\prime}_{*})$ such that
$(v,v_{*},I,I_{*},I^{\prime},I^{\prime}_{*})\in\Omega$ and the same for
$\tilde{\Omega}$. We recover the same macroscopic quantities as well, for
example the density

$\rho(t,x)=\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}mf(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}^{3}}mf(t,x,v,I)\,\text{d}v\,\varphi(I)\,\text{d}I.$

###### Remark 7.1

The approach in borgnakke1975statistical ; desvillettes2005kinetic  consists
in distributing the energy of the incoming molecules simultaneously to the
post-collision relative kinetic energy and to the internal states, whereas, in
our framework, we fix the pre and post-collision internal states and consider
afterwards the set of pre-collision velocity pairs for which this collision is
possible.

This model is generally used with $\varphi(I)=I^{\alpha}$, with $\alpha>-1$.
From Proposition 6.1, we recover that in this case ($\beta=1$ in the
proposition), the number of internal degrees of freedom and heat capacity at
constant volume are

$\delta=2(\alpha+1)\quad\text{and}\quad c_{V}=\frac{5}{2}+\alpha.$

#### The model with discrete energy levels

In order to accurately describe the vibration inside molecules, a model with a
discrete-energy levels description is proposed by Groppi and Spiga in
groppi1999kinetic . The authors consider a finite set of internal energy
levels $(\varepsilon_{n})_{n\in\llbracket 0,N\rrbracket}\in\mathbb{R}^{N+1}$.
The state of a molecule is then $(v,n)\in\mathbb{R}^{3}\times\llbracket
0,N\rrbracket$, and the associated total energy is
$\displaystyle\frac{m}{2}|v|^{2}+\varepsilon_{n}$. The authors define
$\delta_{ij}^{kl}=\frac{4}{m}(\varepsilon_{k}+\varepsilon_{l}-\varepsilon_{i}-\varepsilon_{j})$,
$g=|v-v_{*}|$ and $g^{\prime}=|v^{\prime}-v^{\prime}_{*}|$. A collision is
allowed when $g^{2}-\delta_{ij}^{kl}\geq 0$, with then
$g^{\prime}=\sqrt{g^{2}-\delta_{ij}^{kl}}$.

To recover this model in our framework, we set $\mathcal{E}=\llbracket
0,N\rrbracket$, $\mathcal{A}=\mathcal{P}(\llbracket 0,N\rrbracket)$, $\mu$ the
counting measure on $\llbracket 0,N\rrbracket$ and for $n\in\llbracket
0,N\rrbracket$, $\varepsilon(n)=\varepsilon_{n}$. Indeed, note that

$g^{2}-\delta_{ij}^{kl}\geq
0\iff\frac{1}{4}|v-v_{*}|^{2}+\frac{1}{m}(\varepsilon_{i}+\varepsilon_{j}-\varepsilon_{k}-\varepsilon_{l})\geq
0\iff\Delta(v,v_{*},i,j,k,l)\geq 0,$

where $\Delta$ is defined in equation (2.6). Moreover, the authors give
(formula 2.6 in groppi1999kinetic )

$\text{d}v^{\prime}\,\text{d}v^{\prime}_{*}\,\text{d}\Omega^{\prime}=\mathbf{1}_{g^{2}\geq\delta_{ij}^{kl}}\,\frac{g^{\prime}}{g}\,\text{d}v\,\text{d}v_{*}\,\text{d}\Omega,$

which corresponds in our framework to the Jacobian of the transformation
$(v,v_{*})\mapsto(v^{\prime},v^{\prime}_{*})$, see formula (2.11). The authors
in groppi1999kinetic  give the following post-collision velocities

$\displaystyle v^{\prime}$
$\displaystyle=\frac{v+v_{*}}{2}+\frac{1}{2}\sqrt{g^{2}-\delta_{ij}^{kl}}\,\Omega$
$\displaystyle v^{\prime}_{*}$
$\displaystyle=\frac{v+v_{*}}{2}-\frac{1}{2}\sqrt{g^{2}-\delta_{ij}^{kl}}\,\Omega,$

where $\Omega\in\mathbb{S}^{2}$. Now remark that
$\frac{1}{2}\sqrt{g^{2}-\delta_{ij}^{kl}}=\sqrt{\Delta(v,v_{*},i,j,k,l)}$ and
there exists (assuming $v\neq v_{*}$) $\omega\in\mathbb{S}^{2}$ such that
$\Omega=T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]$. We then have, like
in our framework,

$\displaystyle v^{\prime}$
$\displaystyle=\frac{v+v_{*}}{2}+\sqrt{\Delta(v,v_{*},i,j,k,l)}\,\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right]$
$\displaystyle v^{\prime}_{*}$
$\displaystyle=\frac{v+v_{*}}{2}-\sqrt{\Delta(v,v_{*},i,j,k,l)}\,\,T_{\omega}\left[\frac{v-v_{*}}{|v-v_{*}|}\right].$

The authors in groppi1999kinetic  define the cross-section $\sigma_{ij}^{kl}$,
with the symmetry and micro-reversibility conditions

$\displaystyle\sigma_{ij}^{kl}(v,v_{*},\omega)$
$\displaystyle=\sigma_{ji}^{lk}(v_{*},v,\omega)$
$\displaystyle|v-v_{*}|^{2}\sigma_{ij}^{kl}(v,v_{*},\omega)$
$\displaystyle=|v^{\prime}-v^{\prime}_{*}|^{2}\sigma^{kl}_{ij}(v^{\prime},v^{\prime}_{*},\omega).$

By defining the collision kernel

$b(v,v_{*},i,j,k,l,\omega)=\mathbf{1}_{g^{2}\geq\delta_{ij}^{kl}}\,|v-v_{*}|\sigma_{ij}^{kl}(v,v_{*},\omega),$

$b$ verifies the positivity assumption (if $\sigma_{ij}^{kl}$ is properly
defined) and from the properties assumed for $\sigma_{ij}^{kl}$ we recover

$\displaystyle b(v,v_{*},i,j,k,l,\omega)$
$\displaystyle=b(v_{*},v,j,i,l,k,\omega)$
$\displaystyle|v-v_{*}|b(v,v_{*},i,j,k,l,\omega)$
$\displaystyle=|v^{\prime}-v^{\prime}_{*}|b(v^{\prime},v^{\prime}_{*},k,l,i,j,\omega).$

Finally, with $(\zeta,\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})=(i,j,k,l)$,

$\displaystyle\mathcal{B}(f,f)(v,\zeta)$
$\displaystyle=\iiint_{\mathcal{E}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},\zeta^{\prime})f(v^{\prime}_{*},\zeta^{\prime}_{*})-f(v,\zeta)f(v_{*},\zeta_{*})\Big{)}\;b(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\text{d}\mu^{\otimes
3}(\zeta_{*},\zeta^{\prime},\zeta^{\prime}_{*})$
$\displaystyle=\sum_{j,k,l\in\llbracket
0,N\rrbracket}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\Big{(}f(v^{\prime},k)f(v^{\prime}_{*},l)-f(v,i)f(v_{*},j)\Big{)}\;b(v,v_{*},i,j,k,l,\omega)\;\text{d}\omega\,\text{d}v_{*}$
$\displaystyle=\sum_{j,k,l\in\llbracket
0,N\rrbracket}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\mathbf{1}_{g^{2}\geq\delta_{ij}^{kl}}\,|v-v_{*}|\,\sigma_{ij}^{kl}(v,v_{*},\omega)\,\Big{(}f_{k}(v^{\prime})f_{l}(v^{\prime}_{*})-f_{i}(v)f_{j}(v_{*})\Big{)}\;\text{d}\omega\,\text{d}v_{*},$

with the notation $f_{i}(v)=f(v,i)$. The same macroscopic quantities are
recovered as well, for instance the density

$\rho(t,x)=\int_{\mathcal{E}}\int_{\mathbb{R}^{3}}mf(t,x,v,\zeta)\,\text{d}v\,\text{d}\mu(\zeta)=\sum_{i=0}^{N}\int_{\mathbb{R}^{3}}mf_{i}(t,x,v)\,\text{d}v.$

###  7.2 Combination of the continuous and discrete models

The continuous and discrete models can be combined in our framework. Let us
consider a finite set of internal energy levels
$(\epsilon_{n})_{n\in\llbracket 0,N\rrbracket}\in\mathbb{R}^{N+1}$ and a
weight function $\varphi$ on $\mathbb{R}_{+}$. We can build the following
model

$(\mathcal{E},\mathcal{A},\text{d}\mu(\zeta))=\left(\mathbb{R}_{+}\times\llbracket
0,N\rrbracket,\,\text{Bor}(\mathbb{R}_{+})\otimes\mathcal{P}(\llbracket
0,N\rrbracket),\,\varphi(I)\,\text{d}I\times
1\right),\quad\quad\varepsilon(I,n)=I+\epsilon_{n},$ (7.1)

where $1$ stands for the counting measure. The Boltzmann operator writes

$\displaystyle\mathcal{B}(f,f)(v,I,i)=$ $\displaystyle\sum_{j,k,l\in\llbracket
0,N\rrbracket}\iiint_{(\mathbb{R}_{+})^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\,\Big{(}f_{k}(v^{\prime},I^{\prime})f_{l}(v^{\prime}_{*},I^{\prime}_{*})-f_{i}(v,I)f_{j}(v_{*},I_{*})\Big{)}\;b(\cdot)\;\text{d}\omega\,\text{d}v_{*}\,\varphi(I_{*})\,\text{d}I_{*}\,\varphi(I^{\prime})\,\text{d}I^{\prime}\,\varphi(I^{\prime}_{*})\,\text{d}I^{\prime}_{*}\,.$

Let us set $\epsilon^{0}=\underset{0\leq n\leq
N}{\text{min}}\\{\epsilon_{n}\\}$. From Theorem 6.1, the Maxwellian writes

$\mathcal{M}[\rho,u,T](v,I,i)=\rho\,m^{1/2}\,(2\pi
k_{B}T)^{-3/2}\,Z_{c}\left(\frac{1}{k_{B}T}\right)^{-1}\,Z_{d}\left(\frac{1}{k_{B}T}\right)^{-1}\,\text{exp}\left(-\frac{m|v-u|^{2}}{2k_{B}T}-\frac{I}{k_{B}T}-\frac{\epsilon_{i}-\epsilon^{0}}{k_{B}T}\right),$

where $Z_{c}$ and $Z_{d}$ are the partition functions respectively associated
to the continuous and discrete parts, for all $\beta>0$

$Z_{c}(\beta)=\int_{\mathbb{R}_{+}}e^{-\beta
I}\,\varphi(I)\,\text{d}I,\quad\quad
Z_{d}(\beta)=\sum_{n=0}^{N}e^{-\beta(\epsilon_{n}-\epsilon^{0})}.$

Finally, again from Theorem 6.1, the number of internal degrees of freedom
writes

$\delta(T)=\delta_{c}(T)+\delta_{d}(T)=\frac{2}{k_{B}T}\,\frac{\int_{\mathbb{R}_{+}}I\,\text{exp}\left(-\frac{I}{k_{B}T}\right)\,\varphi(I)\,\text{d}I}{\int_{\mathbb{R}_{+}}\text{exp}\left(-\frac{I}{k_{B}T}\right)\,\varphi(I)\,\text{d}I}+\frac{2}{k_{B}T}\,\frac{\sum_{n=0}^{N}(\epsilon_{n}-\epsilon^{0})\,\text{exp}\left(-\frac{(\epsilon_{n}-\epsilon^{0})}{k_{B}T}\right)}{\sum_{n=0}^{N}\text{exp}\left(-\frac{(\epsilon_{n}-\epsilon^{0})}{k_{B}T}\right)}.$

As seen previously, if $\varphi(I)=I^{\alpha}$ with $\alpha>-1$, then
$\delta_{c}=2(\alpha+1)$. This model with internal energy described by two
different variables, a continuous and a discrete one, may be used to separate
the vibrational energy of a molecule from the rotational part. As suggested in
Herzberg , the rotational energy may be modelled by means of a continuous
variable, while for the vibrational energy a discrete approximation is more
suitable. Note that the number of internal degrees of freedom turns out to be,
as expected, the sum of the vibrational and the rotational ones.

###  7.3 Building a model within the framework

In this subsection, we explain how to build a model for a polyatomic gas
within our framework, with an example. Building an accurate model corresponds
to giving an accurate description of the internal states of a given molecule.
On this subject, the main field to rely on is Molecular Mechanics/Quantum
Chemistry, see for example hehre2003guide . There are various phenomena to be
taken into account, typically rotation, vibration, electronic excitation and
nuclei spin, that could also be correlated. For the sake of simplicity of the
mathematical model, it could be enough to take only rotation and vibration
into account, and assume them to be uncorrelated, but there are physical
problems where this kind of description is too simplistic. The most
appropriate model in a given situation typically depends on the considered
regime and the desired degree of complexity and accuracy. This is the reason
why we wish not to propose one model, but to give examples and insights on how
to build realistic models inside our framework.

#### Quantum description of the internal structure

To our knowledge, the currently best internal description of a molecule is
given by quantum mechanics. Suppose to have a Hamiltonian operator $\hat{H}$.
Let $\\{\lambda_{q}\\}_{q\in Q}$ be the set of its (real) eigenvalues, where
$Q$ is a discrete set (e.g. $\mathbb{N},\mathbb{N}^{2},\dots$). We denote by
$r_{q}$ the dimension of the eigenvector space associated to the eigenvalue
$\lambda_{q}$. The values $\lambda_{q}$ are the possible measurable energy
levels associated to the operator $\hat{H}$, and $r_{q}$ is the so–called
degeneracy of the $q^{\text{th}}$ energy level. We then consider a subset
$\tilde{Q}\subset Q$, the set of energy levels of the _bound states_ (the
physically admissible energy levels), and build the following model

$(\mathcal{E},\mathcal{A},\mu)=(\tilde{Q},\mathcal{P}(\tilde{Q}),(r_{q})_{q}),\qquad\varepsilon(q)=\lambda_{q}\,,$

where the notation $\mu=(r_{q})_{q}$ means that $\mu(\\{q\\})=r_{q}$. As we
explained in subsection 3.2, degeneracy is indeed included in the model via
the measure $\mu$.

We remark that if the set of bound states $\tilde{Q}=Q$ and
inf$\\{\lambda_{q}\\}_{q\in Q}=0$, then we recover the formula of statistical
mechanics for the partition function (see huang1963statistical , Chapter 9)

$Z(\beta)=\sum_{q\in
Q}e^{-\beta\lambda_{q}}\,r_{q}=\text{Tr}(e^{-\beta\hat{H}}),$

where $\text{Tr}(e^{-\beta\hat{H}})$ is the trace of the operator
$e^{-\beta\hat{H}}$.

###### Remark 7.2

The framework we present in this paper, with the Boltzmann equation 3.1 is "at
most" semi-classical, in the sense that the velocity part is described by
classical mechanics. A full quantum mechanical Boltzmann equation was derived
by Waldmann Wa57  and Snider Sn60 . In our framework, we assume that the gas
is isotropic (no polarization effects), and other quantum considerations would
only appear in the internal structure
$(\mathcal{E},\mathcal{A},\mu),\,\varepsilon$ of the molecule.

#### Semi-classical description of the internal structure

A possible approach is to consider only rotation and vibration, describing
independently rotation with classical mechanics and vibration with quantum
mechanics. The description of vibration is thus exactly the model of the
previous paragraph, with a Hamiltonian describing only vibration,

$(\mathcal{E}_{vib},\mathcal{A}_{vib},\mu_{vib})=(\tilde{Q},\mathcal{P}(\tilde{Q}),(r_{q})_{q}),\qquad\varepsilon_{vib}(q)=\lambda_{q}.$

We assume the molecule to be a rigid rotor, that is, in the description of
rotation we assume that no deformation is induced in the molecule. The
rotation-related internal state we consider is thus the angular velocity of
the molecule in a coordinate system attached to the molecule. In general, this
angular velocity lives in $\mathbb{R}^{3}$, however when the molecule is
linear, being symmetric by rotation around its own axis, the contribution of
the angular momentum’s coordinate along this axis can be assumed to be 0 (this
approximation can be found for example in van1951coupling , section 7). Thus,
in the case of a linear molecule, the angular momentum, in a coordinate system
attached to the molecule with one axis being the molecule’s axis, would live
in $\mathbb{R}^{2}$. Moreover, in the linear case, the moment of inertia is
the same in both directions. This way of modelling leads to the following
framework for the description of rotation

$(\mathcal{E}_{rot},\mathcal{A}_{rot},\text{d}\mu_{rot}(z))=\left(\mathbb{R}^{d},\text{Bor}(\mathbb{R}^{d}),\text{d}z\right),\qquad\varepsilon_{rot}(z)=\frac{1}{2}\sum_{i=1}^{d}\mathcal{I}_{i}\,z_{i}^{2},$

where $\mathcal{I}_{i}$ is the moment of inertia along the $i$-th axis, $z$ is
the angular velocity in a coordinate system attached to the molecule, $d=2$
and $\mathcal{I}_{1}=\mathcal{I}_{2}=\mathcal{I}$ if the molecule is linear,
$d=3$ if the molecule is non-linear. The model taking rotation and vibration
into account then writes

$(\mathcal{E},\mathcal{A},\text{d}\mu(\zeta))=\big{(}\mathbb{R}^{d}\times\tilde{Q},\,\text{Bor}(\mathbb{R}^{d})\otimes\mathcal{P}(\tilde{Q}),\,\text{d}z\,r_{q}\big{)},\quad\quad\varepsilon(z,q)=\frac{1}{2}\sum_{i=1}^{d}\mathcal{I}_{i}\,z_{i}^{2}+\lambda_{q},$

From Theorem 6.1, for all $T>0$,

$\text{d}\nu_{T}(z,q)=\text{d}\nu^{rot}_{T}(z)\,\text{d}\nu^{vib}_{T}(q)=\prod_{i=1}^{d}\frac{\text{exp}\left(-\frac{\mathcal{I}_{i}\,z_{i}^{2}}{2k_{B}T}\right)}{\left(\frac{2\pi
k_{B}T}{\mathcal{I}_{i}}\right)^{1/2}}\,\text{d}z\,\,\frac{\text{exp}\left(-\frac{(\lambda_{q}-\lambda^{0})}{k_{B}T}\right)}{\sum_{q^{\prime}\in\tilde{Q}}\text{exp}\left(-\frac{(\lambda_{q^{\prime}}-\lambda^{0})}{k_{B}T}\right)\,r_{q^{\prime}}}\,r_{q}.$

Moreover, setting
$\displaystyle\lambda^{0}=\underset{q\in\tilde{Q}}{\text{inf}}\\{\lambda_{q}\\}$
($\in\mathbb{R}$ by assumption) and using Theorem 6.1 and Proposition 6.1, the
number of internal degrees of freedom turns out to be

$\delta(T)=\delta_{rot}(T)+\delta_{vib}(T)=d+2\,\frac{\sum_{q\in\tilde{Q}}\frac{(\lambda_{q}-\lambda^{0})}{k_{B}T}\,\text{exp}\left(-\frac{(\lambda_{q}-\lambda^{0})}{k_{B}T}\right)\,r_{q}}{\sum_{q\in\tilde{Q}}\text{exp}\left(-\frac{(\lambda_{q}-\lambda^{0})}{k_{B}T}\right)\,r_{q}}.$

#### Example: the $\prescript{1}{}{\emph{H}}\prescript{19}{}{\emph{F}}$ gas

As an example, we propose four models within our framework for the Hydrogen
Fluoride ($\prescript{1}{}{\text{H}}\prescript{19}{}{\text{F}}$) gas. We focus
only on rotation and vibration. We propose two semi-classical models and two
quantum models. For the formulas of eigenvalues presented hereafter, we refer
to the Chapter Spectroscopy Constants of Diatomic Molecules in crchandbook .

1\. Harmonic semi-classical model. We describe the internal states of
$\prescript{1}{}{\text{H}}\prescript{19}{}{\text{F}}$ with a semi-classical
approach. Since this molecule is diatomic, it is linear, so that $d=2$ and
$\mathcal{I}_{1}=\mathcal{I}_{2}$ in the description of rotation, and has only
one mode of vibration. In this simplified model, the Hamiltonian is assumed to
include the harmonic potential (model of the quantum harmonic oscillator).
This leads to the model

$(\mathcal{E},\mathcal{A},\text{d}\mu(\zeta))=\left(\mathbb{R}^{2}\times\mathbb{N},\,\text{Bor}(\mathbb{R}^{2})\otimes\mathcal{P}(\mathbb{N}),\,\text{d}z\times
1\right),\qquad\varepsilon(z,n)=\underset{\text{rotation}}{\underbrace{\frac{1}{2}\mathcal{I}\,|z|^{2}}}+\underset{\text{vibration}}{\underbrace{hc\,\nu_{e}\left(n+\frac{1}{2}\right)}},$

where $\mathcal{I}$ is the moment of inertia associated with
$\prescript{1}{}{\text{H}}\prescript{19}{}{\text{F}}$, $h$ is the Planck
constant (expressed in SI units), $c$ the speed of light (expressed in
$\text{cm}.s^{-1}$) and $\nu_{e}$ the wavenumber (expressed in
$\text{cm}^{-1}$). Defining $T_{vib}=hc\,\nu_{e}/k_{B}$, the number of
internal degrees of freedom writes

$\delta_{1}(T)=\delta^{rot}_{1}(T)+\delta^{vib}_{1}(T)=2+2\,\frac{T_{vib}/T}{\text{exp}(T_{vib}/T)-1}\,.$

2\. Anharmonic semi-classical model. Instead of using the harmonic potential,
the Morse potential is a useful approximation of the actual internuclear
potential, since it allows anharmonicity and an explicit computation of the
eigenvalues of the Hamiltonian. The family of eigenvalues is, up to a constant
(see Morse morse1929diatomic ),

$\left(hc\,\nu_{e}\left(n+\frac{1}{2}\right)-hc\,\nu_{e}x_{e}\left(n+\frac{1}{2}\right)^{2}\right)_{n\in\mathbb{N}}$

where $x_{e}$ represents anharmonicity. Not all eigenvalues correspond to
bound states. We thus restrict the family to the set $\llbracket
0,N_{max}\rrbracket$, where $\displaystyle
N_{max}=\left\lfloor\frac{1}{2x_{e}}\right\rfloor-1$. With this potential, the
model becomes

$\displaystyle(\mathcal{E},\mathcal{A},\text{d}\mu(\zeta))$
$\displaystyle=\left(\mathbb{R}^{2}\times\llbracket
0,N_{max}\rrbracket,\,\text{Bor}(\mathbb{R}^{2})\otimes\mathcal{P}(\llbracket
0,N_{max}\rrbracket),\,\text{d}z\times 1\right),$
$\displaystyle\varepsilon(z,n)$
$\displaystyle=\frac{1}{2}\mathcal{I}\,|z|^{2}+hc\,\nu_{e}\left(n+\frac{1}{2}\right)-\underset{\text{anharmonicity}}{\underbrace{hc\,\nu_{e}x_{e}\left(n+\frac{1}{2}\right)^{2}}}.$

The number of internal degrees of freedom writes

$\delta_{2}(T)=\delta^{rot}_{2}(T)+\delta^{vib}_{2}(T)=2+\delta^{vib}_{2}(T).$

3\. Simplified quantum model. Here we use a quantum description for both
rotation and vibration. To simplify, we describe independently rotation and
vibration, using the rigid-rotor assumption for rotation and the harmonic
potential for vibration. Since
$\prescript{1}{}{\text{H}}\prescript{19}{}{\text{F}}$ is a diatomic molecule,
there is only one mode of vibration. This leads to the model

$(\mathcal{E},\mathcal{A},\mu)=\left(\mathbb{N}\times\mathbb{N},\,\mathcal{P}(\mathbb{N})\otimes\mathcal{P}(\mathbb{N}),\,(2J+1)_{(J,n)\in\mathbb{N}^{2}}\right),\qquad\varepsilon(J,n)=\underset{\text{rotation}}{\underbrace{BJ(J+1)}}+\underset{\text{vibration}}{\underbrace{hc\,\nu_{e}\left(n+\frac{1}{2}\right)}},$

where $B$ is the rotational constant associated with
$\prescript{1}{}{\text{H}}\prescript{19}{}{\text{F}}$. Defining
$T_{vib}=hc\,\nu_{e}/k_{B}$, the number of internal degrees of freedom writes

$\delta_{3}(T)=\delta^{rot}_{3}(T)+\delta^{vib}_{3}(T)=\delta^{rot}_{3}(T)+2\,\frac{T_{vib}/T}{\text{exp}(T_{vib}/T)-1}.$

4\. Improved quantum model. For a better description, we can also take into
account the correlation of rotation and vibration, considering non-rigid
rotation, and using the Morse potential for vibration to allow for
anharmonicity. This would lead to the model

$\displaystyle(\mathcal{E},\mathcal{A},\mu)$
$\displaystyle=\left(\tilde{Q},\,\mathcal{P}(\tilde{Q}),\,(2J+1)_{(J,n)\in\tilde{Q}}\right),$
$\displaystyle\varepsilon(J,n)$ $\displaystyle=\underset{\text{rot-vib
coupling}}{\underbrace{\left(B-\alpha\left(n+\frac{1}{2}\right)\right)}}J(J+1)-\underset{\text{centrifugal
distortion}}{\underbrace{D[J(J+1)]^{2}}}+hc\,\nu_{e}\left(n+\frac{1}{2}\right)-\underset{\text{anharmonicity}}{\underbrace{hc\,\nu_{e}x_{e}\left(n+\frac{1}{2}\right)^{2}}},$

where $\tilde{Q}$ is the set of bound states, defined by

$\tilde{Q}=\left\\{(J,n)\in\mathbb{N}^{2},\;\varepsilon(J,n)\geq\varepsilon(J-1,n)\;\text{
and }\;\varepsilon(J,n-1)\geq\varepsilon(J,n)\right\\}\subset\llbracket
0,J_{max}\rrbracket\times\llbracket 0,N_{max}\rrbracket.$

We denote by $\delta_{4}$ the number of internal degrees of freedom in this
improved case, which cannot be written as the sum of rotational and