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\markboth{\hfil Homogeneous Boltzmann equation \hfil EJDE--2003/Mon.~04}
{EJDE--2003/Mon.~04\hfil M. Escobedo, S. Mischler, \& M. A. Valle \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Monogrpah 04, 2003. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Homogeneous Boltzmann equation in quantum relativistic kinetic theory
 %
\thanks{ {\em Mathematics Subject Classifications:} 82B40, 82C40, 83-02.
\hfil\break\indent
{\em Key words:}  Boltzmann equation, relativistic particles, 
entropy maximization,  \hfil\break\indent
Bose distribution, Fermi distribution, Compton scattering, Kompaneets equation.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted  November 29, 2002. Published January 20, 2003.} }
\date{}
%
\author{Miguel Escobedo, St\'ephane Mischler, \& Manuel A. Valle}
\maketitle

\begin{abstract} 
\noindent
 We consider some mathematical questions about Boltzmann 
 equations for quantum  particles, relativistic or non 
 relativistic.  Relevant particular cases such as Bose, 
 Bose-Fermi, and  photon-electron gases are studied. 
 We also consider some simplifications such as the
 isotropy of the distribution functions and the asymptotic 
 limits (systems where one of the species is at equilibrium). 
 This gives rise to interesting mathematical questions from a 
 physical point of view. New results are presented about the 
 existence  and long time behaviour of the solutions to some 
 of these problems. 
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}

\tableofcontents



\section{Introduction} %1

When quantum methods are applied to molecular encounters, some
divergence from the classical results appear. It is then necessary in some
cases to modify the classical theory in order to account for the quantum
effects which are present in the collision processes;
see  \cite[Sec. 17]{CC}, where the domain of applicability of the classical
kinetic theory is discussed in detail.
In spite of their formal similarity, the equations for classical and quantum
kinetic theory display very different features. Surprisingly,
the appropriate Boltzmann equations, which
account for quantum effects, have received scarce attention in the
mathematical literature.

In this work, we consider some mathematical questions about Boltzmann
equations for quantum particles,  relativistic and  not relativistic.
The general interest in different models involving that kind of equations has
increased recently. This is so because they are supposedly reliable for 
computing
non equilibrium properties of Bose-Einstein condensates
on sufficiently large times and distance scales;
see for example \cite{JP,ST1,ST2} and references therein. We study some
relevant particular cases (Bose, Bose-Fermi, photon-electron gases),
simplifications such as the isotropy of the distribution functions,
and asymptotic limits (systems where one of the species is at
equilibrium) which are important from a physical point of view and give
rise to interesting mathematical questions.

Since quantum and classic or relativistic
particles are involved, we are lead to consider such a
general type of equations.
We first consider the
homogeneous Boltzmann equation for a quantum gas constituted by a single
specie of particles, bosons or fermions. We solve the
entropy maximization  problem under the moments constraint
in the general quantum relativistic case.
The question of the well posedness, i.e. existence, uniqueness, stability of
solutions and of the long time behavior of the solutions is also treated in some
relevant particular cases. One could also consider other qualitative
properties such as regularity, positivity, eternal solution in a purely
kinetic perspective or  study the relation between the
Boltzmann equation and the underlying quantum field theory,
 or a more phenomenological description, such as
the based on hydrodynamics, but we do not go further in these directions.

\subsection{The Boltzmann equations} %1.1

To begin with, we focus our attention on a gas composed of
identical and indiscernible particles.
When two particles with respective momentum $p$ and $p_*$ in
$\mathbb{R}^3$ encounter each other, they collide and we denote $p'$ and
$p'_*$ their new momenta  after the collision. We assume that the
collision is elastic, which means that the total momentum and the total
energy of the system  constituted by this pair of particles are conserved.
More precisely, denoting by $\mathcal{E}(p)$ the energy of one particle
with momentum $p$, we assume that
$$
\begin{gathered}
p' + p'_* = p + p_* \\
\mathcal{E}(p') + \mathcal{E}(p'_*) = \mathcal{E}(p) + \mathcal{E}(p_*).
\end{gathered} \eqno(1.1)
$$
We denote $\mathcal{C}$ the set of all $4$-tuplets of particles
$(p,p_*,p',p'_*) \in \mathbb{R}^{12}$ satisfying (1.1).
The expression of the energy $\mathcal{E}(p)$ of a particle in function of its
momentum $p$ depends on the type of the particle;
$$
\begin{aligned}
&\mathcal{E}(p) = \mathcal{E}_{nr}(p) = {|p|^2 \over 2 \, m} \quad \hbox{for a non
relativistic particle,} \\
&\mathcal{E}(p) = \mathcal{E}_r(p) = \gamma mc^2; \,\gamma = {\sqrt{1+(|p|^2/c^2m^2)}}
\quad \hbox{for a relativistic particle, }\\
&\mathcal{E}(p) = \mathcal{E}_{ph}(p) = c|p| \quad
\hbox{for massless particle such as a photon or neutrino.}
\end{aligned} \eqno(1.2)
$$
Here, $m$ stands for the mass of the particle and $c$ for the velocity of
light. The velocity $v = v(p)$ of a particle with momentum $p$ is
defined by
$v(p) = \nabla_p \mathcal{E}(p)$, and therefore
$$
\begin{aligned}
v(p) &= v_{nr}(p) = {p \over m} \quad \hbox{for a non relativistic
particle,} \\
v(p) &= v_r(p) = {p \over m \gamma}  \quad \hbox{for a relativistic
particle,}\\
v(p) &= v_{ph}(p) = c {p \over |p|} \quad \hbox{for a photon .}
\end{aligned}  \eqno(1.3)
$$

Now we consider a gas constituted by a very large number (of order the
Avogadro number $A \sim 10^{23}/\hbox{mol}$) of a
single specie of identical and indiscernible particles. The very large
number of particles makes  impossible
(or irrelevant) the knowledge of the position
and momentum $(x,p)$ (with
$x$ in a domain $\Omega\subset \mathbb{R}^3$ and $p \in \mathbb{R}^3$) of every 
particle of
the gas. Then, we introduce $f = f(t,x,p) \ge 0$,
the gas density distribution of particles which at
time $t \ge 0$ have position $x \in
\mathbb{R}^3$ and momentum $p \in \mathbb{R}^3$. Under the hypothesis of 
molecular chaos and
of low density of the gas, so that particles collide by pairs  (no collision 
between three or more particles
occurs),  Boltzmann \cite{B} established that
the evolution of a classic (i.e. no quantum nor relativistic) gas  density
$f$ satisfies
$$
\begin{gathered}
{\partial f \over \partial t}  + v(p) \cdot \nabla_x f = Q(f(t,x,.))(p) \\
f(0,.) = f_{\rm in},
\end{gathered} \eqno(1.4)
$$
where $f_{\rm in} \ge 0$ is the initial gas distribution and $Q(f)$ is the
so-called Boltzmann collision
kernel. It describes the change of the momentum of the particles due
to the collisions.

A similar equation was proposed by  Nordheim \cite{N} in 1928  and by
Uehling \&  Uhlenbeck \cite{UU} in 1933 for the description of a quantum
gas, where only the collision term $Q(f)$ had to be changed to take into account 
the quantum
degeneracy of the particles.
The relativistic generalization of the Boltzmann equation including the
effects of collisions was given by  Lichnerowicz and
Marrot \cite{LM} in 1940.


Although this is by no means a review article we may nevertheless give some 
references for the
interested readers. For the classical Boltzmann equation we refer to  Villani's 
recent review
\cite{V} and the rather complete bibliography therein.
Concerning the relativistic kinetic theory, we refer to the monograph \cite{St} 
by J. M. Stewart
and the classical expository text \cite{GLW} by  Groot, Van Leeuwen and  Van 
Weert. A
mathematical  point of view, may be found in the books by Glassey \cite{G}
and Cercignani and  Kremer \cite{CK}. In \cite{J},  J\" uttner gave the
relativistic equilibrium distribution. Then,  Ehlers
in \cite{E}, Tayler \&  Weinberg in \cite{TW} and  Chernikov in \cite{Ch} proved  
the H-theorem for the
relativistic Boltzmann equation. The existence of global classical solutions for 
data close to equilibrium
is shown by R. Glassey and W. Strauss in \cite{GS1}.
The asymptotic stability of the equilibria is studied in
\cite{GS1}, and \cite{GS2}.  For these questions see also the book \cite{G}.
The global existence of renormalized solutions
is proved by Dudynsky and  Ekiel Jezewska in \cite{DE}.
The asymptotic behaviour of the global solutions is
also considered in \cite{An}.

In all the following we make the assumption that the density $f$ only
depends on the momentum. The collision term $Q(f)$ may then be expressed in all 
the
cases described above as
$$
\begin{gathered}
Q(f)(p)=\iiint_{\mathbb{R} ^9} W(p, p_{*}, p', p'_{*})q(f)\, dp_{*}dp'dp'_{*}
\\
q(f)\equiv q(f)(p, p_{*}, p', p'_{*})=[f'f'_{*}(1+\tau f)(1+\tau f_{*})
-ff_{*}(1+\tau f')(1+\tau f'_{*})]\\
\tau \in \{-1, 0, 1\},
\end{gathered} \eqno{(1.5)}
$$
where as usual, we denote:
$$ f=f(p),\quad  f_{*}=f(p_{*}),\quad f'=f(p'),\quad f'_{*}=f(p'_{*}),
$$
and $W$ is a non negative measure called transition rate, which may be written 
in
general as:
$$
W(p, p_{*}, p', p'_{*})=w(p, p_{*}, p', p'_{*})\delta
(p+p_{*}-p'-p'_{*})\delta (\mathcal{E}(p)+\mathcal{E}(p_{*})-\mathcal{E}(p')-
\mathcal{E}(p'_{*}))\eqno{(1.6)}
$$
where $\delta $ represents the Dirac measure. The quantity $Wdp'dp_*'$
is the probability for the initial
state $|p, p_{*}\rangle$ to scatter and become a final state of two particles
whose momenta lie in a small region $dp'\, dp_{*}'$.


The character relativistic or not, of the particles  is taken into
account in the expression of the energy of the particle $\mathcal{E}(p)$
given by (1.2).  The effects due to quantum degeneracy are included in the term 
$q(f)$ when $\tau \not
=0$, and depend on the bosonic or fermionic character of the involved particles. 
These are associated
with the fact that, in quantum mechanics, identical particles cannot be 
distinguished,
not even in principle. For dense gases at low temperature, this kind of terms 
are crucial. However, for non
relativistic dilute gases, quantum degeneracy plays no role and can be safely 
ignored ($\tau =0$).


The function $w$ is directly related to the differential cross section $\sigma $ 
(see (5.11)), a quantity
that is intrinsic to the colliding particles and the kind of interaction between 
them. The calculation of
$\sigma $ from the underlying interaction potential is a central problem in non 
relativistic quantum
mechanics, and  there are a few examples of isotropic interactions (the Coulomb 
potential, the delta
shell,\dots) which have an exact solution. However, in a complete relativistic 
setting or when many-body
effects due to collective dynamics lead to the screening of interactions, the 
description of these in terms
of a potential is impossible. Then, the complete framework of quantum field 
theory (relativistic or not)
must be used in order to perform perturbative computations of the involved 
scattering cross section in $w$.
We give some explicit examples in the Appendix \ref{A3} but let us only mention 
here the case
$w=1$ which corresponds to a non relativistic short range interaction
(see Appendix \ref{A3}). Since the particles
are indiscernible, the collisions are reversible and the two interacting 
particles form a closed
physical system.  We have then:
$$
\begin{aligned}
&W(p, p_{*}, {p'}, {p'_{*}})=W(p_{*}, p, {p'}, {p'_{*}})=W({p'}, {p'_{*}}, p, p_{*})\\
&+\quad   \hbox{Galilean invariance (in the non relativistic case)}\\
&+\quad   \hbox{Lorentz invariance (in the relativistic case).}
\end{aligned} \eqno{(1.7)}
$$
To give a sense to the expression (1.5) under general assumptions on the 
distribution $f$ is not a
simple question in general. Let us only  remark here that
$Q(f)$ is well defined as a measure when $f$ and $w$ are assumed to be 
continuous. But we will
see below that this is not always a reasonable assumption. It is one
of the purposes of this work to clarify this question in part.

The Boltzmann equation reads then very similar, formally at least, in all
the different contexts: classic, quantum and relativistic. In particular
some of the fundamental physically relevant properties of the solutions
$f$ may be formally established in all the cases in the same way:
conservation of the total number of particles, mean impulse and total energy; 
existence of an
``entropy function'' which increases along the trajectory (Boltzmann's
H-Theorem).  For any $\psi = \psi(p)$, the symmetries (1.7), imply the
fundamental and elementary identity
$$
\int_{\mathbb{R}^3} Q(f) \psi \, dp = \frac 14 \iiint_
{\mathbb{R}^{12}} W(p, p_{*}, p', p'_{*}) \,
 \,  q(f)\bigl(\psi + \psi_* - \psi' - \psi'_* \bigr) \, dp dp_* dp' dp'_*.
  \eqno(1.8)
$$
Taking $\psi(p) =1$, $\psi (p)= p_i$ and $\psi (p)=\mathcal{E}(p)$ and using the
definition of $\mathcal{C}$, we obtain that the particle number,
the momentum and the energy of a solution $f$ of the Boltzmann equation (1.5) 
are
conserved along the trajectoires, i.e.
$$
{d \over dt} \int_{\mathbb{R}^3} f(t,p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp =
\int_{\mathbb{R}^3} Q (f) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) \end{pmatrix} 
\, dp = 0,
\eqno(1.9)
$$
so that
$$
\int_{\mathbb{R}^3} f(t,p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp =
\int_{\mathbb{R}^3} f_{\rm in}(p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp.
\eqno(1.10)
$$
The entropy functional is defined by
$$
H(f) := \int_{R^3} h(f(p)) \, dp,
\quad h(f) = \tau^{-1} (1 + \tau f) \ln
(1 + \tau f) - f \ln f. \eqno(1.11)
$$
Taking in (1.8) $\psi = h'(f) = \ln (1 + \tau f) - \ln f$, we get
$$
\int_{\mathbb{R}^3} Q(f) \, h'(f) \, dp = \frac14  \, D (f)
\eqno(1.12)
$$
with
$$
\begin{aligned}
D(f) &= \iiiint_{\mathbb{R}^{12}} W \, e(f) \, dp dp_* dp' dp'_* \\
e(f) &= j\bigl( f f_* (1 + \tau \,f') (1 + \tau \,f'_*) , f' f'_* (1 + \tau \,f) 
(1 + \tau\,f_*) \\
j(s,t) &= (t-s) (\ln \, t - \ln \, s) \ge 0.
\end{aligned} \eqno(1.13)
$$
We deduce from the equation that the entropy is increasing along trajectories,
i.e.
$$
{d \over dt} H (f(t,.))  = \frac14  D(f) \ge 0.\eqno(1.14)
$$

The main qualitative characteristics of $f$ are described by these two
properties: conservation (1.9) and
increasing entropy (1.14). It is therefore natural to expect that as $t$
tends to $\infty$ the function $f$ converges to a function
$f_\infty$ which
 realizes the maximum of the entropy
$H(f)$ under the moments constraint (1.10).
A first simple and heuristic remark is that if $f_\infty$ solves the
entropy maximization problem with
constraints (1.10), there exist Lagrange multipliers $\mu \in \mathbb{R}$,
$\beta^0 \in \mathbb{R}$ and $\beta \in
\mathbb{R}^3$ such that
$$
\langle \nabla H(f_\infty), \varphi\rangle
=  \int_{\mathbb{R}^3} h'(f_\infty) \varphi \, dp  =
\langle \beta^0 \mathcal{E}(p) - \beta \cdot p - \mu,\varphi\rangle \quad
 \forall \varphi,
$$
which implies
$$
\ln (1 + \tau f_\infty) - \ln f_\infty = \beta^0 \mathcal{E}(p) - \beta
\cdot p - \mu
$$
and therefore
$$
f_\infty (p) = {1 \over e^{\nu(p)} - \tau}\quad
\hbox{with }\nu(p):=\beta^0 \mathcal{E}(p) - \beta \cdot p - \mu \,.
\eqno{(1.15)}
$$
The function $f_{\infty }$ is called a Maxwellian when $\tau = 0$, a
Bose-Einstein distribution when
$\tau > 0$ and a Fermi-Dirac distribution when $\tau < 0$.

\subsection{The classical case} %1.2

Let us consider for a moment the case $\tau =0$, i.e. the classic
Boltzmann equation, which has been widely studied.
It is known that for any initial data $f_{\rm in}$  there exists
a unique distribution $f_\infty$ of the form (1.15) such that
$$
\int_{\mathbb{R}^3} f_{\infty }(p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp =
\int_{\mathbb{R}^3} f_{\rm in}(p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp.
\eqno(1.16)
$$

We may briefly recall the main results about the Cauchy problem and
the long time behaviour of the solutions which are
known up to now. We refer to \cite{C,L},
for a more detailed exposition and their proofs.

\begin{theorem}[Stationary solutions] \label{thm1}
For any measurable function $f \ge 0$ such that
$$
\int_{\mathbb{R}^3} f (1, p, {|p|^2 \over 2} ) \, dp = (N,P,E)
\eqno(1.17)
$$
for some $N,E > 0$, $P \in \mathbb{R}^3$, the following four assertions are 
equivalent:
\begin{itemize}
\item[(i)] $f$ is the Maxwellian
$${\mathcal{M}_{N,P,E} = \mathcal{M}[\rho,u,\Theta] = {\rho \over (2 \pi 
\Theta)^{3/2}}\exp \bigl( - {|p-u|^2 \over 2 \Theta} \bigr) }
$$
where $(\rho,u,\Theta)$ is  uniquely determined by
$N = \rho$, $P = \rho u$, and $E = {\rho \over 2} (|u|^2 +3 \Theta)$;

\item[(ii)] $f$ is the solution of the maximization problem
$$
H(f) = \max \{ H(g), \ g \hbox{ satisfies the moments equation } (1.10) \},
$$
where ${ H(g) = - \int_{\mathbb{R}^3} g \log g \, dp}$ stands for
the classical entropy;
\item[(iii)] $Q(f) = 0$;
\item[(iv)] $D(f) = 0$.
\end{itemize}
\end{theorem}

Concerning the evolution problem one can prove.

\begin{theorem} \label{thm2}
Assume that $w = 1$ (for simplicity). For any
initial data $f_{\rm in} \ge 0$ with finite
number of particles, energy and entropy, there exists a unique global solution 
$f \in
C([0,\infty);L^1(\mathbb{R}^3))$ which
conserves the particle number, energy and momentum.
Moreover, when $t \to \infty$, $f(t,.)$ converges to the Maxwellian $M$
with same particle number, momentum an
energy (defined by Theorem \ref{thm1}) and more precisely, for any $m > 0$ there
exists $C_m = C_m(f_0)$ explicitly
computable such that
$$
\| f - M \|_{L^1} \le {C_m \over (1+t)^m}.\eqno{(1.18)}
$$
\end{theorem}

We refer to \cite{Ar1,E,MW,Lu} for existence, conservations and
uniqueness and to \cite{Ar2,V,W,C,TV} for convergence to the equilibrium.
Also note that Theorem \ref{thm2} can be extended (sometimes only partially) to
a large class of cross-section $W$
we refer to \cite{V} for details and references.

\begin{remark} \label{rmk1.1} \rm
The proof of the equivalence (i) - (ii) only involves the entropy $H(f)$
and not the collision integral $Q(f)$ itself.
\end{remark}
\begin{remark} \label{rmk1.2} \rm
To show that (i), (iii) and (iv) are equivalent one
has first to define the quantities $Q(f)$ and $D(f)$ for the functions $f$
belonging to the physical functional space. The first difficulty is to
define precisely the collision integral $Q(f)$, (see Section 3.2).
\end{remark}

\subsection{Quantum and/or relativistic gases} %1.3

The Boltzmann equation looks formally
very similar in the different contexts: classic, quantum and
relativistic, but it actually presents some very different features in each of 
these different contexts.
 The two following remarks give some insight on these differences.

The natural spaces for the density $f$ are the
spaces of distributions $f \ge 0$ such that the ``physical" quantities are
bounded:
$$
\int_{\mathbb{R}^3} f (1 + \mathcal{E}(p)) \, dp < \infty
\quad\hbox{ and }\quad
H(f) < \infty,\eqno{(1.19)}
$$
where $H$ is given by (1.11). This provides the following  different conditions:
$$
\begin{aligned}
&f\in L^1_s\cap {L\log L}\quad \hbox{in the non quantum case, relativistic or
not}\\
&f\in L^1_s\cap L^{\infty }\quad \hbox{in the Fermi case, relativistic or not}\\
&f\in L^1_s \quad \hbox{in the Bose case, relativistic or not,}
\end{aligned}
$$
where
$$
L^1_s=\{f\in L^1(\mathbb{R}^3); \int_{\mathbb{R}^3} \,(1 + |p|^s)\,d|f|(p) 
<\infty\}
\eqno{(1.20)}$$
and $s = 2$ in the non relativistic case, $s = 1$ in the relativistic case.


On the other hand, remember that the density entropy $h$ given by (1.11) is:
$$
h(f) = \tau^{-1} (1 + \tau f)  \ln  (1 + \tau  f) - f  \ln  f.
$$
In the Fermi case we have $\tau =-1$ and then  $h(f) = +\infty$ whenever
$f \notin [0,1]$. Therefore the estimate $H(f)<\infty $ provides a strong
$L^\infty$ bound on $f$. But, in the Bose case,  $\tau =1$. A simple calculus 
argument then
shows that  $h(f)\sim \ln f$ as $f\to \infty $. Therefore  the entropy estimate 
$H(f)<\infty $ does not gives
any additional bound on $f$.

Moreover, and still concerning the Bose case, the following is shown in
\cite{CL}, in the context of the
Kompaneets equation (cf. Section 5). Let $a\in \mathbb{R}^3$ be any fixed vector 
and
$(\varphi _n)_{n\in \mathbb{N}}$  an approximation of the identity:
$$
(\varphi _n)_{n\in \mathbb{N}};\quad \varphi _n\to \delta _a.
$$
Then for any $f\in L^1_2$, the quantity
$H(f+\varphi _n)$ is well defined by (1.11) for all $n\in
\mathbb{N}$ and moreover,
$$N(f+\alpha \varphi _n)\to N(f)+\alpha ,\quad \hbox{and}\quad H(f+\varphi 
_n)\to
H(f)\quad \hbox{as}\quad n\to \infty.\eqno{(1.21)}
$$
See Section 2 for the details. This indicates that the expression of $H$ given 
in (1.11) may be extended to
nonnegative measures and that, moreover, the singular part of the measure does 
not contributes to the
entropy. More precisely, for any non negative measure
$F$ of the form $F=gdp+G$, where
$g\ge 0$ is an integrable function and
$G\ge 0$ is singular with respect to the Lebesgue measure $dp$, we define the 
Bose-Einstein entropy of $F$ by
$$
H(F) := H(g)= \int_{\mathbb{R}^3} \bigl[ (1+g) \ln (1+g) - g \ln g
\bigr] \, dp.\eqno{(1.22)}
$$

The discussion above shows how different is the quantum from the non quantum 
case, and even the Bose from the
Fermi case. Concerning the Fermi gases, the Cauchy
problem has been studied by  Dolbeault \cite{D} and Lions \cite{L}, under the 
hypothesis (H1) which includes the hard sphere
case
$w=1$. As it is indicated by the remark above, the estimates at our disposal in 
this case are even better than
in the classical case. In particular the collision term $Q(f)$ may be defined in 
the same way as in the
classical case. But as far as we know,
no analogue of Theorem \ref{thm1} was known for Fermi gases. The problem for 
Bose gases is
essentially open as we shall see below. Partial results for radially symmetric 
$L^1$
distributions have been obtained by Lu \cite{Lu} under strong cut off 
assumptions on the
function
$w$.

\subsubsection{Equilibrium states, Entropy} %1.3.1


As it is formally indicated by the identity (1.9), the particle number, momentum 
and energy of the
solutions to the Boltzmann equation are conserved along the trajectories. It is 
then very
natural to consider the following entropy maximization problem: given $N>0$, 
$P\in \mathbb{R}^3$ and $E\in \mathbb{R}$, find a
distribution $f$ which maximises the entropy $H$ and whose moments are $(N, P, 
E)$. The solution of this
problem is well known in the non quantum non relativistic case ( and is recalled 
in
Theorem \ref{thm1} above). In \cite{J},
 J\" uttner in [Ju] gave the relativistic Maxwellians.
 The question is also treated by  Chernikov in \cite{Ch}.
For the complete resolution of the moments equation in the relativistic non 
quantum case we refer to
 Glassey \cite{Gl} and  Glassey \& W. Strauss [GS].
 We solve the quantum relativistic case  in \cite{EMV}.
The general result may be stated as follows.

\begin{theorem} \label{thm3}
For every possible choice of $(N, P, E)$ such that the set $K$ defined by
$$
g\in K\quad \hbox{if and only if}\quad  \int_{\mathbb{R}^3} g (1, p, {|p|^2 
\over 2} ) \, dp = (N,P,E),
$$
is non empty, there exists a unique solution
$\overline{f}$ to the entropy maximization problem
$$
\overline{f}\in K,\quad H(\overline{f})=\max\{H(g); \, g\in K\}.
$$
Moreover,
$\overline{f}=f_{\infty }$ given by (1.15) for the nonquantum and Fermi case,
while for the Bose case $\overline{f}=f_{\infty }+\alpha
\delta _{\overline{p}}$ for some $\overline{p}\in \mathbb{R}^3$.
\end{theorem}

It was already observed by Bose and Einstein \cite{B, E1, E2} that for systems 
of bosons in thermal equilibrium  a careful
analysis of the statistical physics of the problem leads to enlarge the class of 
steady distributions to include also the
solutions containing a Dirac mass. On the other hand, the strong uniform bound 
introduced by the Fermi
entropy over the Fermi distributions leads to include in the family of Fermi 
steady states the so called degenerate
states. We present in Section 2 the detailed mathematical results of these two 
facts both for relativistic
and non relativistic particles. The interested reader may find the detailed
proofs in \cite{EMV}.

\subsubsection{Collision kernel, Entropy dissipation, Cauchy Problem} % 1.3.2

Theorem \ref{thm3} is the natural extension to quantum particles of the results 
for  non quantum particles, i.e.
points (i) and (ii) of Theorem \ref{thm1}. The extension of the points (iii) and 
(iv), even for the non relativistic
case, is more delicate. In the Fermi case it is possible to define the collision 
integral $Q(f)$ and the
entropy dissipation $D(f)$ and to solve the problem under some additional
conditions (see Dolbeault \cite{D} and Lions \cite{L}).
We consider this problem and related questions in Section 3.2.


In the equation for bosons, the first difficulty is to define the collision 
integral $Q(f)$
and the entropy dissipation
$D(f)$ in a sufficiently general setting. This question was treated by Lu in 
\cite{Lu} and solved under the
following additional assumptions:
\begin{itemize}
\item[(i)] $f\in L^1$ is radially symmetric.

\item[(ii)] Strong truncation on $w$.
\end{itemize}

These two conditions are introduced in order to give a sense to the collision 
integral.
Unfortunately, the second one is
not satisfied by the main physical examples such as $w=1$ (see Appendix 
\ref{A3}).
Moreover Theorem \ref{thm2.3} shows that the
natural framework to study the quantum Boltzmann equation, relativistic or
not, for Bose gases is the space of non negative measures. This is an
additional difficulty with respect to the non quantum or Fermi
cases. We partly extend the study of Lu to the case where $f$ is a
non negative radially symmetric measure.

\subsection{Two species gases, the Compton-Boltzmann equation} %1.4

Gases composed of two different species of particles,for example bosons and 
fermions, are interesting
by themselves for physical reasons and have thus been considered in the physical 
literature (see the
references below). On the other hand, from a mathematical point of vue, they 
provide simplified
but still interesting versions of Boltzmann equations for quantum particles. 
Their study may
be then a first natural step to understand the behaviour of this type of 
equations.


Let us then call
$F(t,p) \ge 0$ the density of Bose particles and $f(t,p) \ge 0$ that
of Fermi particles. Under the low density assumption, the evolution of the gas
is now given by the following system of Boltzmann equations (see \cite{CC}):
$$
\begin{gathered}
{\partial F \over \partial t} = Q_{1,1}(F,F) + Q_{1,2}(F,f) \quad F(0,.)
= F_{\rm in} \\
{\partial f \over \partial t} = Q_{2,1}(f,F) + Q_{2,2}(f,f) \quad f(0,.)
= f_{\rm in}.
\end{gathered} \eqno(1.23)
$$
The collision terms $Q_{1,1}(F,F)$ and $Q_{2,2}(f,f)$ stand for collisions
between particles of the same specie and are given by (1.2).
The collision terms $Q_{1,2}(F,f)$ and $Q_{2,1}(f,F)$ stands for collisions
between particles of  two different species:
$$
\begin{gathered}
Q_{1,2}(F,f) =  \int_{\mathbb{R}^3}  \int_{\mathbb{R}^3}
 \int_{\mathbb{R}^3} W_{1,2} q_{1,2}\, dp_* dp' dp'_*,\\
q_{1,2}= F' f'_* (1+F) (1 - \tau  \,f_*)
-  F f_* (1 + F') (1 - \tau  f'_*),
\end{gathered}\eqno(1.24)
$$
and $Q_{2,1}(f,F)$ is given by a similar expression. Note that the measure
$W_{1,2}=W_{1,2}(p,p_*,p',p'_*)$ satisfies
the micro-reversibility hypothesis
$$
W_{1,2}(p',p'_*,p,p_*) = W_{1,2}(p,p_*,p',p'_*),
\eqno(1.25)
$$
but not the indiscernibility hypothesis $W_{1,2}(p_*,p,p',p'_*) =
W_{1,2}(p,p_*,p',p'_*)$ as in (1.7) since the two colliding  particles belong 
now
to different species.


We consider in Section 4 some mathematical questions related to the systems
(1.23)-(1.25). We do not perform in detail the general
study of the steady states
since that would be mainly a repetition of what is done in Section 3 and
in \cite{EMV}. Here again, to give a sense to the integral collision $Q_{1,2}$
and $Q_{2,1}$ is the first question to be considered. Since the kernels
$Q_{1,1}(F,F)$ and
$Q_{2,2}(f,f)$ have already been treated in
the precedent section, we focus in the collision terms
$Q_{1,2}(F,f)$ and $Q_{2,1}(f,F)$. Let us notice that in the Fermi case,
$\tau  > 0$, the Fermi density $f$ satisfies an a priori bound in
$L^\infty$. Nevertheless, even with this extra estimate,the problem of existence 
of
solutions and their asymptotic behaviour for generic interactions, even with
strong unphysical truncation kernel and for radially symmetric distributions 
$f$, remains an open
question.


In order to get some insight on these problems, we
consider two simpler situations which are  important from a physical point
of view and still mathematically interesting since, in particular, they display 
Bose condensation in infinite time.
These are the equations describing boson-fermion interactions with fermions at 
equilibrium, and
photon-electron Compton scattering. Of course the deductions of
these two reduced models are well known in the physical literature but we 
believe
nevertheless that it may be interesting to sketch them here. In the first one,
still considered in Section 4, we suppose that the Fermi particles are at rest 
at
isothermal equilibrium. This is nothing but to fix the distribution of
Fermi particles $f$ in the system to be a Maxwellian or a Fermi state.
Without any loss of generality, this may be chosen to be centered at the origin, 
so
that it is radially symmetric. Moreover,
the boson-fermion interaction is  short range and  we
may  consider the ``slow particle interaction'' approximation of the
differential cross section $w=1$ (see Appendix \ref{A3}). The
system reduces then to a single equation which moreover is quadratic and not
cubic. Namely:
$$
{\partial F \over \partial t} =
\int_0^\infty  S (\varepsilon,\varepsilon') \bigl[ F' (1 + F) \,
e^{-\epsilon} - F (1 + F')
\, e^{-\epsilon'} \bigr] \, d\epsilon',\eqno{(1.26)}
$$
for some kernel S (see Section 4).

We prove in Section 4 the following result about existence, uniqueness and 
asymptotic
behaviour of global solutions for the Cauchy problem associated to (1.26) .

\begin{theorem} \label{thm4}
For any initial datum $F_{\rm in} \in L^1_1(\mathbb{R}_+)$, $F_{\rm in}\ge 0$,
there exists a solution $F \in C([0,\infty),L^1_{1/2}))$ to the equation
(1.26) such that
$$
\lim_{t\to 0}\|F(t)-F_{\rm in}\|_{L^1(\mathbb{R}^+)}=0.
$$
Moreover, if $\overline{f}=\mathcal{B}_N$ is the unique solution to the 
maximization
problem
\begin{gather*}
H(\overline{f})=\max\{H(g); \int \,g(\varepsilon)\,\varepsilon^2\,d\varepsilon= 
\int
\,F_{\rm in}(\varepsilon)\,\varepsilon^2\,d\varepsilon=: N\},\,\\
H(F)=\int_0^{\infty }h(f,
\varepsilon)\varepsilon^2d\varepsilon
\end{gather*}
with $h(x, \varepsilon)=(1+x)\ln (1+x)-x\ln x -\varepsilon x$,
$$
\begin{gathered}
 F(t,.) \mathop{\rightharpoonup}_{t \to \infty} \overline{f}
 \hbox{ weakly} \star \hbox{ in } \ \bigl( C_c({\mathbb{R}_+}) \bigr)' \\
\lim_{t\to \infty } \| F(t,.)-\overline{f} \|_{L^1((k_0,\infty))}=0 \quad
\forall k_0 > 0.
\end{gathered}\eqno(4.51)
$$
\end{theorem}

This result shows that the density of bosons $F$ underlies a Bose condensation
asymptotically in infinite time if its initial value is large enough.
The phenomena was already predicted by
Levich \& Yakhot in \cite{LY1, LY2}, and was described as condensation driven by 
the interaction
of bosons with a cold bath (of fermions) (see also Semikoz \& Tkachev
\cite{ST1,ST2}).

\subsubsection{Compton scattering}

In Section 5 we consider the equation describing the photon-electron interaction 
by
Compton scattering. This equation, that we call Boltzmann-Compton
equation, has been extensively studied in the physical and mathematical 
literature
(see in particular the works by Kompaneets \cite{K}, Dreicer \cite{D},
 Weymann \cite{W}, Chapline, Cooper and  Slutz \cite{CCS}). We show how it can
be derived starting from the system which describes the photon electron
interaction via Compton scattering. This interaction is described, in
the non relativistic limit, by the Thomson cross section, (see Appendix 
\ref{A3}).


It is important in this case to start with
the full relativistic quantum formulation since photons are relativistic
particles. Even if, later on, the electrons are considered at non
relativistic classical equilibrium. Finally, The equation has the same form as 
in (1.26) where
the only difference lies in the kernel
$S$.


The possibility of some kind of ``condensation'' for this Compton Boltzmann 
equation was
already considered in physical literature by  Chapline,  Cooper and  Slutz in 
\cite{CCS},
and  Caflisch \&  Levermore \cite{CL} for the Kompaneets equation
(see Section 5 and also \cite{EHV}).


We end with the so called Kompaneets equation, which is the limit of the 
Boltzmann-Compton equation
in the range $|p|, |p'|\ll mc^2$.

\section{The entropy maximization problem} % 2

In this Section we describe the solution to the maximization problem for
the entropy function under the moment
constraint. This problem may be stated as follows.
\begin{quote}
 Given any of the entropies $H$ and of the energies
$\mathcal{E}$ defined in the introduction,
given three quantities $N > 0$, $E > 0$ and  $P \in \mathbb{R}^3$, find $F
\ge 0$ such that
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) \end{pmatrix} F(p) 
\, dp =
\begin{pmatrix} N \\ P \\ E \end{pmatrix}
\eqno(2.1)
$$
and
$$
H(F) = \max\{ H(g): g \mbox{ satisfies  (2.1)}\}. \eqno(2.2)
$$
\end{quote}
We  consider successively the case of a relativistic
non quantum gas, then the case of a Bose-Enstein gas, and last the
case of a Fermi-Dirac gas. For each of these two kind of gases,
we first consider in detail the
relativistic case, where the energy is given by
$$
\mathcal{E}(p) =  \sqrt{ 1\,  + {|p|^2\over c^2m^2}}.
\eqno(2.3)
$$
Then we describe the non relativistic case, $\mathcal{E}(p) = |p|^2/2m$, which 
is
simplest since by Galilean invariance it can be reduced to $P=0$.


In order to avoid lengthy technical details which are unnecessary for our 
purpose here, we
only present the results for each of the different cases. For the detailed
proofs the reader is referred to \cite{EMV}.


The relativistic non quantum case was completely solved by Glassey and
Strauss in \cite{GS1}, see also  Glassey in [Gl, even in
the non homogeneous case with periodic spatial dependence. However,
the proof that we give in \cite{EMV} is different, uses in a crucial way the 
Lorentz
invariance and may be adapted to the quantum relativistic case. Finally, notice 
that we do
not consider this entropy problem for a gas of photons ($\mathcal{E}(p) = |p|$ 
and
$H$  the Bose-Einstein entropy) since it would not have  physical meaning. In
Section 5 we discuss the entropy
problem for a gas constituted of electrons and photons.

\subsection{Relativistic non quantum gas} %2.1

In this subsection, we consider the Maxwell-Boltzmann entropy
$$
H(g) = - \int_{\mathbb{R}^3} g  \ln g \, dp.
\eqno(2.4)
$$
of a non quantum gas. From the heuristic argument presented in the
introduction, the solution to
(2.1)-(2.2) is expected to be a relativistic Maxwellian distribution:
$$
\mathcal{M}(p) = e^{ - \beta^0 p^0 + \beta \cdot p - \mu }.
\eqno(2.5)
$$
The result is the following.

\begin{theorem} \label{thm2.1} \begin{itemize}
\item[(i)] Given $E,N > 0$, $P \in \mathbb{R}^3$, there
exists a least one function $g \ge 0$
which solves the moments equation (2.1) if, and only if,
$$
m^2 c^2 \, N^2 + |P|^2 < E^2.
\eqno(2.6)
$$
When (2.6) holds we will say that $(N,P,E)$ is admissible.

\item[(ii)] For an admissible $(N,P,E)$ there exists at least one
relativistic Maxwellian distribution $\mathcal{M}$  satisfying (2.1).

\item[(iii)] Let $\mathcal{M}$ be a relativistic Maxwellian distribution. For 
any
function $g \ge 0$ satisfying
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, g(p) \, dp =
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \mathcal{M}(p) 
\, dp, \eqno(2.7)
$$
one has
$$
H (g) - H (\mathcal{M}) = H(g|\mathcal{M}) := \int_{\mathbb{R}^3} [g \ln {g 
\over \mathcal{M}} - g + \mathcal{M}
] \, dp. \eqno(2.8)
$$
Moreover, $H(g|\mathcal{M}) \le 0$ and vanishes if, and only if, $g = 
\mathcal{M}$.

\item[(iv)] As a conclusion, for an admissible $(N,P,E)$, the relativistic 
Maxwellian
constructed in (ii)  is the
unique solution to the entropy maximization problem (2.1)-(2.4).
\end{itemize}
\end{theorem}

\subsection{Bose gas} %2.2

We consider now a gas of Bose particles. As it has been said before, Bose 
\cite{B}
and Einstein \cite{E1,E2}, noticed that in this case, the set of steady 
distributions had to include solutions
containing a Dirac mass. It is then necessary to extend the entropy function $H$ 
defined in (1.11) with
$\tau = 1$ to such distributions. The way to do this may be well understood with 
the
following remark from \cite{CL}.

Let $a\in \mathbb{R}^3$ be any fixed vector and $(\varphi _n)_{n\in \mathbb{N}}$  
an
approximation of the identity:
$$
(\varphi _n)_{n\in \mathbb{N}};\quad \varphi _n\to \delta _a.
$$
For any $f\in L^1_2$ and every $n\in \mathbb{N}$ the quantity $H(f+\varphi _n)$
is well defined by (1.11). Moreover,
$$
H(f+\varphi _n)\to H(f)\quad \hbox{as}\quad n\to \infty.
$$
Suppose, for the sake of simplicity
that $\varphi _n\equiv 0$ if $|p-a|\ge 2/n$, then
$$
H(f+ \varphi _n)=\int_{|p-a|\ge 2/n}h(f(p, t), p)\,dp
+\int_{|p-a|\le 2/n}h((f(p, t)+ \varphi_n(p), p)\,dp.
$$
Let us call, $\sigma (z)=(1+z)\ln(1+z)-z\ln z$.
Using $|\sigma (z)|\le c\sqrt{z}$ we obtain
$$
\int_{|p-a|\le 2/n}|\sigma (f(p, t)+ \varphi_n(p))|dp
\le c{2\over \sqrt{ n^3}}\bigl(
\int_{|p-a|\le 2/n}(f(p, t)+ \varphi_n(p)dp\bigr)^{1/2}\to 0.
$$
Then
$$
\big|\int_{\mathbb{R}^3}h(f(p, t), p)dp-\int_{|p-a|\ge 2/n}\! h(f(p, t), 
p)dp\big|
\le \int_{|p-a|\le 2/n}|h(f(p, t), p)|dp\to 0
$$
which completes the proof.

This indicates that the expression of $H$ given in (1.11) may be extended to 
nonnegative measures and
that the singular part of the measure does not contributes to the entropy. More 
precisely,
for any non negative measure
$F$ of the form
$F=gdp+G$, where
$g\ge 0$ is an integrable function and
$G\ge 0$ is singular with respect to the Lebesgue measure $dp$,
we define the Bose-Einstein entropy of
$F$ by
$$
H(F) := H(g)= \int_{\mathbb{R}^3} \bigl[ (1+g) \ln (1+g) - g \ln g
\bigr] \, dp.\eqno{(2.9)}
$$
On the other hand, as we have seen in the Introduction, the
regular solutions to the entropy maximization problem should be the Bose
relativistic distributions
$$
b(p) = {1 \over  e^{ \nu(p) } - 1 }
\quad \hbox{ with } \quad
\nu(p) = \beta^0 p^0 - \beta \cdot p + \mu.
\eqno(2.10)
$$

The following result explains where the Dirac masses have now to be placed
(see \cite{EMV} for the proof).

\begin{lemma} \label{lm2.2}
The Bose relativistic distribution $b$ is non
negative and belongs to $L^1(\mathbb{R}^3)$
if, and only if, $\beta^0 > 0$, $|\beta| < \beta^0$ and $\mu \ge \mu_\mathbf{b}$ 
with
$\mu_\mathbf{b} := - m c \mathbf{b}$, $\mathbf{b} > 0$ and $\mathbf{b}^2= 
(\beta^0)^2 - |\beta|^2$.
In this case, all the moments of $b$ are well defined. Finally,
$$
\nu(p) > \nu(p_{m,c}) \ge 0 \quad \forall p \not= p_{m,c} :=
{m c \beta \over \sqrt{\beta^{0 \, 2} - |\beta|^2}}.
\eqno(2.11)
$$
\end{lemma}

We define now the generalized Bose-Einstein relativistic distribution 
$\mathcal{B}$ by
$$
\mathcal{B}(p) = b + \alpha \delta_{p_{m,c}}
= {1 \over  e^{ \nu(p) } - 1 } + \alpha \delta_{p_{m,c}}
\eqno(2.12)
$$
with
$$
\nu(p) = \beta^0 p_0 - \beta \cdot p + \mu > \nu(p_{m,c}) \ge 0
\quad \forall p \not= p_{m,c},
\eqno(2.13)
$$
and the condition $\alpha \nu(p_{m,c}) = 0$.


\begin{theorem} \label{thm2.3}
\begin{itemize}
\item[(i)] Given $E,N > 0$, $P \in \mathbb{R}^3$, there
exists at least one measure $F \ge 0$
which solves the moments equation (2.1) if, and only if,
$$
m^2 c^2 \, N^2 + |P|^2 \le E^2.
\eqno(2.14)
$$
When (2.14) holds we will say that $(N,P,E)$ is a admissible.

\item[(ii)] For any admissible $(N,P,E)$ there exists at least one
relativistic Bose-Einstein distribution $\mathcal{B}$ satisfying (2.1).

\item[(iii)]  Let $\mathcal{B}$ be a relativistic  Bose-Einstein distribution. 
For any
measure $F \ge 0$ satisfying
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, dF(p) =
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, 
d\mathcal{B}(p),
\eqno(2.15)
$$
one has
$$
H(F) - H(\mathcal{B}) = H_1(g|b) + H_2(G|b)
\eqno(2.16)
$$
where
$$
\begin{gathered}
H_1(g|b) := \int_{\mathbb{R}^3} \bigl( (1+g)  \ln {1+g \over 1+b} - g  \ln {g
\over b} \bigr) \, dp\,,\\
H_2(G|b) := - \int_{\mathbb{R}^3} \nu(p) \, dG(p). \end{gathered}
\eqno(2.17)
$$
Moreover, $H(F|\mathcal{B}) \le 0$ and vanishes if, and only if, $F = 
\mathcal{B}$. In
particular,
$H(F) < H(\mathcal{B})$ if $F \not= \mathcal{B}$.

\item[(iv)] As a conclusion, for an admissible $(N,P,E)$,
the relativistic Bose-Einstein distribution constructed in (ii) is the
unique solution to the entropy maximization problem
(2.1)-(2.3), (2.9).
\end{itemize}\end{theorem}


\subsubsection{Nonrelativistic Bose particles}
For non relativistic particles  the energy is
$\mathcal{E}(p)=|p|^2/2m$. By
Galilean invariance the problem (2.1) is then equivalent to the following 
simpler one: given three
quantities
$N>0$,
$E>0$, $P\in \mathbb{R}^3$ find $F(p)$ such
that
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p-{P\over N} \\ |p-{P\over N}|^2/(2m)
\end{pmatrix}  F(p) \, dp =
\begin{pmatrix} N \\ 0 \\ E-(|P|^2/(2mN)) \end{pmatrix}.\eqno{(2.18)}
$$
It is rather simple, using elementary calculus, to prove that for any $E, N>0$,
$P\in \mathbb{R}^3$ there exists a distribution of the from
$$
F(p)={1\over e^{a|p-{P\over N}|^2+b^+}-1}-b^-\delta _{{P\over N}}\eqno{(2.19)}
$$
with
$a\in \mathbb{R}$, $b\in \mathbb{R}$, $\nu \in \mathbb{R}$, $b^+=\max (b, 0)$, 
$b^-=-\max (-b, 0)$
which satisfies (2.18).
Once such a solution (2.18) of (2.19) is obtained, the following
Bose-Einstein distribution
$$
\mathcal{B}(p)={1\over e^{\nu (p)}-1}+\alpha \delta _{p_{m,c}},\quad
\nu (p)=a|p|^2-{2a\over N}\cdot p+(b^++{a|P|^2\over N^2}) \eqno{(2.20)}
$$
solves (2.1).
This shows that for non relativistic  particles Theorem \ref{thm2.3} remains 
valid
under the unique following change: the statements (i) and (ii) have to be
replaced by
\begin{itemize}
\item[(i')] For every $E, N>0$, $P\in \mathbb{R}^3$,
there exists one relativistic Bose Einstein distribution defined by (2.19)
corresponding to these moments, i.e. satisfying (2.1).
\end{itemize}
Statements (iii) and (iv) of Theorem \ref{thm2.3} remain unchanged.

\subsection{Fermi-Dirac gas} %2.3

In this subsection we consider the entropy maximization problem for the
Fermi-Dirac entropy
$$
H_{FD}(f) := - \int_{\mathbb{R}^3} \bigl( (1-f) \ln (1-f) + f \ln f \bigr) \, 
dp.
\eqno(2.21)
$$
In particular, this implies the constraint $ 0 \le f \le 1$ on the density
$f$ of the gas.

From the heuristics argument presented in the introduction, we know that
the solution $\mathcal{F}$ of
(2.1)-(2.3), (2.21) is the Fermi-Dirac distribution
$$
\mathcal{F}(p) = {1 \over  e^{ \nu(p) } + 1 }
\quad \hbox{ with } \quad
\nu(p) = \beta^0 p^0 - \beta \cdot p + \mu.
\eqno(2.22)
$$
We also introduce the ``saturated" Fermi-Dirac (SFD) density
$$
\chi(p) = \chi_{\beta^0,\beta}(p) = {\bf 1}_{ \{ \beta^0 p^0 - \beta
\cdot p \le 1 \} } =
{\bf 1}_\mathcal{E} \hbox{ with } \mathcal{E} = \{ \beta^0 p^0 - \beta \cdot p 
\le 1 \} ,
\eqno(2.23)
$$
with $\beta \in \mathbb{R}^3$ and $\beta^0 > |\beta|$.

Our main result is the following.

\begin{theorem} \label{thm2.4}
\begin{itemize}
\item[(i)] For any $P$ and $E$ such that $|P| < E$
there exists an unique SFD state
$\chi = \chi_{P,E}$ such that $P(\chi) = P$ and $E(\chi) = E$. This one
realizes the maximum of particle number for
given energy $E$ and mean momentum $P$. More precisely, for any $f$ such
that $0 \le f \le 1$ one has
$$
P(f) = P, \quad E(f) = E \quad\hbox{ implies }\quad N(f) \le N(\chi_{P,E}).
\eqno(2.24)
$$
As a consequence, given $(N,P,E)$ there exists $F$ satisfying the
moments equation (2.1) if, and only
if, $E > |P|$ and $0 \le N \le N(\chi_{P,E})$. In this case, we say that
$(N,P,E)$ is admissible.

\item[(ii)] For any $(N,P,E)$ admissible there exists a Fermi-Dirac state
$\mathcal{F}$ (``saturated" or not) which
solves the moments equation (2.1).

\item[(iii)] Let $\mathcal{F}$ be a Fermi-Dirac state. For any $f$ such that $0 
\le f
\le 1$ and
$$
\int_{\mathbb{R}^3} f(p) \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, dp =
\int_{\mathbb{R}^3} \mathcal{F}(p) \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} 
\, dp,
$$
one has
$$
\begin{aligned}
H_{FD} (f) - H_{FD} (\mathcal{F}) =& H_{FD} (f | \mathcal{F}) \\
:=&
\int_{\mathbb{R}^3} \bigl( (1-f) \ln { 1-f \over 1-\mathcal{F}} - f \ln {f 
\over \mathcal{F}
} \bigr) \, dp.
\end{aligned} \eqno(2.25)
$$

\item[(iv)] As a conclusion, for any admissible $(N,P,E)$, the
relativistic Fermi-Dirac distribution
constructed in (ii) is the unique solution to the entropy
maximization problem
(2.1)-(2.3), (2.21).
\end{itemize}
\end{theorem}

The new difficulty with respect to the classic or the Bose case is to be
managed with the constraint $0\le f\le 1$.

\subsubsection{Nonrelativistic Fermi-Dirac particles}
Here again, since the energy is $\mathcal{E}(p)=|p^2|/2m$ the problem (2.1)
is equivalent to (2.18): given three quantities
$N>0$, $E>0$, $P\in \mathbb{R}^3$ find $f(p)$ such
that $0\le f\le 1$ and satisfying (2.18).
One may then check that for non relativistic  particles Theorem \ref{thm2.4}
remains valid under the unique following change: the
statements (i) and (ii) have
to be replaced by
\begin{itemize}
\item[(i')] For every $E, N>0$, $P\in \mathbb{R}^3$, satisfying
$5E\ge 3^{5/3}(4\pi )^{2/3}\, N^{5/3}$
there exists a non relativistic fermi Dirac state, saturated or not, defined by
$$
\mathcal{F}(p)=\begin{cases}
{1\over e^{\nu (p)}+1}, \mbox{where }\\
\nu (p)=a|p|^2-{2a\over N}P\cdot p+(b+{a|P|^2\over N^2})
& \mbox{if }E>{3^{5/3}(4\pi )^{2/3}\over 5}N^{5/3},\\
{\bf 1}_{\{|p-{P\over N}|\le c\}}
&\hbox{if }  E={3^{5/3}(4\pi )^{2/3}\over 5}N^{5/3}
\end{cases}
$$
corresponding to these moments, i.e., satisfying (2.1).
\end{itemize}
Statements (iii) and (iv) of Theorem \ref{thm2.4} remain unchanged.

\section{The Boltzmann equation for one single specie of quantum particles} %3

We consider now the homogeneous Boltzmann equation for quantum non relativistic 
particles, and treat
both Fermi-Dirac and Bose-Einstein particles.  We begin with the Fermi-Dirac 
Boltzmann equation for
which we may slightly improve the existence result of  Dolbeault [JD]áand Lions 
\cite{L}.
We also state a very simple (and weak) result concerning the long time behavior 
of solutions.
We finally consider the Bose-Einstein Boltzmann equation. We discuss the work of 
Lu \cite{Lu} and
slightly extend some of its results to the natural framework of measures

Consider the non relativistic quantum Boltzmann equation
$$
\begin{gathered}
{\partial f \over \partial t}  = Q(f)
= \iiint_{\mathbb{R}^9} w \delta_{\mathcal{C}} q(f) \, dv_* dv' dv'_*, \\
q(f)=\bigl( f' f'_* (1 + \tau \,f) (1 + \tau \,f_*)-
f f_* (1 + \tau \,f') (1 + \tau \,f'_*) \bigr)
\end{gathered}\eqno(3.1)
$$
where $\tau = \pm 1$ and $\mathcal{C}$ is the non relativistic collision 
manifold
of $\mathbb{R}^{12}$, $(\mathcal{C})$:
$$
\begin{gathered}
v_* + v'_* = v + v_* \\
{|v_*|^2 \over 2} + {|v'_*|^2 \over 2}= {|v|^2 \over 2} + {|v_*|^2 \over 2}.\\
\end{gathered} \eqno(3.2)
$$
We assume, without any loss of generality in this Section that the mass
$m$ of the particles is one.

\subsection{The Boltzmann equation for Fermi-Dirac particles} %3.1

We first want to give a mathematical sense to the
collision operator $Q$ in (3.1) under the physical natural bounds on the
distribution $f$.
Of course, if $f$ is smooth (say $C_c(\mathbb{R}^3)$) and $w$ is smooth (for 
instance $w=1$) the collision term
$Q(f)$ is defined in the distributional sense as it has been mentioned in the 
Introduction.
But, as we have already seen, the physical space for the densities of Fermi-
Dirac particles
is  $L^1_2 \cap L^\infty(\mathbb{R}^3)$.

To give a pointwise sense to the formula (3.1), we first recall the following
elementary argument from \cite{Gl}. After integration with respect to the $v'_*$
variable in (3.1) we have
$$
Q(f) = \int \!\! \int_{\mathbb{R}^6} w q(f)  \delta_{ \{ {|v'|^2 \over 2} + 
{|v+v_*-v'|^2 \over 2} - {|v|^2
\over 2} - {|v_*|^2 \over 2} = 0 \} }  \, dv_* dv'.
$$
By the change of variable $v' \to (r,\omega)$ with
$r \in \mathbb{R}$, $\omega \in S^2$, and $v' = v + r \omega$,
and using Lemma \ref{A1.1}, we obtain
$$
\begin{aligned}
Q(f) &= {1 \over 2} \int_{\mathbb{R}^3} \!\! \int_{S^2} \!\! \int_0^\infty w 
q(f)
\delta_{ \{ r (r - (v_*-v) \cdot \omega )  = 0 \} } \,  r^2 \,dr d\sigma
dv_* \\
&=  \int_{\mathbb{R}^3} \!\! \int_{S^2} w {|(v_*-v,\omega)| \over 2} q(f)  \,
d\omega dv_*,
\end{aligned} \eqno(3.3)
$$
where $v'$ and $v'_*$ are defined by
$$
v' = v + (v_*-v,\omega)  \omega, \quad
v'_* = v_* - (v_*-v,\omega) \omega. \eqno(3.4)
$$
Formula (3.3) gives a pointwise sense to $Q(f)$, say for $f \in 
C_c(\mathbb{R}^3)$.

To extend the definition of $Q(f)$ to measurable functions we
make the following assumptions on the cross-section:
\begin{gather*}
B = {1 \over 2} w |(v_*-v,\omega)|
\ \hbox{ is a function of }v_* - v \hbox{ and } \omega ,\tag{3.5}\\
B \in L^1(\mathbb{R}^3 \times S^2). \tag{3.6}
\end{gather*}
Though (3.5) is a natural assumption in view of Section 3.1, (3.6) is a strong 
restriction, in particular it does not hold when $w = 1$.
With these assumptions, first introduced in \cite{D}, we now explain how to give 
a sense to the collision
term $Q(f)$ when $f \in L^1 \cap L^\infty$.
In one hand, since $\Phi \ : \  (v,v_*,\omega) \  \mapsto \ (v',v'_*,\omega)$ 
(with $v'$, $v'_*$ given by
(3.4)) is a  $C^1$-diffeomorphism on  $\mathbb{R}^3 \times \mathbb{R}^3 \times 
S^2$ with jacobian $\hbox{Jac} \Phi = 1$,
we clearly have that $(v,v_*,\omega) \  \mapsto \ f' f'_*$ is a measurable 
function of
$\mathbb{R}^3 \times \mathbb{R}^3 \times S^2$.
On the other hand, performing a change of variable, we get
$$
\begin{aligned}
\iiint_{\mathbb{R}^3 \times \mathbb{R}^3 \times S^2}  B f' f'_* \, dv dv_* 
d\omega
&= \iint_{\mathbb{R}^3 \times \mathbb{R}^3} f f_* \Bigl( \int_{S^2}  B \, 
d\omega \Bigr) (v_* - v) \, dv dv_* \\
&\le \| f \|_{L^1}á\|áf \|_{L^\infty} \|áB \|_{L^1} < \infty,
\end{aligned}
$$
and by the Fubini-Tonelli Theorem,
$$
\int_{S^2}  B f' f' \, d\omega \ \in \ L^1(\mathbb{R}^3 \times \mathbb{R}^3).
$$
That gives a sense to the gain term
$$
Q^+(f) = \int_{\mathbb{R}^3} (1-f) (1-f_*)
\Bigl( \int_{S^2} B f' f' \, d\omega \Bigr) \, dv_*
$$
as an $L^1$ function. The same argument gives a sense to the loss term
$Q^-(f)$ as an $L^1$ function.

Note that, under the assumptions $B \in L^1_{\rm loc}(\mathbb{R}^3 \times S^2)$
and
$$
{1 \over 1+|z|^2} \iint_{B_R^v \times S^2}\!\!\! B(z+v,\omega) \, d\omega dv
\ \mathop{\longrightarrow}_{|z| \to \infty}  0 \quad \forall R > 0,
\eqno(3.7)
$$
one may give a sense to $Q^\pm(f)$ as a function of $L^1(B_R)$ ($\forall R > 0$) 
for any
$f \in L^1_2 \cap L^\infty$, see \cite{L}. In particular, the cross-section $B$ 
associated to $w = 1$
satisfies (3.7).


Finally, we can make a third assumption on the cross-section, namely
$$
0 \le B(z,\omega) \le (1+|z|^\gamma) \zeta(\theta), \hbox{ with } \gamma \in (-
5,0),
\quad \int_0^{\pi/2} \theta \zeta(\theta) \, d\theta < \infty.
\eqno(3.8)
$$
This assumption allows singular cross-sections, both in the $z$ variable and the
$\theta$ variable, near the origin. In that case, the collision term may be 
defined as a distribution
as follows:
$$
\int_{\mathbb{R}^3} Q(f) \varphi \, dv
= \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f f_* (1 - f' - 
f'_*) \, B \, K_\varphi \,
dv dv_*  d\omega
\eqno(3.9)
$$
with the notation
$$
K_\varphi = \varphi'+\varphi'_*-\varphi-\varphi_*. \eqno(3.10)
$$
To see that (3.9) is well defined we note that (see e.g. \cite{V}),
$$
| K_\varphi |\le C_\varphi |v-v_*|^2 \theta,
\eqno(3.11)
$$
from where,
$$
| f f_* (1 - f' - f'_*) \, B \, K_\varphi | \le
C_\varphi \theta \zeta(\theta) f f_* (|v-v_*|^2 + |v-v_*|^{\gamma+2})
\in L^1(\mathbb{R}^3 \times \mathbb{R}^3 \times S^2).
\eqno(3.12)
$$
To see this last claim, we just put $g(z) = |z|^a$; if $a \in (-3,0]$ one has $g 
\in L^1 + L^\infty$ and
therefore  $f (f \star g) \in L^1$ when $f \in L^1 \cap L^\infty$, and if $a \in 
[0,2]$ then
writing
$g(v-v_*) \le (1+|v|^2) (1 + |v_*|^2)$ we see that $f (f \star g) \in L^1$ when 
$f \in L^1_2$.


\begin{theorem} \label{thm3.1}
Assume that one of the conditions (3.6), (3.7) or (3.8) holds. Then, for any 
$f_{\rm in} \in
L^1_2(\mathbb{R}^3)$ such that $0 \le f_{\rm in} \le 1$ there exists a solution
$f \in C([0,+\infty);L^1(\mathbb{R}^3))$ to
equation (3.1). Furthermore,
$$
\begin{aligned}
&\int_{\mathbb{R}^3} f(t,v) \, dv = \int_{\mathbb{R}^3} f_{\rm in} \, dv =: N, 
\\
&\int_{\mathbb{R}^3} f(t,v) v \, dv = \int_{\mathbb{R}^3} f_{\rm in}(v) v \, dv 
=: P, \\
&\int_{\mathbb{R}^3} f(t,v) {|v|^2 \over 2} \, dv \le \int_{\mathbb{R}^3} f_{\rm 
in}(v) {|v|^2 \over 2} \, dv =: E,
\end{aligned}
\eqno(3.13)
$$
and
$$
\int_0^\infty \tilde D (f) \, dt \le C(f_{\rm in}),
\eqno(3.14)
$$
where
$$
\tilde D (f) := \iiint_{\mathbb{R}^3 \times \mathbb{R}^3 \times S^2} 
\!\!\!\!\!\!\!\! B_2 (f' f'_* (1 - f) (1 - f_*) - f f_* (1 - f') (1 - f'_*))^2 
\,
d\omega dv_* dv.
\eqno(3.15)
$$
\end{theorem}


\begin{remark} \label{rmk3.2} \rm
 In the non homogeneous context, the existence of solutions has been proved
by  Dolbeault \cite{D} under the assumption (3.6) and by  Lions \cite{L} under
the assumption (3.7). Existence in
a classical context under assumption (3.8) (without Grad's cut-off) has been
established by  Arkeryd \cite{Ar2}, Goudon \cite{G} and  Villani \cite{V}.
Finally, bounds on modified entropy dissipation term of the
kind of (3.15) have been introduced by Lu \cite{Lu} for the
Boltzmann-Bose equation.
\end{remark}

Concerning the behavior of the solutions we prove the following result.

\begin{theorem} \label{thm3.3}
For any sequence $(t_n)$ such that $t_n \to +\infty$ there
exists a subsequence $(t_{n'})$ and a stationary solution $\mathcal{S}$ such 
that
$$
f(t_{n'}+.,.) \ \mathop{\rightharpoonup}_{n' \to \infty} \ \mathcal{S}
\quad \hbox{ in } C([0,T];L^1 \cap L^\infty (\mathbb{R}^3) - weak) \quad \forall 
T > 0
\eqno(3.16)
$$
and
$$
\int_{\mathbb{R}^3} \mathcal{S} \, dv =  N, \quad
\int_{\mathbb{R}^3} \mathcal{S} v \, dv = P, \quad
\int_{\mathbb{R}^3} \mathcal{S} {|v|^2 \over 2} \, dv \le E.
\eqno(3.17)
$$
\end{theorem}

\begin{remark} \label{rmk3.4} \rm
 By stationary solution we mean a function $\mathcal{S}$ satisfying
$$
\mathcal{S}' \mathcal{S}'_* (1-\mathcal{S}) (1-\mathcal{S}_*) =
\mathcal{S} \mathcal{S}_* (1-\mathcal{S}') (1-\mathcal{S}'_*).
\eqno(3.18)
$$
Of course, such a function is, formally at least, a Fermi-Dirac distribution, we 
refer to \cite{D} for
the proof of this claim in a particular case.
Theorem \ref{thm3.3} also holds for the non homogeneous Boltzmann-Fermi 
equation, when, for example, the
position space is the torus, and under assumption (3.7) on $B$
(in order to have existence).
\end{remark}

\paragraph{Open questions:}
\begin{enumerate}
\item Is Theorem \ref{thm3.1} true under assumption (3.8) for all $\gamma \in (-
5,-2]$ ?

\item Is it possible to prove the entropy identity (1.14) instead of the 
modified
entropy dissipation bound (3.14)? Of course (1.14) implies the dissipation 
entropy bound (3.14) as it
will be clear in the proof of Theorem \ref{thm3.1}.

\item Is any function satisfying (3.18) a Fermi-Dirac distribution?

\item Is it true that
$$
\sup_{[0,\infty)} \int_{\mathbb{R}^3} f(t,v) |v|^{2 + \epsilon} \, dv \le 
C(f_{\rm in},\epsilon)
$$
for some $\epsilon > 0\,$? Notice that with such an estimate one could prove the 
conservation of the energy
(instead of (3.17)). If we could also give a positive answer to the question 3, 
we could then prove the
convergence to the Fermi-Dirac distribution
$\mathcal{F}_{N,P,E}$.

\item  Finally, is it possible to improve  the convergence (3.16), and prove for 
instance strong $L^1$
convergence?
\end{enumerate}

\paragraph{Proof of Theorem \ref{thm3.1}}
 Suppose that $B$ satisfies (3.6), (3.7) or (3.8) and define
$B_\epsilon = B {\bf 1}_{\theta > \epsilon}{\bf 1}_{\epsilon < |z| < 
1/\epsilon}$. Notice that $B_\epsilon$ satisfies
(3.6), $0 \le B_\epsilon \le B$ and $B_\epsilon \to B$ a.e.
From \cite{D} there exists a sequence of solutions $(f_\epsilon)$ to (3.1) 
corresponding to $(B_\epsilon)$.
Moreover, for any $\epsilon > 0$, the solution $f_\epsilon$ satisfies (3.13) and 
(1.14).
As it is shown by Lions in \cite{L}:
$$
\begin{aligned}
&a_\epsilon {f'_\epsilon} {f'_{\epsilon*}} (1 - f_\epsilon - f_{\epsilon *} )
\ \mathop{\rightharpoonup}_{n' \to \infty}
a {f' f'_*} (1 - f - f_* )
\quad L^1 \ \hbox{ weak}, \\
&a_\epsilon f_\epsilon f_{\epsilon*} {(1 - f'_\epsilon - f'_{\epsilon *} )}
\ \mathop{\rightharpoonup}_{n' \to \infty}
a {f f_* (1 - f' - f'_* )}
\quad L^1 \ \hbox{ weak},
\end{aligned}
\eqno(3.19)
$$
for any sequence $(f_\epsilon)$ such that $f_\epsilon \rightharpoonup f$ in $L^1 
\cap L^\infty$,
and any sequence $(a_\epsilon)$ satisfying (3.8) uniformly in $\epsilon$ and 
such that $a_\epsilon \to a$ a.e.
In particular, under the assumption (3.8) on $B$ and taking $a_\epsilon = 
B_\epsilon, a = B$, it is possible to
pass to the limit $\epsilon \to 0$ in the equation (3.1).
That gives existence of a solution $f$ to the Fermi-Boltzmann equation for $B$ 
satisfying (3.8).


Now, for $B$ satisfying (3.8), we just write
$$
\begin{aligned}
&\int_{\mathbb{R}^3} Q_\epsilon(f_\epsilon) \varphi \, dv\\
&= \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f_\epsilon 
f_{\epsilon *} {(1 - f_\epsilon' - f'_{\epsilon *})}
\, B_\epsilon  \, K_\varphi \,  {\bf 1}_{ \{ \theta \ge \delta, |v-v_*| \ge 
\delta \} } \, dv dv_*  d\omega
\\
&+ \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f_\epsilon 
f_{\epsilon *} {(1 - f_\epsilon' - f'_{\epsilon *})}
\, B_\epsilon \, K_\varphi \,  \bigl( {\bf 1}_{ \{ \theta \le \delta, |v-v_*| 
\ge \delta \}} +
{\bf 1}_{ \{ |v-v_*| \le \delta \}} \bigr) \, dv dv_*  d\omega \\
& \equiv \mathcal{Q}_{\delta,\epsilon} + r_{\delta,\epsilon}.
\end{aligned}
$$
For the first term $\mathcal{Q}_{\delta,\epsilon}$ we easily pass to the limit 
$\epsilon \to 0$ using (3.19).
For the second term $r_{\delta,\epsilon}$, using (3.11), we have
$$
\begin{aligned}
&\lim_{\epsilon \to 0}ár_{\delta,\epsilon}\\
&\le  \lim_{\epsilon \to 0} C_\varphi \int_{\mathbb{R}^3}\!\!\! 
\int_{\mathbb{R}^3}\!\!\!\! f_\epsilon f_{\epsilon *} (|v-v_*|^2 +
|v-v_*|^{\gamma+2})
\Bigl( \int_{S^2}  \theta \zeta(\theta) {\bf 1}_{ \{ \theta \le \delta \} } \, 
d\omega \Bigr) \, dv
dv_* \\
&+ \lim_{\epsilon \to 0} C_\varphi \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3} 
f_\epsilon  f_{\epsilon *} (|v-v_*|^2 +
|v-v_*|^{\gamma+2})  {\bf 1}_{ \{ |v_*-v| \le \delta \} }
\Bigl( \int_{S^2}  \theta \zeta(\theta) \, d\omega \Bigr) \, dv dv_*\\
&\le C_{\varphi,f} \int_{S^2}  \theta \zeta(\theta) {\bf 1}_{ \{ \theta \le 
\delta \} }
\, d\omega\\
&\quad+ C_{\varphi,\zeta} \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3} 
f_\epsilon f_{\epsilon *} (|v-v_*|^2 +
|v-v_*|^{\gamma+2})  {\bf 1}_{ \{ |v_*-v| \le \delta \} } \, dv dv_*.
\end{aligned}
$$
Therefore,
$$
\begin{aligned}
&\lim_{\epsilon \to 0} \int_{\mathbb{R}^3} Q_\epsilon(f_\epsilon) \varphi \, 
dv\\
&= \int_{\mathbb{R}^3}\!\!\int_{\mathbb{R}^3}\!\! \int_{S^2} f f_* {(1 - f' - 
f'_*)}
\, B  \, K_\varphi \,  {\bf 1}_{ \{ \theta \ge \delta, |v-v_*| \ge \delta \} } 
\, dv dv_*  d\omega
+ r_\delta,
\end{aligned}
$$
with $r_\delta \to 0$ when $\delta \to 0$. Since we also have
$$
\int_{\mathbb{R}^3} Q(f) \varphi \, dv
= \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f f_* (1 - f' - 
f'_*)
\, B  \, K_\varphi \,  {\bf 1}_{ \{ \theta \ge \delta, |v-v_*| \ge \delta \} } 
\, dv dv_*  d\omega
+ \tilde r_\delta,
$$
with $\tilde r_\delta \to 0$ when $\delta \to 0$, we conclude that
$$
\lim_{\epsilon \to 0} \int_{\mathbb{R}^3} Q_\epsilon(f_\epsilon) \varphi \, dv
= \int_{\mathbb{R}^3} Q(f) \varphi \, dv
$$
and $f$ is a solution to (3.1). We refer to \cite{L,V} for more details.
To establish (3.14), we take $a_\epsilon = a = \sqrt{B_\delta}$ in (3.19) and
we then have
$$
\begin{aligned}
\sqrt {B_{\delta }}[ f'_\epsilon f'_{\epsilon*}
(1 - f_\epsilon -& f_{\epsilon *} )
-f_\epsilon f_{\epsilon*} (1 - f'_\epsilon - f'_{\epsilon *} )] \\
& \mathop{\rightharpoonup}_{\epsilon \to 0}
\sqrt {B_{\delta }} [ f' f'_* (1 - f - f_* ) - f f_* (1 - f' - f'_* ) ].
\end{aligned}
\eqno(3.20)
$$
Using the elementary inequality
$$
(b-a)^2 \le j(a,b) \quad \forall \ a,b \in [0,1],
$$
we have for $\epsilon \in (0,\delta]$
$$
\int_0^\infty \tilde D_\delta(f_\epsilon) \, dt
\le \int_0^\infty \tilde D_\epsilon(f_\epsilon) \, dt
\le \int_0^\infty D_\epsilon(f_\epsilon) \, dt \le C(f_{\rm in}).
\eqno(3.21)
$$
Gathering (3.20) and (3.21) we obtain, using the convexity of $s \mapsto s^2$,
$$
\int_0^\infty \tilde D_\delta(f) \, dt
\le  \liminf_{\epsilon \to 0}á\int_0^\infty \tilde D_\delta(f_\epsilon) \, dt
\le C(f_{\rm in}),
$$
and we recover (3.15) letting $\delta \to 0$. \hfill$\square$\smallskip


\paragraph{Proof of Theorem \ref{thm3.3}}
 Let consider $f_n = f(t+t_n,.)$ as in the statement of the Theorem 
\ref{thm3.3}.
We know that there exists $n'$ and $\mathcal{S}$ such that
$f_{n'} \ \mathop{\rightharpoonup}_{n' \to \infty}  \mathcal{S}$ weakly in
$L^1 \cap L^\infty ((0,T) \times \mathbb{R}^3)$ for all $T > 0$, and we only 
have to
identify the limit $\mathcal{S}$. On one hand, $\mathcal{S}$
satisfies the moments equation (3.17). On the other hand by (3.21) and
lower semicontinuity we get
$$
\int_0^T \tilde D_\delta (\mathcal{S}) \, ds \le \liminf \int_0^T \tilde D 
(f_{n'}) \, ds
\le \liminf \int_{t_{n'}}^{T+t_{n'}} \tilde D (f) \, ds = 0
$$
for any $\delta > 0$, so that
$$
\mathcal{S}' \mathcal{S}'_* (1 - \mathcal{S}) (1 - \mathcal{S}_* ) - \mathcal{S} 
\mathcal{S}_* (1 - \mathcal{S}') (1 - \mathcal{S}'_* )
=
\mathcal{S}' \mathcal{S}'_* (1 - \mathcal{S} - \mathcal{S}_* ) - \mathcal{S} 
\mathcal{S}_* (1 - \mathcal{S}' - \mathcal{S}'_* ) =0
$$
for a.e. $(v,v_*,\omega) \in \mathbb{R}^3 \times \mathbb{R}^3 \times S^2$. In 
particular,
${\partial \mathcal{S} \over \partial t} = Q(\mathcal{S}) = 0$
and $\mathcal{S}$ is a constant function in time. We finally improve the 
convergence
of $f_{n'}$ to $\mathcal{S}$ and establish (3.16).
To this end we note that for any  $\psi \in C_c(\mathbb{R}^3)$,
$$
{d \over dt} \int_{\mathbb{R}^3} f_{n'} \psi \, dv = \int_{\mathbb{R}^3} 
Q(f_{n'}) \psi \, dv
$$
and the right hand side is bounded in $L^1(0,T)$. Therefore, 
$\int_{\mathbb{R}^3} f_n \psi \, dv$ is bounded
in $BV(0,T)$.  We can then extract a second subsequence
(not relabeled) such that
${ \bigl(\int_{\mathbb{R}^3} f_{n'} \psi \, dv \bigr) }$
converges strongly $L^1(0,T)$ and a.e. on
$(0,T)$ and the limit is obviously
${ \int_{\mathbb{R}^3} \mathcal{S} \psi \, dv }$.
Let  $\tau \in (0,T)$ be such that ${ \bigl(\int_{\mathbb{R}^3} f_{n'} \psi \, 
dv \bigr) }
\to { \int_{\mathbb{R}^3} \mathcal{S} \psi \, dv }$. We deduce that
$$
\int_{\mathbb{R}^3} f_{n'}(t,.) \psi \, dv = \int_{\mathbb{R}^3} f_{n'}(\tau) 
\psi \, dv
- \int_\tau^t \int_{\mathbb{R}^3} Q(f_{n'}) \psi \, dv ds
$$
and using P. L. Lions' result we get
$$
\int_0^T \int_{\mathbb{R}^3} Q(f_{n'}) \psi \, dv ds \to \int_0^\tau 
\int_{\mathbb{R}^3} Q(\mathcal{S}) \psi \, dv ds = 0,
$$
and therefore
$$
\sup_{[0,T]} \bigl| \int_{\mathbb{R}^3} f(t_{n'}) \psi \, dv - 
\int_{\mathbb{R}^3} \mathcal{S} \,
 \psi \, dv \bigr| \ \to \ 0.
$$
\hfill$\square$

\subsection{Bose-Einstein collision operator for isotropic density} %3.2

We consider now the Boltzmann equation for Bose-Einstein particles and take
$\tau = 1$ in (3.1).
We start with an elementary remark. Assume that $w = 1$ and consider a sequence 
$(f_\epsilon)$ defined by
$f_\epsilon(v) := \epsilon^{-3} f(v/\epsilon)$ for a given $0 \le f \in 
C_c(\mathbb{R}^3)$, $f \not\equiv 0$.
An elementary change of variables leads to
$$
\|Q^\pm(f_\epsilon) \|_{L^1} = {1 \over \epsilon^3} \|Q^\pm(f) \|_{L^1} \to 
+\infty.
\eqno
$$
Therefore, no a priori estimate of the form
$$
\| Q(f) \| \le \Phi (\|f \|_{L^1}), \hbox{ with }\Phi \in C(\mathbb{R}_+),
\eqno
$$
can be expected. In particular we will not be able to give a sense to the kernel
$Q(f)$ under the only physical bound $f \in M^1_2(\mathbb{R}^3)$ for such a $w$.


This first remark motivates the two following simplifications, originally 
performed by Lu \cite{Lu}
to give a sense to $Q(f)$: we assume that the density is isotropic and we make a 
strong (and unphysical)
truncation assumption on $w$.  We use them here, in a slightly different way 
that we believe to be simpler.


We then assume, until the end of this Section, that the density $f$ only depends 
on the quantity $|v|$, and
denote
$f(v) = f(|v|) = f(r)$ with
$r = |v|$. For a given function $q = q (r,r_*;r',r'_*)$ we define
$$
\mathcal{Q} [q] (v) : = \int_{\mathbb{R}^3} \!\! \int_{\mathbb{R}^3} \!\! 
\int_{\mathbb{R}^3} w q \delta_{\mathcal{C}} \, dv_* dv'
dv'_*.
\eqno(3.22)
$$
By introducing the new function
$$
\hat{w} (r,r_*,r',r'_*) := \iiint_{S^2 \times S^2 \times S^2}
w(v,v_*,v',v'_*) \delta_{\{ v+v_* = v' + v'_* \} } \, d\sigma_* d\sigma' 
d\sigma'_*
\eqno(3.23)
$$
and the notation $v_* = r_* \sigma_*$, $v_* = r' \sigma'$, $v'_* = r'_* 
\sigma'_*$,
$d\hat{r}_* = r_*^2 \, dr_*$, $d\hat{r}' = {r'}^2 \, dr'$, $d\hat{r}'_* = 
{r'_*}^2 \, dr'_*$,
$$
\widehat{\mathcal{C}}= \{(r,r_*,r',r'_*) \in \mathbb{R}_+^4, \  r^2 + r_*^2 = 
{r'}^2 + {r'}_*^2 \},
\eqno(3.24)
$$
we may write $\mathcal{Q}[q]$ in the simple way
$$
\mathcal{Q} [q] (r) = \iiint_{\mathbb{R}_+^3} \hat{w} 
\,\delta_{\widehat{\mathcal{C}}} \,q \, d\hat{r}_* d\hat{r}' d\hat{r}'_*.
\eqno(3.25)
$$
Let us emphasize that $\hat{w}$ is indeed a function of $r = |v|$ (and not on 
the whole variable $v$)
since, for any
$R \in SO(3)$, we have
$$
\begin{aligned}
&\iiint_{S^2 \times S^2  \times S^2}
w(R v,v_*,{v'},{v'_*}) \delta_{\{ R v+v_* = v' + v'_* \} } \, d\sigma_*
{d\sigma'} {d\sigma'_*} \\
&= \iiint_{S^2 \times S^2 \times S^2}
w(R v,R v_*,R {v'},R {v'_*}) \delta_{\{ R v+R v_* = R v' + R v'_* \} } \, d\sigma_*
{d\sigma' d\sigma'_*} \\
&= \iiint_{S^2 \times S^2 \times S^2}
w( v,v_*,{v'},{v'_*}) \delta_{\{ v+v_* = v' + v'_* \} } \, d\sigma_*
{d\sigma' d\sigma'_*}
\end{aligned}
$$
by the change of variables $(\sigma_*,\sigma',\sigma'_*) \to (R \sigma_*,R 
\sigma',R \sigma'_*)$ and
the rotation invariance of $w$. Moreover, $\hat{w}$ clearly satisfies the 
property
$$
\hat{w} (r,r_*,r',r'_*) =  \hat{w} (r_*,r,r',r'_*) =  \hat{w} (r',r'_*,r,r_*).
\eqno(3.26)
$$
     %%%%%%%%%   L^1   %%%%%%%

\begin{lemma} \label{lm3.5}
Assume that $w$ is such that $B$ defined by (3.5) satisfies
$$
\sup_{z \in \mathbb{R}^3} {1 \over 1 + |z|^s} \int_{S^2} B(z,\omega) \, d\omega
< \infty \eqno(3.27)
$$
with $s = 2$ and that $\hat{w}$ defined by (3.26) satisfies
$$
\hat{w} (r+r_*+r'+r'_*) \in L^\infty(\mathbb{R}^4_+).
\eqno(3.28)
$$
Then for any isotropic function $f \in L^1_2(\mathbb{R}^3)$ the kernel 
$Q^\pm(f)$
is well defined and
$$
\|Q^\pm (f) \|_{L^1} \le  C_B \|f \|^2_{L^1_2(\mathbb{R}^3)} + C_{\hat{w}} \|f 
\|^3_{L^1(\mathbb{R}^3)}.
\eqno(3.29)
$$
\end{lemma}
A refined version of the bound (3.29) has been used, by Lu \cite{Lu}, in order 
to prove a global existence
result when $s = 0$ in (3.27). For $s = 1$, X. Lu also proves an existence 
result under the
additional assumption that $B$ has the  particular form:
$B(z,\omega) = |z|^\gamma \zeta(\theta)$ with $\gamma \in [0,1]$, $\zeta \in 
L^1$.
Here condition (3.28) has to be understood as a truncation assumption
near the origin
$$
\exists B_0 \in (0,\infty), \quad B(z,\omega) \le B_0 (\cos \theta)^2 \sin 
\theta |z|^3,
\eqno(3.30)
$$
introduced in \cite{Lu}. In order to clarify the assumption (3.28) let us state
the following result.

\begin{lemma} \label{lm3.6}
\begin{enumerate}
\item For $w = 1$,
$$
\hat w = {4 \pi^2 \over r \, r_* \, r' \, r'_*} \min (r,r_*,r',r'_*).
\eqno(3.31)
$$
\item As a consequence, any cross-section $w$ such that
$$
\exists w_0 \in (0,\infty), \quad
w \le w_0 ((|v'-v| |v'_* - v|) \wedge 1)
\eqno(3.32)
$$
satisfies (3.27)-(3.28).
\end{enumerate}
\end{lemma}

Note that condition (3.32) is exactly the X. Lu's assumption (3.30) near the 
origin. This condition
kills the interaction between particles with low energy. We emphasize that this 
assumption is never
satisfied by the physically relevant cross-sections.

\paragraph{Proof of Lemma \ref{lm3.5}} We may write
$$
f' f'_* (1 + f) (1 + f_*) - f f_* (1 + f') (1 + f'_*)
= f' f'_* (1 + f + f_*) - f f_* (1 + f' + f'_*).
\eqno(3.33)
$$
Then, we have to define $\mathcal{Q} [q]$ for two kinds of terms $q$: for the 
quadratic terms $q = f f_*$
and
$q = f' f'_*$ and for cubic terms $q = f' f'_* f$, $q = f' f'_* f_*$,
$q = f f_* f'$ and $q = f f_* f'_*$. The quadratic terms may be defined in the 
same way
that for the Fermi-Dirac Boltzmann equation in Section 3.1 thanks to assumption 
(3.28) and they are
bounded by the first term in the right hand side of estimate (3.29).

We then focus on the cubic terms. We define them by performing one integration 
more in the
representation formula (3.25) (with respect to  one of the variables $r_*$, 
$r'$, $r'_*$) but we
still use the formula (3.25) in order to preserve the symmetries of 
$\mathcal{Q}$.

Let us first assume that moreover, $f \in C_c(\mathbb{R}^3)$. Performing in 
(3.25) the integration in
the $r_*$ variable we get, using Lemma \ref{lmA1.1},
$$
\mathcal{Q}[q] (r) = \iint_{\mathbb{R}_+^2} \hat{w} {r_* \over 2} {\bf 1}_{ \{ 
{r'}^2+ {r'_*}^2 \ge r^2 \} }
\,  q \, d\hat{r}' d\hat{r}'_*.
\eqno(3.34)
$$
where  $r_* = \sqrt{{r'}^2+{r'_*}^2-r^2}$.
It is then clear that (3.33) defines $\mathcal{Q}[q]$ as measurable function.

Moreover, we write
$$
\int_{\mathbb{R}_+} \mathcal{Q} [q] (r) d\hat{r} = \iiiint_{\mathbb{R}_+^4} 
\hat{w} \,\delta_{\widehat{\mathcal{C}}} \,q
\, d\hat{r} d\hat{r}_* d\hat{r}' d\hat{r}'_*
\eqno
$$
and first, perform an integration with respect to the variable which does not
appear in $q$ (namely, $r_*$ if $q = f' f'_* f$,  $r$
if $q = f' f'_* f_*$, and so on) and we get (using Lemma \ref{lmA1.1} again)
$$
\begin{aligned}
&\int_{\mathbb{R}_+} \mathcal{Q} [q] (r) d\hat{r}\\
&= \iiint_{\mathbb{R}_+^3} \hat{w}(r_1,r_2,r_3,r_4) {r_4 \over 2} {\bf 1}_{ \{ 
r_1^2+r_2^2 \ge r_3^2 \} }  \,  f(r_1) f(r_2) f(r_3) \,
 d\hat{r}_1 d\hat{r}_2 d\hat{r}_3 \\
& \le \| \hat{w} {(r+r_*+r'+r'_*)} \|_{ L^\infty(\mathbb{R}^4_+)} \|f 
\|^3_{L^1(\mathbb{R}^3)},
\end{aligned} \eqno(3.35)
$$
where in the first inequality we have used the notation
$r_4 = \sqrt{ r_1^2 + r_2^2 - r_3^2 }$.
We finally deduce (3.29) using a density-continuity argument.
\hfill$\square$ \smallskip


\paragraph{Proof of Lemma \ref{lm3.6}}
 We start by proving (3.31). Using that
$$
\delta (v+v_*-v'-v'_*) = \int_{\mathbb{R}^3} e^{i (z, v+v_*-v'-v'_*)} { dz \over 
(2\pi)^3}
=\int_0^\infty \!\! \int_{S^2} e^{i (z, v+v_*-v'-v'_*)} \, d\sigma {r^2 \, dr  
\over (2\pi )^3}
$$
where $z =|z| \sigma$ and $\epsilon = |z|$ in polar coordinates,
we obtain
$$
\begin{aligned}
&\int_{S^2}\!\int_{S^2}\!\int_{S^2} \delta {(v+v_*-v'-v'_*)} \, {d\sigma_* d\sigma'} 
{d\sigma'_*} \\
& = \int_0^\infty \int_{S^2}\!\int_{S^2}\!\int_{S^2}\!\int_{S^2}
e^{i (z, v+v_*-v'-v'_*)} \, {d\sigma d\sigma_* d\sigma' d\sigma'_*} {\epsilon^2 \, 
d\epsilon
\over(2\pi)^3} \\
& = \int_0^\infty {\epsilon^2 \, d\epsilon \over(2\pi)^3} \int_{S^2} e^{i (z, 
v)} \, d\sigma
\int_{S^2} e^{i (z,v_*)} \, d\sigma_* \int_{S^2} e^{i (z, v')} \, {d\sigma'}
\int_{S^2} e^{i (z, v'_*)} \, {d\sigma'_*} \\
&={16 \pi \over r \, r_* \, r' \, r'_*} \int_0^\infty {d\epsilon \over (2\pi)^3 
\epsilon^2} {\sin }(\epsilon \, r) {\sin }(\epsilon \, {r_*}) {\sin }(\epsilon \, {r'}) 
{\sin (\epsilon \, r'_*)}.
\end{aligned}
$$
Then (3.31) follows by an lengthy elementary trigonometric computation.
It is clear, thanks to (3.25), that (3.32) implies (3.27).
We then have to prove that (3.32) implies (3.28).
For any  $r,r_*,r',r'_*$ given, we set
$m_1 = \min (r,r_*,r',r'_*)$,
$m_2 = \min (\{ r,r_*,r',r'_*\}\backslash \{ m_1 \})$,
$m_3 = \min (\{ r,r_*,r',r'_*\}\backslash \{ m_1,m_2 \})$
and finally, $m_4 = \max ( r,r_*,r',r'_*)$.
Since $|v-v'| = |v_*-v'_*|$, $|v-v'_*| = |v_*-v'|$ and
$\{ r,r_* \}\not= \{ m_1,m_2 \}$, $\{ r,r_* \}\not= \{ m_3,m_4 \}$,
the assumption (3.32) leads to
$$
\begin{aligned}
w &\le w_0 \bigl\{ \bigl[ 4 \min \bigl( \max(r,r'), \max(r_*,r'_*) \bigr) 
\min\bigl(
\max(r,r'_*),
\max(r_*,r')\bigr) \bigr] \wedge 1 \bigr\} \\
&\le w_0  \bigl\{ \bigl[ 4 \, m_2 \, m_3 \,  \bigr] \wedge 1 \bigr\}.
\end{aligned}
$$
Gathering (3.31) and this last inequality  we deduce
$$
\hat{w} (r+r_*+r'+r'_*) \le
{4 \pi^2 \, m_1 \over r \, r_* \, r' \, r'_*} (4 \, w_0 \, m_2 \, m_3) 
(r+r_*+r'+r'_*)
\le 64 \pi^2 \, w_0,
$$
and that concludes the proof.
\hfill$\square$ \smallskip

We want now to extend the previous arguments and give a sense to $Q(F)$ for $F 
\in M^1(\mathbb{R}^3)$. For the
quadratic term that problem has been solved by Povzner in \cite{P}. For the 
cubic term we write for a
radial measure $dF = f \, r^2 \, dr$
$$
\langle \mathcal{Q}[q],\phi\rangle = \langle F \otimes F \otimes F , 
B_{\hat{w}}[\psi]
\rangle \eqno(3.36)
$$
with
$$
B_{\hat{w}}[\psi](r_1,r_2,r_3) := \hat{w}(r_1,r_2,r_3,r_4) {r_4 \over 2}
{\bf 1}_{ \{ r_1^2+r_2^2 \ge r_3^2 \}} \psi
$$
and $r_4 = \sqrt{ r_1^2 + r_2^2 - r_3^2 }$ and $\psi (r_1,r_2,r_3) = \phi(r_    
i)$ $i = 1,2,3$ or $4$
depending on $q$.
Under the condition
$$
B_{\hat{w}}[1] \in C (\mathbb{R}_+^3)
\eqno(3.37)
$$
we clearly have $B_{\hat{w}}[\psi] \in C_c (\mathbb{R}_+^3)$ for any $\phi \in 
C_c(\mathbb{R}_+)$ and then  the right
hand side term of (3.36) is well defined.
The assumption
$$
B \in C(\mathbb{R}^3 \times S^2), \eqno(3.38)
$$
which is satisfied by the hard potentials, guaranties that the
quadratic term is well defined and that $\hat{w} \in C(E  \backslash \{á0 \})$ 
where
$E = \{ (r_1,r_2,r_3,r_4) \in \mathbb{R}^4_+, \, r_1^2 + r_2^2 - r_3^2 - r_4^2 = 
0 \}$.
The condition (3.37) is then satisfied if moreover
$$
\lim_{ (r_1,r_2,r_3) \to 0}  \hat{w}(r_1,r_2,r_3,r_4) \, r_4 = 0\,.
$$
From the computations performed in the proof of Lemma \ref{lm3.6}. it is clear 
that the last condition holds
if we assume, for instance
$$
w  \le w_0 \,  |v'- v|^\gamma \wedge 1, \quad \gamma > 1.
\eqno(3.39)
$$
As a conclusion, under assumption (3.27), (3.38), (3.39) we may define
the collision kernel for general non negative bounded and isotropic measures.
From all the above one easily deduces the following existence result for (3.1)
with $\tau =1$ in the framework of non negative bounded measures.

\begin{theorem} \label{thm3.7}
 Assume $w$ satisfies (3.39). Let $F_{\rm in} \in M^1_{\rm rad}(\mathbb{R}^3)$,
$F_{\rm in}\ge 0$. Then, there exists a unique global solution
$F = g + G \in C([0,\infty),M^1_{\rm rad}(\mathbb{R}^3))$ to (3.1) with
$\tau=1$.
\end{theorem}

\begin{remark} \label{rmk3.8} \rm
It is straightforward to check that in the radially
symmetric case Theorem \ref{thm2.4} gives:
\end{remark}

\begin{theorem} \label{rmk2.5'}
Let $0 \le F \in M^1_{\rm rad}(\mathbb{R}^3)$ an isotropic measure on 
$\mathbb{R}^3$ such
that
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ |v|^2 / 2 \end{pmatrix} \, dF(v) =
\begin{pmatrix} N  \\ E \end{pmatrix}
\quad \hbox{and of course} \quad
\int_{\mathbb{R}^3} v \, dF(v) = 0,\eqno{(3.41)}
$$
with $N > 0$, $E > 0$.
Therefore the two following assertions are equivalent:
\begin{itemize}
\item[(i)] $F$ is the Bose-Einstein distribution $\mathcal{B}[N,0,E]$,

\item[(ii)] $F$ is the solution of the maximization problem:
$$\mathcal{H}(F) = \max \{ \mathcal{H}(F') \mbox{ with $F'$ satisfying (3.41)}.
$$
\end{itemize}
\end{theorem}


\paragraph{Open questions:}
In the $L^1$ setting X. Lu has proved in \cite{Lu}:
\begin{itemize}
\item[(i)] For any $(t_n)$ such that $t_n \to \infty$ there exists $m' \le m$, 
$E' \le E$ and a subsequence
$(t_{n'})$ such that
$$
g(t_{n'})  \rightharpoonup  b_{m',0,E'} \quad \hbox{biting } L^1_{\rm rad} \ 
\hbox{áweak.}
$$

\item[(ii)] For a given $E$ there exists $N_c = N_c(E)$ such that if
$N(f_{\rm in}) < N_c$ and $E(f_{\rm in}) = E$ then
$$
g(t)  \rightharpoonup  \mathcal{B}_{N,0,E} = b_{N,0,E} \quad L^1_{\rm rad} \
\hbox{weak.}
$$
\end{itemize}
Where the distributions $b$ and $\mathcal{B}$ are defined in (2.22) and (2.25).
The two following questions are then natural.

\noindent
1. Is it possible to  construct (global?) solutions to (3.1) for $\tau =1$ 
without
the strong truncation condition (3.39); for instance for $w =1$? What is the 
qualitative behavior of
such  solutions ?

\noindent
2. Is it possible to prove that, under the strong truncation condition (3.39),
$$
F(t) \rightharpoonup \mathcal{B}_{N,0,E} \quad
\hbox{weakly } \sigma (M^1,C_c)
\quad\hbox{ and }\quad
g \to b_{N,0,E} \quad\hbox{strongly}L^1(\mathbb{R}^3 \backslash \{ 0 \})
$$
as it may be expected from the stationary analysis?
If $N(g_{\rm in}) < N_c$, is it possible to prove strong convergence instead of 
result (ii) in Theorem
3.5?

\begin{remark} \label{rmk3.9}\rm
The Boltzmann-Compton equation, introduced in Section 5 below, is a
particular case of (3.1) with $\tau =1$. It has been proved in
\cite{EM1,EM2} that it also has
global solutions $F=g+G\in C([0,\infty),M^1_{\rm rad}(\mathbb{R}^3))$, where $g$ 
is the regular and $G$ the
singular part of the measure $F$ with respect to the Lebesgue measure. Moreover 
it was proved that the
Boltzmann-Compton may be splitted as a system of two coupled equations for the 
pair $(g, G)$. This
allows in particular for a detailed study of the asymptotic behavior of the 
solutions. Let us
briefly show that this is not true for the general isotropic solutions
$F=g+G$ of the equation (3.1) unless $G$ is one single Dirac mass.
\end{remark}

Let us write $F = g +G$ with
$g$ regular with respect to the Lebesgue measure  ($g \prec dv$) and $G$ 
singular with respect to
the Lebesgue measure ($G \perp dv$). We have then
\begin{align*}
&F' F'_* \,(1 + F) (1+F_*) - F F_* (1 + F') (1+F'_*) =\\
&= (1+g) (1+ g_* + G_*) (g' + G') (g'_* + G'_*) + G (1+F_*) F'
 F'_* \\
& \quad- g (g_* + G_*) (1+ g' + G') (1 + g'_* + G'_*) - G F_* (1 +F') (1+F'_*)\\
&= (1 + g) \bigl\{ (1+g_*) \, g' \, g'_* + (1+g_*) \, g' \, G'_* + (1+g_*) \, G' 
\, g'_* +
(1+g_*) \, G' \, G'_* \\
&\quad + G_* \, g' \, g'_* + G_* \, g' \, G'_* + G_* \, G' \, g'_* + G_*
\, G' \, G'_*\bigr\} \\
&\quad - g \bigl\{ g_* (1 + g') (1 + g'_*) + g_* (1 + g')
\, G'_* + g_* \, G' (1 + g'_*) + g_* \, G' \, G'_* \\
&\quad + G_* (1 + g') (1 + g'_*) + G_* (1 + g') \, G'_* + G_* \,
G' (1 + g'_*)+ G_* \, G' \, G'_* \bigr\} \\
&\quad + G (1+F_*) F' F'_* - G F_* (1 +F') (1+F'_*)\\
&= q(g) + q_1(g,G) + q_2(g,G) + q_3(G,g),
\end{align*}
with
\begin{align*}
q(g) &:= (1+g) (1+g_*) \, g' \, g'_* - g \, g_* (1+g') (1+g'_*),\\
q_1(g,G) &:=  G_* [ (1+g) \, g' \, g'_* - g (1+g') (1+g'_*) ]\\
&\quad+ G' [ (1+g) (1+g_*) \, g'_* - g \, g_* (1+g'_*)\\
&\quad+ G'_* [ (1+g) (1+g_*) \, g' - g \, g_* (1+g'),\\
q_2(g,G) &:= G_* \, G'[ (1+g) \, g'_* - g (1+g'_*) ]\\
&\quad+ G_* \, G'_* [(1+g) \, g' - g (1+g') ] \\
&\quad+ G' \, G'_* [ (1+g) (1+g_*) - g \, g_* ], \\
q_3(G) &:= (1+g) \,G_* \, G' \, G'_*  - g \, G_* \, G' \, G'_*, \\
q_4(G,g) &:= G (1+F_*) F' F'_* - G F_* (1 +F') (1+F'_*).
\end{align*}
Defining
$$
Q_i(g,G) =
\iiint_{\mathbb{R}^9} w \delta_\mathcal{C} q_i(g,G) \, dv_* dv' dv'_*,
$$
we may write
$$
Q(F) = Q(g) + Q_1(g,G) +  Q_2(g,G) + Q_3(G) + Q_4(g,G).
$$
The key result in all our analysis is the following result.

\begin{lemma} \label{3.10}
Assume (3.27), (3.38) and (3.39).
For any $F \in M^1_{\rm rad}(\mathbb{R}^3)$, $F_{\rm in}\ge 0$, we have
\begin{align*}
&Q(g), \ Q_1(g,G), \ Q_2(g,G) \in L^1_{\rm rad}(\mathbb{R}^3), \\
&Q_4(g,G) \in M^1_{\rm rad}(\mathbb{R}^3) \hbox{ and } Q_4(g,G) \perp dv.
\end{align*}
For any finite sum of Dirac masses $G$ the kernel $Q_3(G)$ is also finite sum of 
Dirac masses and, moreover, if $G$
is not single Dirac mass then $\hbox{supp} \, G \backslash \{ 0 \}$ is strictly 
contained in $\hbox{supp} \,
Q_3(G)$.  Finally, there exists $G$ singular such that $Q_3(G)$ is regular.
\end{lemma}

\paragraph{Proof of Lemma \ref{lm3.5}}
We already know from the above that
$Q(g) \in L^1_{\rm rad}(\mathbb{R}^3)$ and $Q_1(g,G)$, $Q_2(g,G)$, $Q_4(g,G)$, 
$Q_3(G) \in M^1_{\rm rad}(\mathbb{R}^3)$.
Let us denote by $q$ the first term in $q_1(g,G)$ and write, for any  $\phi$
\[
\langle \mathcal{Q}[q], \phi\rangle = \iiiint_{\mathbb{R}^4_+} \hat{w} 
\,\delta_{\widehat{\mathcal{C}}}
\, g' \, g'_* \phi \, dG_* d\hat{r}' d\hat{r}'_*
= \iint_{\mathbb{R}^2_+} \psi\, g' \,  d\hat{r}' \, dG(r_*),
\]
with
$$
\psi = \int_{\mathbb{R}_+} \hat{w} \, r {\bf 1}_{ \{{r'}^2 + {r'_*}^2 \ge r^2_* 
\} }
\,\phi(r) \, g'_* \,  d\hat{r}'_*
$$
where in the expression of $\psi$ we have used the notation $r = \sqrt{ {r'}^2 + 
{r'_*}^2 - r^2_*}$.
Observe that, by assumption, $(r_*,r',r'_*) \mapsto \hat{w} \, r {\bf 1}_{ 
\{{r'}^2 + {r'_*}^2 \ge r^2_*
\}}$ is continuous so that $(r_*,r') \mapsto \psi$ is continuous for any $\phi 
\in L^\infty_{\rm rad}(\mathbb{R}^3)$.
Moreover, taking $\phi = {\bf 1}_A$ for any set $A$ with Lebesgue measure equal 
zero we get $\psi = 0$,
so that the Radon-Nykodim Theorem implies that $\mathcal{Q}[q] \in L^1$.
By the same way we prove that all the terms in $Q_1(g,G)$ and $Q_2(g,G)$
belongs to $L^1_{\rm rad}(\mathbb{R}^3)$.

It is clear that taking $q =  F' F'_* - F_* - F_* F' - F_* F'_*$ we have
$\mathcal{Q}[q] \in C_b(\mathbb{R}^3)$ so that $Q_4(g,G) \perp dv$.
For given $R_*,R',R'_* \ge 0$ we set
$q = \delta_{R_*}(r_*) \delta_{R'}(r') \delta_{R'_*}(r'_*)$ and we
verify
$$
\langle \mathcal{Q}[q], \phi \rangle =  \phi(R) \, R \hat{w}(R,R_*,R',R'_*) {\bf 
1}_{ \{{R'}^2 + {R'_*}^2 \ge R^2_* \}}
$$
where $R$ is defined by $R = \sqrt{ {R'}^2 + {R'_*}^2 - R^2_*}$ if ${R'}^2 + 
{R'_*}^2 \ge R^2_*$.
In particular, if $G = \alpha \delta_0$ then $Q_3(G) = 0$ and if
$G = \alpha \delta_a$ with $a,\alpha \not = 0$ then
$Q_3(G) = \beta \delta_a$ with $\beta > 0$.
If
$$
G = \sum_{a \in E} \alpha_a \delta_a
$$
then
$$
Q_3(G) = \sum_{b \in E'} \beta_b \delta_b,
$$
with $E' = \{ b \ge 0, \ \exists a_*,a',a'_* \in E \ s.t. (b,a_*,a',a'_*) \in 
\widehat{\mathcal{C}} \}$.
That proves the claim of the Lemma since $E'$ strictly contain $E$ if $E$ is not 
a single point.

Finally, let $G$ be a measure supported by $\sqrt{C}$, where $C$ is a Cantor 
set, constructed for example as
follows.  For every $n\in \mathbb{N}$ consider the set
$$C_n := \{ x \in [0,1], \exists k \in \mathbb{N}, 2 \, k \le 3^n \, x \le 2 \, 
k + 1 \},$$
so  that $C_n \searrow C$ as $n\to \infty $. Then $g_n := |C_n|^{-1} {\bf 
1}_{C_n}\rightharpoonup H$, a singular non
negative  measure  whose support is  $C$. If we take now $G := H \circ s$ with 
$s:\mathbb{R} \to \mathbb{R}$, $s(r) =
\sqrt{r}$ we have, for any $\phi \in C_{rad,c}(\mathbb{R}^3)$:
$$
\langle Q_3(G) = \int \!\! \int \!\! \int_{\mathbb{R}_+^3} dG(r_*) \, dG(r') \, 
dG(r'_*) [ \phi(r) \, r \hat{w} {\bf 1}_{ \{{r'}^2 + {r'_*}^2 \ge r^2_* \} } ].
$$
Since
$$
\{ z \in \mathbb{R}_+; \ \exists \epsilon_*,\epsilon',\epsilon'_* \in C z = 
\epsilon'+\epsilon'_*-\epsilon_* \}
= \mathbb{R}_+
$$
we see that $\langle Q_3(G);\phi\rangle >  0$ for any non negative and not 
vanishing $\phi$, and thus $Q_3(G)$ has a
regular
part which is not equal to $0$.
\hfill$\square$

\begin{remark} \label{rmk3.11} \rm
If we assume that for every time $t>0$, $G(t)\equiv \alpha(t)\,\delta _{0}$,
then the equation (3.1) with $\tau =1$ may be split into a coupled system of
equations for the pair $(g, \alpha )$.
\end{remark}

\begin{remark} \label{rmk3.12}\rm
The equation (3.1) with $\tau =1$ has deserved some interest in the recent
physical literature in the context of Bose condensation. We particularly refer 
here to the works by
Semikoz \& Tkachev \cite{ST1,ST2}, and by Josserand \& Pomeau \cite{JP} due to 
their strong
mathematical point of view (but see also the references therein). In these two 
articles the authors present a
possible ``scenario'' to describe the occurrence of Bose condensation in finite 
time, based on the isotropic
version of the equation (3.1) with
$\tau =1$. Their arguments are based on formal asymptotics and, in
\cite{ST1,ST2}, also supported by numerical simulations.
\end{remark}

\section{Boltzmann equation for two species} %sec 4.

In this Section, we consider a system of two coupled homogeneous equations
describing a Bose gas interacting with a heat bath, chosen to be a
Fermi gas. This is a particular case of a gas composed of two species and has
already been considered in the physical literature (see references below). One 
of the
species are Bose particles and the other are either Fermi-Dirac particles or
non quantum particles. In this Section we only deal with non
relativistic particles. Relativistic particles, in particular
photons will be considered in the next Section. In order to avoid lengthy 
repetitions,
we do not specify the energy  (relativistic or not relativistic) unless
necessary.

From a mathematical point of view, the study of the Boltzmann equation for
a gas of Bose particles  is rather difficult, as we have already seen it in the 
preceding Section.
But it is  possible to derive, from the Boson-Fermion interaction system, a 
physically
relevant model which turns out to be a sort of  ``linearization''
of equation (3.1). This model is then simpler and gives some insight into the
behavior of the Boltzman equations for quantum particles. It describes the 
interaction
of Bose particles with isotropic distribution and non quantum Fermi particles at
isotropic equilibrium with non truncated cross-section. The equation is now 
quadratic
instead of cubic and its mathematical analysis is easier. We prove that  Bose- 
Einstein
condensation takes place in infinite time, in contrast with the finite time
condensation which is expected for the Bose-Bose interaction equation. Similar 
results
had previously been obtained in the physical literature, using formal and 
numerical
methods for similar situations (cf. \cite{LY1,LY2, ST1,ST2}).

We thus consider in what follows a gas composed of two species of particles.
The first are Bose particles.  The second are either Fermi particles or their
non quantum approximation but will always be designed as Fermi particles.
We suppose that when a Bose particle of momentum $p$ collides with a Fermi
particle of momentum $p_*$ they
undergo an elastic collision, so that the total momentum and the
total energy of the system
constituted by that pair of particles are conserved. More precisely,
denoting by
$\mathcal{E}_1(p)$ the energy of Bose particles with momentum $p$ and by
$\mathcal{E}_2(p_*)$ the energy of Fermi particles
with momentum $p_*$ we assume that after collision the particles have
momentum $p'$ (for Bose particles)
and $p'_*$ (for Fermi particles) which satisfy
$\mathcal{C}_{12}$:
$$\begin{gathered}
p' + p'_* = p + p_* \\
\mathcal{E}_1(p') + \mathcal{E}_2(p'_*) = \mathcal{E}_1(p) + \mathcal{E}_2(p_*). 
\\
\end{gathered}  \eqno(4.1)
$$

The gas is described by the density $F(t,p) \ge 0$ of Bose particles and
the density $f(t,p) \ge 0$ of
Fermi particles. We assume that the evolution of the gas is given by the
following Boltzmann equation (see for instance \cite{CC})
$$\begin{gathered}
{\partial F \over \partial t} = Q_{1,1}(F,F) + Q_{1,2}(F,f), \quad F(0,.)
= F_{\rm in}, \\
{\partial f \over \partial t} = Q_{2,1}(f,F) + Q_{2,2}(f,f), \quad f(0,.)
= f_{\rm in}.
\end{gathered} \eqno(4.2)
$$
The collision terms $Q_{1,1}(F,F)$ and $Q_{2,2}(f,f)$ stand for collisions
between particles of the same specie and therefore are given by (1.5).
The collision terms $Q_{1,2}(F,f)$ and $Q_{2,1}(f,F)$ stand for collisions
between particles of the
two different species, they are given by
$$
\begin{gathered}
Q_{1,2}(F,f) =  \int_{\mathbb{R}^3} \!\! \int_{\mathbb{R}^3} \!\! 
\int_{\mathbb{R}^3} w_{1,2} \delta_{\mathcal{C}_{1,2}} q_{1, 2}
 \, dp_* dp' dp'_*,\\
q_{1,2} = F' f'_* (1+F) (1 + \tau  \,f_*)
-  F f_* (1 + F') (1 + \tau  f'_*)
\end{gathered}\eqno(4.3)
$$
with $\tau = -1$ when the second specie is composed of true Fermi
particles and $\tau  = 0$ when the second specie is constituted of non quantum
particles. The collision kernel $Q_{2,1}(f,F)$ is given by an obvious similar 
expression.
The measure 
$w_{1,2}\delta_{\mathcal{C}_{1,2}}=w_{1,2}(p,p_*,p',p'_*)\delta_{\mathcal{C}_{1,
2}}$
satisfies the
micro-reversibility hypothesis
$$
w_{1,2}(p',p'_*,p,p_*)\delta_{\mathcal{C}_{1,2}} = 
w_{1,2}(p,p_*,p',p'_*)\delta_{\mathcal{C}_{1,2}},
\eqno(4.4)
$$
but not the indiscernibility $w_{1,2}(p_*,p,p',p'_*)\delta_{\mathcal{C}_{1,2}} =
w_{1,2}(p,p_*,p',p'_*)\delta_{\mathcal{C}_{1,2}}$ as in (1.7)
since the two species are now different. When both
energies are non relativistic
$w_{1,2}$ is invariant by Galilean transformations, and when both
energies are relativistic it is invariant by Lorentz transformations.
In the mixed case of one non relativistic specie and one
relativistic specie, the situation is a little more complicated and we
postpone the analysis to the next Section.


We start with some simple formal properties of the solutions of (4.2).
Thanks to symmetry (4.4),
performing a change of variables $(p',p'_*,p,p_*) \to (p,p_*,p',p'_*)$ we
get the fundamental formula:
for any $\psi = \psi(p)$,
$$\begin{aligned}
\int_{\mathbb{R}^3} Q_{1,2}(F,f) \psi \, dp = &{1 \over 2} 
\iiiint_{\mathbb{R}^{12}}
w_{1,2} \delta_{\mathcal{C}_{1,2}}  \bigl( F' f'_* (1+F) (1 +\tau
\,f_*) \\
&-  F f_* (1 + F') (1 +\tau  f'_*) \bigr) \bigl(\psi -
\psi' \bigr) \, dp\, dp_* dp'\, dp'_*\,.
\end{aligned}\eqno(4.5)
$$
A similar formula holds for $Q_{2,1}(f,F)$.


After integration of the two  equations of (4.2) separately, and using (1.7) and 
(4.5) we
formally get the particle number conservation
of each specie:
$$
\int_{\mathbb{R}^3} F(t,p) \, dp = \int_{\mathbb{R}^3} F_{\rm in}(p) \, dp\,, 
\quad
\int_{\mathbb{R}^3} f(t,p) \, dp = \int_{\mathbb{R}^3} f_{\rm in}(p) \, dp.
 \eqno(4.6)
$$
Multiplying both equations in (4.2) by $\psi(p) = p$, summing up and
using (1.7), (4.5) and $w_{1,2} =
w_{2,1}$ we obtain the global momentum conservation
$$
\int_{\mathbb{R}^3} \bigl( F(t,p) + f(t,p) \bigr) p  \, dp
= \int_{\mathbb{R}^3} \bigl( F_{\rm in}(p) + f_{\rm in}(p) \bigr) p  \, dp.
\eqno(4.7)
$$
Multiplying the first equation of (4.2) by $\mathcal{E}_1$, the second
equation of (4.2) by $\mathcal{E}_2$, using (1.7), (4.5) and the collision
invariance (4.1) we
get the global energy conservation
$$
\int_{\mathbb{R}^3} \bigl( F(t,p) \mathcal{E}_1(p) + f(t,p) \mathcal{E}_2(p) 
\bigr)\, dp
= \int_{\mathbb{R}^3} \bigl( F_{\rm in}(p) \mathcal{E}_1(p) + f_{\rm in}(p) 
\mathcal{E}_2(p) \bigr)
\, dp\,.
\eqno(4.8)
$$
Finally, we define the entropy of the system by
$$
H_{\mathcal{S}}(F,f) := H_1 (F) + H_{\tau } (f)
\eqno(4.9)
$$
with $H_\tau$ given by (1.1). Multiplying the first equation by $h_1'(F)$
and the second equation by
$h'_{\tau } (f)$ we obtain using (4.5)
$$
{d \over dt} H_{\mathcal{S}}(F,f) = D_\mathcal{S}(F,f) := \frac14  D_1(F) + {1 
\over 2}
D_{1,2} (F,f) +
\frac14  D_{\tau }(f) \ge 0,
 \eqno(4.10)
$$
where $D_1(F)$ and $D_{\tau }(f)$ are the usual dissipation entropy
production of one specie, and
$D_{1,2} (F,f)$ is the mixed dissipation entropy production given by
$$
\begin{aligned}
&D_{1,2} (F,f) :=\\
&\iiiint_{\mathbb{R}^{12}} w_{1,2} \delta_{\mathcal{C}_{1,2}}
j\bigl( {F' f'_* (1+F) (1 +\tau  f_*) , F'_* (1+F') (1 +\tau  f'_*) }
\bigr) \, dp dp_* {dp' dp'_*}.
\end{aligned}
\eqno(4.11)
$$

We list now several questions that one may naturally consider about the
system (4.2).

1. For the single quantum equation, one may consider the maximization
entropy problem under particle number, momentum and energy restriction i.e.: for 
any
given
$N_B, N_F, E > 0$,
$P
\in
\mathbb{R}^3$, to find a pair of functions
$(F,f)$ such that
$$
\begin{gathered}
\int_{\mathbb{R}^3} dF(p)= N_B, \quad \int_{\mathbb{R}^3} f(p) \, dp = N_F\\
 \int_{\mathbb{R}^3} p (F(p) +f(p)) \, dp = P, \quad
\int_{\mathbb{R}^3} (F(p) \mathcal{E}1(p) + f(p) \mathcal{E}_2(p)) \, dp = E.
\end{gathered}\eqno(4.12)
$$
and satisfying
$$
H_\mathcal{S}(F,f) = \max_{G,g s.t. (4.12)} H_\mathcal{S}(G,g),
\quad\hbox{with } H_\mathcal{S}(G,g) = H_{BE}(G) + H_\tau(g).
\eqno(4.13)
$$
We believe that a complete analysis of this problem can be done using
the same ideas exposed in the Section 2 and used in \cite{EMV}.

2. One  can also address the well posedness of the Cauchy problem for the
system (4.2). Of course the situation
here is the same as for the Boltzmann Bose equation. The
analysis performed in the Section 3.2 may be readily extended to prove the 
existence
of solutions under the assumption of isotropy of the distribution and with Lu's 
truncation
on the cross sections. A simpler question would be to consider the case when
$Q_{1,1}(F,F)$ and $Q_{2,2}(f,f)$
vanish and to address the well posedness of the Cauchy problem in this case.
Even  when $\tau  > 0$ (which gives an $L^\infty$ a priori bound on the Fermi
density $f$) we do not know if it is
possible to give a sense to the collision terms $Q_{1,2}(F,f)$ and
$Q_{2,1}(f,F)$ without Lu's truncation  on the cross sections.

We do not try to go further in any of these two directions and consider instead
the following question. In the study of gases formed by Bose and Fermi 
particles, it is particularly relevant
to consider the case where the Fermi particles are at equilibrium and where the
collisions between Bose particles can not distort significantly their 
distribution
function (cf. \cite{LY1,LY2}). This moreover constitutes a first important 
simplification
from a mathematical point of view. The system (4.2) reduces then to a unique 
equation on
the Bose distribution $F$. Moreover this equation is quadratic and not cubic
(c.f. sub Section 4.1)

A second simplification arises if we consider non relativistic, isotropic 
densities
and we assume  on physical grounds, that $w _{1,2}$ is constant, (c.f below). In 
that
case, we keep the same quadratic structure for the
equation on the Bose distribution $F$, but we obtain an explicit and quite 
simple
cross-section (c.f. sub Section 4.2).

In both situations, our main concern is to understand if it is possible to
obtain a global existence result
without the Lu's truncation on the cross sections and then to describe the long 
time
asymptotic behaviour of the solutions.

\subsection{Second specie at thermodynamical equilibrium} %4.1

Let us assume that $f = \mathcal{F}$ is at thermodynamical
equilibrium, which means that it is a Fermi or a
Maxwellian distribution defined by
$$
\mathcal{F}(p) = {1 \over e^{\nu(p)} -\tau }, \quad
\nu(p) = \beta^0 \mathcal{E}_2(p) - \beta \cdot p + \mu.\eqno{(4.14)}
$$
This greatly simplifies the situation since the system (4.2) reduces now to a 
single
equation for the Bose distribution
$F$ which reads
$$
{\partial F \over \partial t} = Q_{BQ}(F) := 
Q_{1,2}(F,\mathcal{F})\eqno{(4.15)}
$$
with
$$
Q_{BQ}(F) =  \int_{\mathbb{R}^3} S(p,p') \bigl( F' (1 + F) \, e^{- \beta^0 
\mathcal{E}_1(p)}
- F (1 + F') \, e^{- \beta^0 \mathcal{E}_1(p')} \bigr) \,  dp' ,
\eqno(4.16)
$$
$$
S(p,p') = \iint_{\mathbb{R}^3 \times \mathbb{R}^3} \!\!\! \!\!\! w_{1,2} 
\delta_{\mathcal{C}_{12}}
e^{\beta^0 \mathcal{E}_1(p)} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, dp_* 
dp'_*.
\eqno(4.17)
$$
To establish (4.16)-(4.17) we have used the elementary
identity
$$
e^{\beta^0 \mathcal{E}_1(p)} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*)
= e^{\beta^0 \mathcal{E}_1(p')} \mathcal{F}_* (1 +\tau  \mathcal{F}'_*)
\eqno(4.18)
$$
which holds on $\mathcal{C}_{12}$. Notice that, using the micro-reversibility
symmetry (4.4) and the identity
(4.18) we have
\begin{align*}
S(p',p) &= \iint_{\mathbb{R}^3 \times \mathbb{R}^3} \!\!\! \!\!\!
w_{1,2}(p',p_*,p,p'_*) \delta_{\mathcal{C}_{12}}
e^{\beta^0 \mathcal{E}_1(p')} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, dp_* 
dp'_* \\
&= \iint_{\mathbb{R}^3 \times \mathbb{R}^3} \!\!\! \!\!\! w_{1,2}(p,p_*,p',p'_*)
\delta_{\mathcal{C}_{12}}
e^{\beta^0 \mathcal{E}_1(p)} \mathcal{F}_* (1 +\tau  \mathcal{F}'_*)
\, dp_* dp'_* \\
&= S(p,p'),
\end{align*}
so that $S$ is symmetric.


Using the symmetry above one can observe that, at least formally, a
solution $F$ of equation (4.15) still
satisfies the qualitative properties
$$
\int_{\mathbb{R}^3} F \, dp = \int_{\mathbb{R}^3} F_{\rm in} \, dp
\eqno(4.19)
$$
and
$$
{d \over dt} H_{BQ}(F)  = D_{BQ} (F),\eqno{(4.20)}
$$
with
$$
H_{BQ}(F) = \int_{\mathbb{R}^3} \bigl((1+F)  \ln (1+F) - F \ln F - F \beta^0
\mathcal{E}_1(p) \bigr) \, dp\eqno{(4.21)}
$$
and
$$
D_{BQ} (F) = \iint_{\mathbb{R}^3 \times \mathbb{R}^3} S (p,p') \, j\bigl( F' (1 
+ F) \, e^{- \beta^0 \mathcal{E}_1(p)}  - F (1 + F') \, e^{- \beta^0 
\mathcal{E}_1(p')} \bigr) \,
dpdp'.\eqno{(4.22)}
$$
In other words, the particle number is preserved along the trajectories and 
$H_{BQ}$ is
a Lyapunov function (the relative entropy $H_{BQ}$ is a decreasing function 
along
the trajectories).


\subsubsection{Non relativistic particles, fermions at isotropic Fermi Dirac
equilibrium}

It is possible, under further simplifications of the model, to obtain more 
explicit
expressions of the cross section
$S$. As a first step in that direction we consider nonrelativistic
particles.  We also assume, without any loss of generality, that the two 
particles
have the same mass $m=1$ from where their energies are
$\mathcal{E}_i(p)=\mathcal{E}(p)
= |p|^2/2$, $i=1, 2$. We assume moreover that fermions are at isotropic 
equilibrium (see (2.39)):
$$\mathcal{F}(p)={1\over e^{\beta ^0{|p|^2\over 2\ }+b}-\tau }\quad
\hbox{for some}\,\theta 0, b\ge 0.$$
We introduce now the Carleman parametrization of the
collision manifold (4.1). Starting
by performing the $dp_*$ integration one finds
$$
S(p,p') =\! \int_{\mathbb{R}^3} w_{1,2} (p,p_*,p',p'_*) \delta_{|p'|^2 + 
|p'_*|^2
- |p|^2 - |p_*|^2 = 0}
\, e^{\beta ^0{|p|^2\over2 }} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, 
dp'_*,
\eqno(4.23)
$$
where now $p_*$ is defined by
$$
p_* := p' + p'_* - p_*.
\eqno(4.24)
$$
An elementary computation shows that
$$
|p'|^2 + |p'_*|^2 - |p|^2 - |p_*|^2 = - 2 (p'_* - p) \cdot (p'-p),
$$
so that in (4.23) $p'_*$ describes all the plane $E_{p,p'}$ orthogonal
to $p'-p$ and containing  $p$. Then, using the distributional Lemma 
\ref{lmA1.1}, we get
$$
S(p,p') = \int_{E_{p,p'}} {w_{1,2} (p,p_*,p',p'_*)  \over |p'-p|}
\, e^{\beta ^0{|p|^2\over2 }} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, 
dE(p'_*),
\eqno(4.25)
$$
where $dE(p'_*)$ stands for the Lebesgue measure on $E_{p,p'}$ and $p_*$ is
again defined thanks to
(4.24).

A further simplification may be performed, noticing that, on physical ground 
(cf.
Appendix \ref{A3}), the
boson-fermion interaction is a short range interaction and that the velocities 
of
the particles undergoing scattering are small . This allows to consider that
$w_{1,2}$ is constant, assuming without loss of generality that $w_{1,2}=1$. We 
then
obtain a more explicit expression of $S$ in (4.25). Namely:
$$
S(p,p') = \int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over 2 }}\over |p'-p|}
 {1\over e^{\beta ^0{|p'_*|^2\over 2 }+b} -\tau } {e^{\beta ^0{|p_*|^2\over 2 
}+b}\over e^{\beta ^0{|p_*|^2\over 2 }+b}-\tau }\, dE(p'_*).
$$
Note that
$${e^{b}\over 1+e^{b}}\le {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta 
^0{|p_*|^2\over 2
}+b}-\tau }<1$$
and
\begin{align*}
&\delta {e^{b}\over 1+e^{b}}\int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over
2}}\over |p'-p|} \, e^{-\beta ^0{|p'_*|^2\over 2 }-b} \, dE(p'_*)\\
&\le S(p,  p')<\int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over 2 }}\over
|p'-p|}  \, e^{-\beta ^0{|p'_*|^2\over 2}-b} \, dE(p'_*).
\end{align*}
Let us consider then the function
$$
S_0(p,p')\equiv\int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over 2 }}\over
|p'-p|}
 \, e^{-\beta ^0{|p'_*|^2\over 2 }} \, dE(p'_*)$$
In the orthonormal basis
$(\vec {k},\vec{i},\vec{j})$ where
$\vec{k} := (p'-p)/|p'-p|$, $p'_\star$ may be written as follows
$p'_\star = p + s \vec{i} + t \vec{j}$ with
$s,t \in \mathbb{R}$.
\begin{align*}
|p'_\star|^2 &=  |p|^2 + (s + p \cdot i)^2 - (p \cdot i)^2 + (t + p \cdot
j)^2 - (p \cdot j)^2 \\
&= \Bigl(p ,{p' - p \over |p'-p|} \Bigr)^2 + (s + p \cdot i)^2 + (t + p
\cdot j)^2,
\end{align*}
and, integrating (4.25), we find
\begin{align*}
S_0(p,p') &= {e^{\beta ^0{|p|^2\over 2}}  \over |p'-p|} \int \!\! 
\int_{\mathbb{R}^2}
e^{-\beta ^0|p'_*|^2/2} \, dsdt \\
&= {2 \pi  \over \beta^0 |p'-p|} \exp \Bigl[ {\beta ^0 \over 2 } \Bigl( |p|^2 - 
\bigl(p ,{p' - p \over
|p'-p|} \bigr)^2 \Bigr) \Bigr].
\end{align*}
This can also be written in the symmetric form
$$
S_0(p,p') = {2 \pi  \over \beta ^0 |p'-p|} \exp \Bigl[ {\beta ^0 \over 4 } 
\Bigl( |p|^2 + |p'|^2 - \bigl(p ,{p' - p
\over |p'-p|} \bigr)^2
- \bigl(p' ,{p' - p \over |p'-p|} \bigr)^2 \Bigr) \Bigr].\eqno{(4.26)}
$$

\begin{remark} \label{rmk4.1}\rm
If one considers the system
(4.2) when collisions between Boson particles are very weak (so
that they may be neglected) and collisions between Fermi particles are very
strong (in such a way that the
distribution of Fermi particles goes through thermodynamical
equilibrium very rapidly) we may consider
as relevant the following scaling
$$
\begin{aligned}
{\partial F_\epsilon \over \partial t} &= \epsilon \, 
Q_{1,1}(F_\epsilon,F_\epsilon) +
Q_{1,2}(F_\epsilon,f_\epsilon) \\
{\partial f_\epsilon \over \partial t} &= Q_{2,1}(f_\epsilon,F_\epsilon) + 
{1\over\epsilon}
\, Q_{2,2}(f_\epsilon,f_\epsilon),
\end{aligned} \eqno(4.27)
$$
in the limit $\epsilon \to 0$. Notice that the particle number conservation 
(4.6)
and energy conservation (4.8) provide the a priori bounds
$$
\sup_{\epsilon > 0} \sup_{t \in [0,\infty)} \int_{\mathbb{R}^3} F_\epsilon(t,p) 
(1 +
\mathcal{E}_1(p)) \, dp, \quad
\int_{\mathbb{R}^3} f_\epsilon(t,p) (1 + \mathcal{E}_2(p)) \, dp \le 
C.\eqno{(4.28)}
$$
Moreover, since
$$
{d \over dt} H_{\mathcal{S}}(F_\epsilon,f_\epsilon) = {\epsilon \over 4} 
D_1(F_\epsilon) + {1 \over
2} D_{1,2} (F_\epsilon,f_\epsilon) +
{1 \over 4 \epsilon} \, D_{\tau }(f_\epsilon),
$$
and all the terms at the right hand side are positive, we obtain
$$
\int_0^\infty D_{\tau }(f_\epsilon) \le \epsilon \, C(F_{\rm in},f_{\rm 
in}).\eqno{(4.29)}
$$
Formally, these bounds imply that, up to the extraction of a subsequence,
we have
$$
f_\epsilon \to \mathcal{F}, \quad F_\epsilon \to F,\eqno{(4.30)}
$$
where $\mathcal{F}$ has same momentum that $f_{\rm in}$ and $D_{\tau 
}(\mathcal{F}) = 0$ so
that $\mathcal{F}$ is the Fermi or Maxwellian
distribution associated to $f_{\rm in}$ given by (4.4) and $F$ solves the
Quadratic Bose equation (4.15)-(4.17).
\end{remark}

\begin{remark} \label{rmk4.2} \rm
It is important to notice that there is
no conservation of the energy for equation (4.15)-(4.16). The conserved
quantity in the system (4.2) is the total energy of the bosons-fermions gas
but not of any of the two species as it is shown by (4.8).
\end{remark}

\paragraph{Open questions:}
\begin{enumerate}
\item Establish rigorously (4.30).

\item Solve the Cauchy problem (4.15)-(4.17) for physically relevant $S$.
\end{enumerate}

\subsection{Isotropic distribution and second specie at the
thermodynamical equilibrium} %4.2

We now consider the Fermi particles
at non relativistic isotropic equilibrium and assume that the Bose distributions
are also isotropic. In other words we suppose that
$$
f(p, t)\equiv f(|p|)= {1\over e^{\beta ^0{|p|^2\over 2 }+b}-\tau }\quad
\hbox{and}\quad F(p, t)=F(|p|, t).\eqno{(4.31)}
$$
The interaction of an isotropic gas of Bose particles and Fermi particles at
equilibrium has been considered by Levich and  Yakhot in \cite{LY1} and
\cite{LY2}, by means of formal arguments. They were interested in particular
in the occurrence of Bose Einstein
condensation. Numerical simulations showing Dirac mass formation in infinite 
time for a
related equation have been obtained by  Semikoz and  Tkachev in \cite{ST2,ST2}.
The Fermi distributions considered in that case are  quantum, isotropic,
saturated Fermi Dirac distributions (SFD in Section 2). This corresponds to the 
choice
$$f(p)\equiv f(|p|)={\bf 1}_{\{0\le |p|\le \mu \}}$$
for some $\mu >0$, and gives rise to a slightly different equation than ours.

The Cauchy problem for the resulting quadratic Bose equation is a
``linearized model'' of the Bose-Bose interaction equation considered in
Section 3, where moreover, the function $S(p, p')$ may be calculated  
explicitly.

Similar equations have been considered in \cite{EM1,EM2}. Nevertheless, the 
global
existence results obtained in these references do not apply to our case, because
the  collision kernel does not fulfill the required hypothesis. Therefore, we 
end
this Section proving  global existence of solutions, with integrable
initial data, to our problem. Finally, the long time behaviour of these global 
solutions
may be addressed exactly as in \cite{EM2} and then, we only state the result for 
the sake of
completeness.

We start with the following:

\begin{proposition} \label{prop4.3}
If the function $F$ is radially symmetric then so is $Q_{BQ}(F)$.
In that case we write $Q_{BQ}(F)(p)=Q_{BQ}(F)({{|p|^2\over 2\theta}})$
by abuse of notation. Moreover,
$$
Q_{BQ}(F) (\varepsilon) =
\int_0^\infty S \bigl[ F' (1 + F) \, e^{-\epsilon} - F (1 + F') \,
e^{-\epsilon'} \bigr] \,  d\epsilon',
\eqno(4.32)
$$
with $S(\epsilon,\epsilon') = \Sigma(\epsilon,\epsilon') / \sqrt{\epsilon}$ and
$$
\Sigma (\epsilon,\epsilon') =
\int_{\max(0,\epsilon-\epsilon')}^\infty \, e^{\epsilon-\epsilon'_*} 
{e^{\varepsilon' +
\varepsilon'_*- \varepsilon+b}\over  e^{\varepsilon' +\varepsilon'_*-
\varepsilon+b}-\tau }\min(\sqrt{\epsilon},\sqrt{\epsilon'+\epsilon'_*-
\epsilon},\sqrt{\epsilon'},\sqrt{\epsilon'_*}) \,
d\epsilon'_*.\eqno{(4.33)}
$$
\end{proposition}

\paragraph{Proof}
It follows from (4.17) and (4.31) that
\begin{align*}
S(p,p')&= e^{\beta ^0{|p|^2\over2 }} \int\int_{\mathbb{R}^3 \times \mathbb{R}^3}
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0}\,\delta _{p'+p'_*-p-p_*=0}\\
&\hskip 3cm \times {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta
^0{|p'_*|^2\over 2 }+b}-\tau } {dp_* dp'_*\over e^{\beta ^0{|p_*|^2\over 2 }+b}-
\tau }
\\
&=\int_0^{\infty }\int_0^{\infty }
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0}{e^{\beta ^0{|p|^2\over2 }}\over e^{\beta ^0{|p'_*|^2\over 
2
}+b}-\tau } {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta ^0{|p_*|^2\over 2
}+b}-\tau }\\
&\hskip 3cm \times
\int_{S^2}\int_{S^2}\delta _{p'+p'_*-p-p_*=0}\,d\omega _*d\omega
'_*|p'_*|^2d|p'_*\|p_*|^2d|p_*|.
\end{align*}
Therefore,
\begin{align*}Q_{BQ}(F)=&\int_0^{\infty }\int_0^{\infty }\int_0^{\infty }
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0} {e^{\beta ^0{|p|^2\over2 }}\over e^{\beta 
^0{|p'_*|^2\over 2
}+b}-\tau } \,  {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta ^0{|p_*|^2\over 2
}+b}-\tau }\\
& \times  [F'(1+F)e^{-\beta ^0{|p|^2\over 2 }}-
 F(1+F')e^{-\beta ^0{|p'|^2\over 2 }}]\cdot \\
& \times \int_{S^2}\int_{S^2}\int_{S^2}\delta _{p'+p'_*-p-p_*=0}
\,d\omega _*
d\omega '_* d\omega '|p'_*|^2d|p'_*\|p_*|^2d|p_*\|p'|^2d|p'|.
\end{align*}
As we have already seen in Section 3,
$$
\int_{S^2}\int_{S^2}\int_{S^2}\delta _{p'+p'_*-p-p_*=0}\,d\omega _* d\omega '_*
d\omega '={4 \pi ^2\over |p\|p'\|p_*\|p_*'|}\min \bigl[|p|,|p'|, |p_*|, |p_*'|
\bigr]
$$
from where,
\begin{align*}
Q_{BQ}(F)=&{4 \pi ^2\over |p|}\int_0^{\infty }
[F'(1+F)e^{-\beta ^0{|p|^2\over 2 }}-F(1+F')e^{-{|p'|^2\over
2\theta }}]e^{\beta ^0{|p|^2\over2}}\\
&\Big\{\int_0^{\infty }\int_0^{\infty}
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0} {e^{\beta ^0{|p|^2\over2 }}\over e^{\beta 
^0{|p'_*|^2\over 2
}+b}-\tau } \,  {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta ^0{|p_*|^2\over 2
}+b}-\tau }\\
&\min \bigl[|p|,|p'|, |p_*|,
|p_*'|\bigr]\,|p'_*|d|p'_*\|p_*|d|p_*|\Big\}|p'|d|p'|\\
=&\int_0^{\infty }
\big[F'(1+F)e^{-\beta ^0{|p|^2\over 2 }}-F(1+F')e^{-\beta ^0{|p'|^2\over
2 }}\big]\,\overline{S}(|p|,|p'|)|p'|d|p'|,
\end{align*}
\begin{align*}
\overline{S}(|p|, |p'|)&={4 \pi ^2\over |p|}\int_0^{\infty }\int_0^{\infty}
\delta_{|p'|^2 + |p'_*|^2- |p|^2 - |p_*|^2 = 0} {e^{\beta ^0{|p|^2\over2 }}\over
e^{\beta ^0{|p'_*|^2\over 2}+b}-\tau } \,  {e^{\beta ^0{|p_*|^2\over 2 }+b}\over
e^{\beta ^0{|p_*|^2\over 2 }+b}-\tau }\\
&\hskip 3cm \times \min[|p|,|p'|, |p_*|,
|p_*'|]|p'_*|d|p'_*\|p_*|d|p_*|\\
&={4 \pi ^2\over |p|}\int_0^{\infty }{e^{\beta ^0{|p|^2\over2 }}\over e^{\beta 
^0{|p'_*|^2\over
2 }+b}} {e^{\beta ^0{{|p'|^2 + |p'_*|^2- |p|^2}\over 2 }+b}\over e^{\beta 
^0{{|p'|^2 +
|p'_*|^2- |p|^2}\over 2 }+b}-\tau }{\bf 1}_{\{|p'|^2+|p'_*|^2\ge |p^2|\}} \\
&\hskip 3cm \times\min \bigl[|p|,|p'|,
\sqrt{|p'|^2 + |p'_*|^2- |p|^2}, |p_*'|\bigr]\,|p'_*|d|p'_*|
\end{align*}
By defining
$$
\varepsilon=\beta ^0{|p|^2\over 2 },
\quad \varepsilon'=\beta ^0{|p'|^2\over 2},
\varepsilon_*=\beta ^0{|p_*|^2\over 2 },
\quad \varepsilon'_*=\beta ^0{|p'_*|^2\over 2 },
$$
we have
\begin{align*}
\overline{S}(\sqrt {2\varepsilon/\beta ^0 }, \sqrt {2\varepsilon'/\beta ^0})
&=4\pi ^2\sqrt{2}(\beta ^0)^{-3/2}e^{-b} \int_0^{\infty
}e^{\varepsilon-\varepsilon'_*}{e^{\varepsilon' +
\varepsilon'_*- \varepsilon+b}\over  e^{\varepsilon' +\varepsilon'_*-
\varepsilon+b}-\tau}{\bf 1}_{ \{ \epsilon' + \epsilon'_* \ge
\epsilon \} }\\
&\quad\times {\min(\sqrt{\epsilon},\sqrt{\varepsilon' +
\varepsilon'_*- \varepsilon},\sqrt{\epsilon'},\sqrt{\epsilon'_*})\over \sqrt 
{\varepsilon}}
d\varepsilon'_*
\equiv S(\varepsilon, \varepsilon').
\end{align*}
Finally
$$
Q_{BQ}(F)= \int_0^\infty \bigl[ F' (1 + F) \, e^{-\epsilon} - F (1 + F') \,
e^{-\epsilon'} \bigr] \, S \,  d\epsilon'
$$
with $S(\epsilon,\epsilon') = \Sigma(\epsilon,\epsilon') / \sqrt{\epsilon}$ and
$$
\Sigma (\epsilon,\epsilon') =
\int_{\max(0,\epsilon-\epsilon')}^\infty \, e^{\epsilon-\epsilon'_*} 
{e^{\varepsilon' +
\varepsilon'_*- \varepsilon+b}\over  e^{\varepsilon' +\varepsilon'_*-
\varepsilon+b}-\tau }\min(\sqrt{\epsilon},\sqrt{\epsilon'+\epsilon'_*-
\epsilon},\sqrt{\epsilon'},\sqrt{\epsilon'_*}) \,
d\epsilon'_*.
$$
By a change of variables in the integral definition of $\Sigma$  see
that it is a symmetric
function.
We may then write equation (4.15) as
$$
\sqrt{\epsilon} {\partial F \over \partial t} =
\int_0^\infty  \Sigma \bigl[ F' (1 + F) \, e^{-\epsilon} - F (1 + F')
\, e^{-\epsilon'} \bigr] \, d\epsilon'.
$$
and perform the usual change of variable
$$
\sqrt{\epsilon} F \to F, \quad b(\epsilon,\epsilon') = {\Sigma 
(\epsilon,\epsilon') \over
\sqrt{\epsilon} \sqrt{\epsilon'}}
$$
to obtain the more suitable form
$$
{\partial F \over \partial t} =
\int_0^\infty  b(\epsilon,\epsilon') \bigl[ F' (\sqrt{\epsilon} + F) \, e^{-
\epsilon}
- F (\sqrt{\epsilon'} + F') \, e^{-\epsilon'} \bigr] \, d\epsilon',
\eqno(4.34)
$$
Notice that $b$ is singular near the origin and as a consequence we can not
apply the results obtained in \cite{EM1, EM2}.

 We first want to give a precise mathematical sense
to the collision term in (4.34). Notice that it can be written t in the 
following way
\begin{align*}
Q(F) = &\int_0^\infty b(\epsilon,\epsilon') \sqrt{\epsilon} \, e^{-\epsilon} F' 
\, d\epsilon'
- F \int_0^\infty b(\epsilon,\epsilon') \sqrt{\epsilon'} \, e^{-\epsilon'} \, 
d\epsilon' \\
&+ \int_0^\infty b(\epsilon,\epsilon') (e^{-\epsilon} - e^{-\epsilon'} ) \,  F 
F' \,
d\epsilon'.
\end{align*}

\begin{lemma} \label{lm4.4}
The function
$$
\ell(\epsilon) := \int_0^\infty b(\epsilon,\epsilon') \sqrt{\epsilon'} \, e^{-
\epsilon'} \,
d\epsilon'.
\eqno(4.37)
$$
satisfies $\ell \in C([0,\infty))$, $\ell\le C_1(1+\sqrt{\varepsilon})$ for
some positive constant $C_2$, $\ell
\to C_2(b, \tau )$ when $\epsilon
\to 0$ and
$\ell \sim \gamma \sqrt{\epsilon}$
when $\epsilon \to \infty$, with $\gamma > 0$. Moreover, we have
$$
\chi(\epsilon,\epsilon') := ( e^{-\epsilon} - e^{-\epsilon'}) 
b(\varepsilon,\varepsilon')
\in C_b([0,\infty) \times [0,\infty)).
\eqno(4.38)
$$
As a conclusion, for any $F$ such that $(1 +\sqrt{\epsilon}) F \in L^1$,
$Q(F)$ belongs to $L^1$ and the map $F\mapsto Q(F)$ is continuous from
$L_{1/2}^1$ to $L^1$.
\end{lemma}


\paragraph{Proof of Lemma \ref{lm4.4}}
In order to prove the properties of $\ell$ we write
$$
\ell(\epsilon)
= {1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta_b (\epsilon') \, e^{-
\epsilon'} \, d\epsilon'
+  {1 \over \sqrt{\epsilon}} \int_0^\epsilon \xi _b (\epsilon, \varepsilon') \, 
e^{-\epsilon'} \, d\epsilon'
$$
where,
\begin{gather*}
\zeta _b(z)=\int_0^{\infty }e^{z-k}{e^{k+b}\over
e^{k+b}-\tau }\min(\sqrt{z},\sqrt{k})\,dk \\
\xi _{b}(z, y)=\int_0^{\infty }e^{z-k}{e^{k-z+y+b}\over
e^{k-z+y+b}-\tau }\min(\sqrt{z},\sqrt{k})\,dk.
\end{gather*}
Note that
$$
\zeta _b(z)\le \int_0^{\infty }e^{z-k}\min(\sqrt{z},\sqrt{k})dk= \gamma
(z)e^{z}+\sqrt {z}\equiv\zeta (z)
$$
with $\gamma (x)=\int_0^{x}e^{-y}\sqrt{y}dy$,
and $\xi _b(z, y)\le \zeta (z)$.
Assume first that $\varepsilon\to 0$. Then,
$$
{1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta_b (\epsilon') \, e^{-\epsilon'} 
\, d\epsilon' \le
{1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta (\varepsilon') e^{-\epsilon'} \, 
d\epsilon'\to 0\quad
\hbox{as}\,\varepsilon\to 0.
$$
On the other hand,
\begin{align*}
&{1 \over \sqrt{\epsilon}} \int_\epsilon ^{\infty }\xi _b (\epsilon, 
\varepsilon') \,
e^{-\epsilon'} \,d\epsilon'\\
&={1\over \sqrt{\epsilon}}\int_{\varepsilon}^{\infty
}e^{-\varepsilon'}\int_0^{\varepsilon}e^{\varepsilon-\varepsilon'_*}
{e^{\varepsilon'+\varepsilon'_*-\varepsilon+b}\over
e^{\varepsilon'+\varepsilon'_*-\varepsilon+b}-\tau }\sqrt
{\varepsilon'_*} d\varepsilon'_*d\varepsilon'+\\
&\quad+\int_{\varepsilon}^{\infty }e^{-\varepsilon'}\int_{\varepsilon}^{\infty }
e^{\varepsilon-\varepsilon'_*}{e^{\varepsilon'+\varepsilon'_*-
\varepsilon+b}\over
e^{\varepsilon'+\varepsilon'_*-\varepsilon+b}-\tau }\sqrt
{\varepsilon'_*} d\varepsilon'_*d\varepsilon'\\
&\quad
\to \int_0^{\infty }e^{-\varepsilon'}\int_0^{\infty }
e^{-\varepsilon'_*}{e^{\varepsilon'+\varepsilon'_*+b}\over
e^{\varepsilon'+\varepsilon'_*+b}-\tau }\sqrt
{\varepsilon'_*} d\varepsilon'_*d\varepsilon'=C_2(b, \tau )\quad
\hbox{as}\,\varepsilon\to 0.
\end{align*}
Suppose now that $\varepsilon\to \infty $. We first note that,
$$
{1 \over \sqrt{\epsilon}} \int_0^\epsilon \xi _b (\epsilon, \varepsilon') \, 
e^{-\epsilon'}
\, d\epsilon'\le {1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta (\varepsilon')
\, e^{-\epsilon'}
\, d\epsilon'\to 0\quad \hbox{as}\,\ \varepsilon\to \infty.
$$
Finally,
\begin{align*}
{1\over \sqrt{\varepsilon}}\int_0^{\varepsilon}e^{-\varepsilon'}\zeta
_b(\varepsilon')d\varepsilon'
=&{1\over\sqrt{\varepsilon}}\int_0^{\varepsilon}
\int_0^{\varepsilon'}e^{-k}{e^{k+b}\over e^{k+b}-\tau }
\sqrt{k}dk\,d\varepsilon'\\
&+{1\over \sqrt{\varepsilon}}\int_0^{\varepsilon}
\sqrt{\varepsilon'}\int_{\varepsilon'}^{\infty}
e^{-k}{e^{k+b}\over e^{k+b}-\tau }dkd\varepsilon'\equiv I_1+I_2
\end{align*}
and,
\begin{gather*}
I_2\le {1\over
\sqrt{\varepsilon}}
\int_0^{\varepsilon}\sqrt{\varepsilon'}e^{-\varepsilon'}d\varepsilon'=O({1\over
\sqrt{\varepsilon}}),\\
\lim_{\varepsilon\to \infty }{1\over \varepsilon}\int_0^{\varepsilon}
\int_0^{\varepsilon'}e^{-k}\sqrt{k}dkd\varepsilon'=\int_0^{\infty }e^{-k}
\sqrt{k}dk\equiv \gamma
\end{gather*}
To prove (4.38) we just have to consider the two cases:
If $\min(\epsilon,\epsilon') \to 0$ then
$$
\chi \ \sim \ { e^{-\max} - e^{-\min} \over \sqrt{\max} }  \
\mathop{\longrightarrow}_{\max \to 0,\infty} \ 0
\quad\hbox{ uniformly in } \min.
$$
If $\min(\epsilon,\epsilon') \to \infty$ then $\epsilon,\epsilon' \to \infty$ 
and
$$
|\chi| \ \le \ { e^{-\min} - e^{-\max} \over \sqrt{\max} \sqrt{\min} }
\, e^{\min} \le 1.
$$
\hfill$\square$ \smallskip

The particle number of the solutions to (4.34) is constant in time (at least
formally) and this gives a first a
priori bound in $L^1$. But we have still need to control higher moments of $F$.
We show in the next Lemma how this may be formally done for polynomial momments.
Exponential moments may be controlled in a  similar way.

\begin{lemma} \label{lm4.5}
Let $F$ be a solution to equation (4.34). Then, there exists
a positive constant $C$ such that, formally:
$$
{d \over dt} Y_1(F) + {\gamma \over 4} \int_0^\infty F (1 + \sqrt{\epsilon})
\epsilon \, d\epsilon \le C_{\rm in},
\eqno(4.39)
$$
with
$$
Y_\theta (F) := \int_0^\infty \epsilon^\theta \, d|F|(\epsilon).
\eqno(4.40)
$$
\end{lemma}

\paragraph{Proof of Lemma \ref{lm4.5}}
Let $F$ be a solution to (4.34). A formal straightforward calculation gives
\begin{align*}
{d \over dt} Y_1(F)
&= \int_0^\infty \!\!\! \int_0^\infty  b F F' [e^{-\epsilon} -
e^{-\epsilon'}] \epsilon \,
d\epsilon'd\epsilon \\
&\quad
+ \int_0^\infty \!\!\! \int_0^\infty  b [ \sqrt{\epsilon} F' \epsilon \,
e^{-\epsilon}
- \sqrt{\epsilon'} F \, e^{-\epsilon'} \epsilon ] \, d\epsilon'd\epsilon \\
&= \int_0^\infty \!\!\! \int_0^\infty { b \over 2} F \,  F' [e^{-\epsilon} - 
e^{-\epsilon'}] ( \epsilon - \epsilon') \,d\epsilon'd\epsilon \\
&\quad
+ \int_0^\infty \!\!\! \int_0^\infty  b \sqrt{\epsilon'} F \, e^{-\epsilon'} 
[\epsilon' - \epsilon] \, d\epsilon'd\epsilon.
\end{align*}
Since the first term is non positive, we just have to manage with the
second term.
In order to estimate it we compute
\begin{align*}
I(\epsilon)
&:= \int_0^\infty b \sqrt{\epsilon'} \, e^{-\epsilon'} \,  [ \epsilon' - 
\epsilon] \,
d\epsilon'  \\
&= \int_0^\epsilon {\zeta(\epsilon') \over \sqrt{\epsilon}} \, e^{-\epsilon'} \,  
[ \epsilon' -
\epsilon] \, d\epsilon'
+  {\zeta(\epsilon) \over \sqrt{\epsilon}} \int_\epsilon^\infty \,  e^{-
\epsilon'} \,  [
\epsilon' - \epsilon] \, d\epsilon' = I_1 + I_2.
\end{align*}
On the one hand,
$$
I_2 = {\zeta(\epsilon) \over \sqrt{\epsilon}} \Bigl( \int_\epsilon^\infty 
\epsilon' \,
e^{-\epsilon'} \, d\epsilon'
- \epsilon \int_\epsilon^\infty \, e^{-\epsilon'} \, d\epsilon' \Bigr)\\
= {\zeta(\epsilon) \over \sqrt{\epsilon}} \, e^{-\epsilon} = e^{-\epsilon} + 
{\gamma
(\epsilon) \over \sqrt{\epsilon}} \le C.
$$
On the other hand,
$$
I_1 \le {1 \over \sqrt{\epsilon}} \int_0^\epsilon \gamma(\epsilon') (\epsilon'-
\epsilon) \,
d\epsilon'
\le {\gamma(m) \over \sqrt{\epsilon}} \int_m^\epsilon (\epsilon'-\epsilon) \, 
d\epsilon' =
{\gamma(m) \over \sqrt{\epsilon}} \bigr(-{\epsilon^2 \over 2} + m \epsilon - 
{m^2
\over 2} \bigl).
$$
Since $\gamma(m) \to \gamma$ when $m \to \infty$ and the term of greater
order have a minus sign,
we obtain
$$
I \le C - {\gamma \over 4} (1+\sqrt{\epsilon}) \epsilon,
$$
and (4.39) follows. \hfill$\square$

We state now the global existence result for the equation (1.26).

\begin{theorem} \label{thm4.6}
For any initial datum $F_{\rm in} \in L^1_1(\mathbb{R}_+)$, $F\ge 0$, there 
exists a
solution $F \in C([0,\infty),L^1_{1/2}))$ to the equation
$$
{\partial F\over \partial t}=Q_{BQ}(F),\quad
\hbox{for}\quad \varepsilon>0,\,\,t>0
$$
with $Q_{BQ}$ defined in (4.32), and such that
$$
\lim_{t\to 0}\|F(t)-F_{\rm in}\|_{L^1(\mathbb{R}^+)}=0.$$
\end{theorem}

\paragraph{Proof of Theorem \ref{thm4.6}}
We first introduce the regularized problem:
$$
{\partial F_n \over \partial t} =
\int_0^\infty  b_n(\epsilon,\epsilon') \bigl[ F'_n (\sqrt{\epsilon_n} + F_n) \,
e^{-\epsilon}
- F_n (\sqrt{\epsilon'_n} + F'_n) \, e^{-\epsilon'} \bigr] \, d\epsilon',
\eqno(4.41)
$$
with
$$
b_n(\epsilon,\epsilon') = {\zeta(\epsilon \wedge \epsilon' \wedge n)  \over 
\sqrt{\epsilon
\wedge \epsilon' \wedge n} \sqrt{\epsilon \vee \epsilon' \vee 1/n} }, \quad
\epsilon_n = \epsilon \vee {1 \over n}.
$$
The existence of a solution $F_n$ to this equation follows from the existence
result in [EM] since $b_n \in L^\infty$. We obtain a priori estimates on the
sequence $(F_n)$ in the two following Lemmas and then pass to the limit
using Lemma \ref{lm4.4}

\begin{lemma} \label{lm4.7}
The sequence of solutions $(F_n)$ satisfies
$$
{d \over dt} Y_1(F_n)
+ {\gamma \over 3} \int_0^\infty F_n \sqrt{\epsilon} (\epsilon \wedge n) \, 
d\epsilon
\le C_{\rm in} + Y_1(F_n).
\eqno(4.42)
$$
for $n$ large enough. This implies that $Y_1(F_n)$ is bounded in
$L^\infty(0,T)$ and \hfill \break
$\int_0^\infty F_n \sqrt{\epsilon} (\epsilon \wedge n) \, d\epsilon$ is bounded
in $L^1(0,T)$ for any $T > 0$.
\end{lemma}


\paragraph{Proof of Lemma \ref{lm4.7}}
 By the same computation as in Lemma \ref{lm4.5}, which
makes now perfectly sense by the properties of $F_n$, we deduce
$$
{d \over dt} Y_1(F_n) = \int_0^\infty \!\!\! \int_0^\infty { b_n \over 2}
F_n \,  F'_n [e^{-\epsilon} - e^{-\epsilon'}] ( \epsilon - \epsilon') 
\,d\epsilon'd\epsilon
+ \int_0^\infty F_n \, I_n \, d\epsilon.
$$
The first term in the right hand side is non positive and the second satisfies:
\begin{align*}
I_n(\epsilon)
&:= \int_0^\infty b_n \sqrt{\epsilon'_n} \, e^{-\epsilon'} \,  [ \epsilon' - 
\epsilon ]
\, d\epsilon'  \\
&\le \int_0^\epsilon { \gamma (\epsilon' \wedge n) \sqrt{\epsilon'_n} \over
\sqrt{\epsilon_n} \sqrt{\epsilon' \wedge n} }
\, e^{\epsilon' \wedge n - \epsilon'} [\epsilon'-\epsilon] d\epsilon'
+ { \zeta(\epsilon \wedge n) \over \sqrt{\epsilon \wedge n} } \, e^{-\epsilon}.
\end{align*}
Note that the last term is bounded.
Let fix $\ell \ge 1$ such that $\gamma(\ell) > 2 \gamma/3$. For $n$ and
$\epsilon$ such that $\epsilon \wedge n \ge
\ell$ we write
\begin{align*}
&\int_0^\epsilon { \gamma (\epsilon' \wedge n) \sqrt{\epsilon'_n}
 \over \sqrt{\epsilon_n}\sqrt{\epsilon' \wedge n} }
 e^{\epsilon' \wedge n-\epsilon'} [\epsilon'-\epsilon] d\epsilon'\\
&\le {2 \gamma \over 3 \sqrt{\epsilon}} \int_0^{\epsilon} {\sqrt{\epsilon'} 
\over
\sqrt{\epsilon' \wedge n}}
\, e^{\epsilon'\wedge n - \epsilon'} (\epsilon'-\epsilon) \, d\epsilon' + \gamma 
\int_0^\ell  {\epsilon \over \sqrt{\epsilon'
\wedge n}} \, d\epsilon' \\
&\le {2 \gamma \over 3 \sqrt{\epsilon}} \int_0^{\epsilon \wedge n} (\epsilon'-
\epsilon) \, d\epsilon'
+ \gamma \, 2 \sqrt{\ell} \epsilon
\le 2 \sqrt{\ell} \gamma \epsilon - {\gamma \over 3} \sqrt{\epsilon} (\epsilon 
\wedge n).
\end{align*}
We then conclude as in the proof of Lemma \ref{lm4.5}. \hfill$\square$

\begin{lemma} \label{lm4.8}
 The set $(F_n)$ is a Cauchy sequence
in $C([0,T];L^1_{1/2}(\mathbb{R}_+))$ for any $T > 0$.
\end{lemma}

\paragraph{Proof of Lemma \ref{lm4.8}} We just compute
$$
{\partial \over \partial t} (F_m-F_n) = Q_n(F_m) - Q_n(F_n) + Q_m(F_m) -
Q_n(F_m)
$$
and
$$
\begin{aligned}
&{d \over dt} \| F_m-F_n \|_{L^1_{1/2}}\\
&\le \int_0^\infty |F_m - F_n| \int_0^\infty b_n {\sqrt{\epsilon_n'}} \, e^{-
{\epsilon'}} [{\sqrt{\epsilon} - \sqrt{\epsilon'} }] \, {d\epsilon'} \, d\epsilon
\\
&+ \int_0^\infty\!\!\!\int_0^\infty |{F_m F'_m - F_n F'_n}| |e^{-\epsilon} - e^{-
\epsilon'} \, b_n (1 +{\sqrt{\epsilon}}) \, {d\epsilon'}d\epsilon \\
&+ \int_0^\infty \!\!\! \int_0^\infty  (b_m - b_n) F_m \,  {F'_m} |e^{-\epsilon} - 
e^{-\epsilon'}| ( 1 + {\sqrt{\epsilon}} ) \,{d\epsilon'}d\epsilon\\
&+ \int_0^\infty \!\!\! \int_0^\infty  |b_m {\sqrt{\epsilon'_m}} - b_n 
{\sqrt{\epsilon'_n}}| F_m \, e^{-\epsilon'}
\,   [ 2 + {\sqrt{\epsilon'} + \sqrt{\epsilon}}] \, {d\epsilon'}d\epsilon,\\
&\equiv J_1+J_2+J_3+J_4
\end{aligned} \eqno(4.43)
$$
with $b_{m,n} = b_m - b_n$.
We now estimate each of the terms $J_i$, $i=1,2,3,4$ separately.


\noindent{\sl 1. Estimate of $J_1$.} The first term is nothing but
$$
J_1(n) = \int_0^\infty |F_m - F_n| \, I_n \, d\epsilon
$$
with
\begin{align*}
I_n(\epsilon):=& \int_0^\infty b_n \sqrt{\epsilon'_n} \, e^{-\epsilon'} [ 
\sqrt{\epsilon'} - \sqrt{\epsilon} ] \, d\epsilon' \\
\le& {\zeta(\epsilon\wedge n) \over \sqrt{\epsilon\wedge n}} 
\int_\epsilon^\infty
e^{-\epsilon'} (\sqrt{\epsilon'} - \sqrt{\epsilon}) \, d\epsilon' =: I'_n.
\end{align*}
For $\epsilon \ge 1$, integration by parts gives
$$
I'_n = {\zeta(\epsilon\wedge n) \over \sqrt{\epsilon\wedge n}} 
\int_\epsilon^\infty
{e^{-\epsilon'} \over
\sqrt{\epsilon'}}
\, d\epsilon' \le {\zeta(\epsilon\wedge n) \over \sqrt{\epsilon\wedge n}} \, 
e^{-\epsilon}
\in L^\infty (\mathbb{R}_+).
$$
Since $I'_n$ is bounded for $\epsilon \le 1$, there exists a constant $K_1 > 0$
such that
$$
J_1 \le K_1 \| F_m - F_n \|_{L^1}.
\eqno(4.44)
$$
{\sl 2. Estimate of $J_2$.} We notice that
\begin{align*}
\chi_n &:= b_n (e^{-\epsilon} - e^{\epsilon'}) \\
&\le {\zeta(\epsilon \wedge \epsilon') \over \sqrt{\epsilon \wedge \epsilon'}}
{e^{-\epsilon} - \epsilon^{-\epsilon'} \over \sqrt{\epsilon \vee \epsilon'}} 
{\bf 1}_{
\epsilon \wedge \epsilon' \le 1}
+ {\zeta(\epsilon \wedge \epsilon' \wedge n) \over \sqrt{\epsilon \wedge 
\epsilon'
\wedge n}}\max(e^{-\epsilon},\epsilon^{-\epsilon'} ) {\bf 1}_{ \epsilon \vee 
\epsilon'
\ge 1}
\end{align*}
is uniformly bounded in $\mathbb{R}_+^2$ as we have already shown in the proof
of Lemma \ref{lm4.4}. Then,  we deduce
$$
\begin{aligned}
J_2 \!\le& \| \chi_n\|_{L^\infty_{\epsilon,\epsilon'}} [ \| (F_m - F_n) 
(1+\sqrt{\epsilon}) \|_{L^1} \| F_m
\|_{L^1} + \| F_m - F_n \|_{L^1} \| F_m (1+\sqrt{\epsilon}) \|_{L^1} \\
\le& K_2 \| F_m \|_{L^1_{1/2}} \,  \| F_m - F_n \|_{L^1_{1/2}},
\end{aligned}\eqno{(4.45)}
$$
for some constant $K_2 > 0$.


\noindent{\sl 3. Estimate of $J_3$.} Since $x \mapsto \zeta(x)/\sqrt{x}$ is
increasing (at least for the
large values of $x$) we have $0 \le b_n \le b_m$. We then notice that
\begin{align*}
0 \le b_m - b_n = &{\zeta(\epsilon \wedge \epsilon') \over \sqrt{\epsilon \wedge 
\epsilon'}} \bigl( {1 \over \sqrt{\epsilon \vee \epsilon' \vee 1/m}} - \sqrt{n} 
\bigr) {\bf
1}_{\epsilon \vee \epsilon' < 1/n} \\
&+ {1 \over \sqrt{\epsilon \vee \epsilon'}} \bigl( {\zeta(\epsilon \wedge 
\epsilon'
\wedge m) \over \sqrt{\epsilon \wedge \epsilon'
\wedge m}} - {\zeta(n) \over \sqrt{n}} \bigr) {\bf 1}_{\epsilon \wedge \epsilon'
n}.
\end{align*}
On the one hand, we have
\begin{align*}
&(b_m - b_n) |e^{-\epsilon} - e^{-\epsilon'}| {\bf 1}_{\epsilon \vee
\epsilon' < 1/n}\\
&\le
{\zeta(\epsilon \wedge \epsilon') \over \sqrt{\epsilon \wedge \epsilon'}} \,  {1 
\over
\sqrt{\epsilon \vee \epsilon' \vee 1/m}}
| e^{-\epsilon} - e^{-\epsilon'} | {\bf 1}_{\epsilon \vee \epsilon' < 1/n} \\
&\le \| {\zeta(x) \over \sqrt{x}} \|_{L^\infty (0,1)}
 \sqrt{\epsilon \vee \epsilon'}  {\bf 1}_{\epsilon \vee \epsilon' < 1/n}
\le {K_3 \over\sqrt{n}}  {\bf 1}_{\epsilon \vee \epsilon' < 1/n}\,.
\end{align*}
On the other hand
\begin{align*}
(b_m - b_n) |e^{-\epsilon} - e^{-\epsilon'}| {\bf 1}_{\epsilon \wedge \epsilon' 
\ge n}
&\le
{\zeta(\epsilon \wedge \epsilon' \wedge m) \over \sqrt{\epsilon \wedge \epsilon' 
\wedge m}} {1 \over \sqrt{\epsilon \vee \epsilon'}}
| e^{-\epsilon} - e^{-\epsilon'} | {\bf 1}_{\epsilon \wedge \epsilon' \ge n} \\
&\le {2 \over \sqrt{\epsilon \vee \epsilon'}} \| {\zeta(x) \, e^{-x} \over
\sqrt{x}} \|_{L^\infty}
{\bf 1}_{\epsilon \wedge \epsilon' \ge n} \le {K_3 \over\sqrt{n}} {\bf
1}_{\epsilon \wedge \epsilon' > n}
\end{align*}
for some constant $K_3 > 0$. We deduce that
$$
J_3 \le {K_3 \over \sqrt{n}} \, Y_0(F_m) \| F_m \|_{L^1_{1/2}}.\eqno{(4.46)}
$$
{\sl 4. Estimate of $J_4$.} Since $0 \le \sqrt{\epsilon'_n} \, b_n \le
\sqrt{\epsilon'_m} \, b_m$ for $n$ large
enough, we may write
\begin{align*}
&\sqrt{\epsilon'_m} \, b_m - \sqrt{\epsilon'_n} \, b_n\\
&\le {\sqrt{\epsilon' \vee 1/m}  \over \sqrt{\epsilon \vee \epsilon' \vee 1/m}} 
{\zeta(\epsilon \wedge \epsilon') \over
\sqrt{\epsilon \wedge \epsilon'}} {\bf 1}_{\epsilon \vee \epsilon' < 1/n}
+ {\sqrt{\epsilon'} \over \sqrt{\epsilon \vee \epsilon'}} {\zeta(\epsilon \wedge 
\epsilon'
\wedge m) \over
\sqrt{\epsilon \wedge \epsilon' \wedge m}} {\bf 1}_{\epsilon \wedge \epsilon' > 
n}.
\end{align*}
On the one hand we compute
$$
\begin{aligned}
\int_0^\infty &(b_m \sqrt{\epsilon_m'} - b_n \sqrt{\epsilon_n'} ) \,
e^{-\epsilon'} [2 + \sqrt{\epsilon} +
\sqrt{\epsilon'}] {\bf 1}_{\epsilon \vee \epsilon' < 1/n}  \, d\epsilon' \le \\
&\le 4 \int_0^{1/n} (b_m \sqrt{\epsilon_m'} - b_n \sqrt{\epsilon_n'} )  \,
d\epsilon'
\le 4 \| {\zeta(x)  \over \sqrt{x}} \|_{L^\infty(0,1)}  {1 \over n}.
\end{aligned} \eqno(4.47)
$$
On the other hand,
\begin{align*}
&\int_0^\infty (b_m \sqrt{\epsilon_m'} - b_n \sqrt{\epsilon_n'} ) \,
e^{-\epsilon'} [2 + \sqrt{\epsilon} + \sqrt{\epsilon'}] {\bf 1}_{\epsilon \wedge 
\epsilon' \ge n} \,
d\epsilon' \\
&\le \int_0^\epsilon {\sqrt{\epsilon'} \over \sqrt{\epsilon}} {\zeta(\epsilon' 
\wedge m) \over \sqrt{\epsilon'\wedge m}} \,  e^{-\epsilon'} [2 + 
\sqrt{\epsilon} + \sqrt{\epsilon'}] \, d\epsilon' {\bf 1}_{\epsilon \ge n} \\
&\quad + {\zeta(\epsilon \wedge m) \over \sqrt{\epsilon\wedge m}} 
\int_\epsilon^\infty  \,
e^{-\epsilon'} [2 + \sqrt{\epsilon} + \sqrt{\epsilon'}] \, d\epsilon' {\bf 
1}_{\epsilon \ge n} \\
&\le \int_0^\epsilon {\zeta(\epsilon' \wedge m) \over \sqrt{\epsilon'\wedge m}} 
\,
e^{-\epsilon'} \,  \sqrt{\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon' {\bf 1}_{\epsilon 
\ge n}
+ \| {\zeta(x) \over \sqrt{x}} \,  e^{-x} \|_{L^\infty} (3 + 2 \sqrt{\epsilon}) 
{\bf 1}_{\epsilon \ge n}.
\end{align*}
To estimate the first term in the last right hand side, we
successively consider the cases
$\epsilon \le m$ and $\epsilon \ge m$ and make use of the following elementary
estimate
$$
\int_m^\epsilon (\epsilon')^\gamma \, e^{-\epsilon'} \, d\epsilon' \le 
(1+m^\gamma) \, e^{-m},
$$
with $\gamma = 1/2$ and $\gamma = 1$. When $\epsilon \le m$ we have
$$
\int_0^\epsilon {\zeta(\epsilon' \wedge m) \over \sqrt{\epsilon'\wedge m}} \,
e^{-\epsilon'} \,  \sqrt{\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon'
=\! \int_0^\epsilon \zeta(\epsilon' ) \,  e^{-\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon'
\le 4 \| \zeta(x) \,  e^{-x} \|_{L^\infty} \epsilon.
\eqno(4.48)
$$
Now, when $\epsilon \ge m$, we get
\begin{align*}
&\int_0^\epsilon {\zeta(\epsilon' \wedge m) \over \sqrt{\epsilon'\wedge m}} \,
e^{-\epsilon'} \,  \sqrt{\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon'
= \\
&= \int_0^m \zeta(\epsilon' ) \,  e^{-\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon'
+{\zeta( m) \over \sqrt{ m}}
\int_m^\epsilon e^{-\epsilon'} \,  \sqrt{\epsilon'} [3 + {\sqrt{\epsilon'} \over
\sqrt{\epsilon}}] \, d\epsilon' \\
&\le 4 \| \zeta(x) \,  e^{-x} \|_{L^\infty} \, m +
\| {\zeta(x) \over \sqrt{x}} \,  e^{-x} \|_{L^\infty} (1 + \sqrt{m}),
\end{align*}
using (4.48).
As a conclusion, we have proved that
$$
\int_0^\infty (b_m \sqrt{\epsilon_m'} - b_n \sqrt{\epsilon_n'} ) \,
e^{-\epsilon'} [2 + \sqrt{\epsilon} + \sqrt{\epsilon'}] {\bf 1}_{\epsilon \wedge 
\epsilon' \ge n} \,
d\epsilon'
\le K_4' ((\epsilon\wedge m) + \sqrt{\epsilon} + 1) {\bf 1}_{\epsilon \ge n} .
\eqno(4.49)
$$
Therefore, combining (4.47) and (4.49), we get
$$
J_4 \le {K_4 \over \sqrt{n}} \int_0^\infty F_m (1 + \sqrt{\epsilon} (\epsilon
\wedge m)) \, d\epsilon.
\eqno(4.50)
$$
From (4.42)-(4.46), (4.50) and Lemma \ref{lm4.7} we obtain
$$
{d \over dt} \| F_m - F_n \|_{L^1_{1/2}} \le A(t) \| F_m - F_n
\|_{L^1_{1/2}} + {B(t) \over \sqrt{n}}
$$
with $A \in L^\infty(0,T)$ and $B \in L^1(0,T)$ for any $T > 0$.
We end the proof using the Gronwall Lemma. \hfill$\square$\smallskip

The equation (4.15)-(4.32) is of the form given by equation (1.1) in
\cite{EM1,EM2}:
$$
\varepsilon^2{\partial F\over \partial t}(\varepsilon)=\int_0^{\infty 
}b(\varepsilon,
\varepsilon')(F'(1+F)e^{-\varepsilon}-F(1+F')e^{-\varepsilon'})d\varepsilon'
$$
with $\varepsilon^2{\varepsilon'}^2b(\varepsilon, \varepsilon')=S(\varepsilon, 
\varepsilon')$.
We then refer to \cite[Theorem 2]{EM2} for the proof of the following
proposition.


\begin{proposition} \label{prop4.9}
Let $0 \le F \in M^1_{\rm rad}([0, \infty )$  such that
$$
M(F):=\int_{0}^{\infty } v^2 dF(v) = N\ge 0\eqno{(4.51)}
$$
Then the following two assertions are equivalent:

\noindent (i) $F=\mathcal{B}_N$ with
$$
\varepsilon^2 \mathcal{B}_N(\varepsilon)=
\begin{cases} {\varepsilon^2\over e^{\varepsilon+\mu }-1},&
 \hbox{if }\quad N\le N_0\equiv \int_0^{\infty }{\varepsilon^2
 d\varepsilon\over e^{\varepsilon}-1},\\
{\varepsilon^2\over e^{\varepsilon}-1}+(N-N_0)\delta _0 & \hbox{otherwise}.
\end{cases}
$$
(ii) $F$ is the solution of the maximization problem:
$$
\mathcal{H}(F) = \max \{ \mathcal{H}(F'), \; F' \hbox{ satisfying } (4.51) \},
$$
where the entropy $H$ given by (4.21) reads:
$$
H(F)=\int_0^{\infty }[(1+F)\ln (1+F)-F\ln F -\varepsilon 
F]\varepsilon^2d\varepsilon.
\eqno{(4.52)}
$$
\end{proposition}

\begin{theorem}[Asymptotic behaviour] \label{thm4.10}
For every $ F_{\rm in}\in L_1^1(\mathbb{R}^+)$, $F_{\rm in}\ge 0$, let 
$N=M(F_{\rm in})$, and
$F\in C([0,\infty ), L^1_{1/2}(\mathbb{R}^+))$ the corresponding solution to 
(4.33)
with initial data $F_{\rm in}$. Then
$$
\begin{gathered}
 F(t,.) \mathop{\rightharpoonup}_{t \to \infty} \mathcal{B}_N
\ \hbox{weakly} \,\star \hbox{ in } \ \bigl( C_c({\mathbb{R}_+}) \bigr)' \\
\lim_{t\to \infty } \| g(t,.)-\mathcal{B}_N \|_{L^1((k_0,\infty))}=0 \quad
\forall k_0 > 0.
\end{gathered} \eqno(4.53)
$$
\end{theorem}
The proof of this theorem is the same as that of \cite[theorem 6]{EM2};
thus we skip it.

\section{The collision integral for relativistic quantum particles} %5

In this section, we follow the books by de Groot, van Leeuwen
and van Weert \cite{GLW} and  Glassey \cite{Gl}

\subsection{Parametrizations} %5.1

In this Section we introduce the Boltzmann equation for relativistic particles.
A particle is now determined by the pair $(X,P)$ representing the position and 
momentum in the time-position
and momentum space $\mathbb{R}^4 \times \mathbb{R}^4$ where we write $X = 
(X^\mu)$, with $X^0 = t$, $(X^1,X^2,X^3) = x$,
$P = (P^\mu)$ with $(P^1,P^2,P^3) = p$. Moreover, since the particle has to be 
on its mass shell, we have
$$P^0\equiv p^0: = \sqrt{ |p|^2 + m^2 c^2 }.\eqno{(5.1)}
$$
where $c$ is the speed of the light. The gas is described by its density $F = 
F(X,P)$.  In this context, the Boltzmann equation as it may be
found in \cite{GLW,Gl} is
$$\langle P,\nabla _{X}F\rangle = \mathcal{Q}(F),\eqno{(5.2)}
$$
where the collision kernel reads
$$
\mathcal{Q}(F)(P) = 
\int_{\mathbb{R}^4}\!\!\int_{\mathbb{R}^4}\!\!\int_{\mathbb{R}^4} \mathcal{W} 
q(F) \delta_{P+P_*-P'-P_*'}
\chi(P_*) \chi(P') \chi(P'_*) \, dP'dP'_*dP_*.\eqno{(5.3)}
$$
Here, $\nabla _{X}=(c^{-1}\partial _t,\nabla _x)$ and
$\langle\cdot ,\cdot\rangle$ represents the Lorentz inner product in 
$\mathbb{R}^4$
and we use the abbreviation
$|P|^2 = P \cdot P$ for any $P \in \mathbb{R}^4$. We refer to the Appendix 
\ref{A2}
for more details on the Lorentz space. Notice nevertheless that
$$
\langle P,\nabla _{X}F\rangle
=\sum_{\mu ,  \nu =1}^4\eta _{\mu \nu}P^{\mu }(\nabla _{X}F)^{\nu},\quad
\hbox{and}\quad
(\nabla_{X} F)^{\nu}=(c^{-1}\partial _t,-\nabla _x)
$$
Moreover, $\mathcal{W} =  \mathcal{W} (P,P_*,P',P'_*)$ is a given non negative 
function related to the differential cross-section $\sigma $, see (5.10) and
(5.11) below, and as before,
$$
q(F) = F' F'_* (1+ \tau F) (1 + \tau F_*) - F F_* (1+ \tau F') (1 + \tau 
F'_*)\eqno{(5.4)}
$$
with $\tau \in \{-1, 0, 1\}$, $F = F(X,P)$, $F_* = F(X,P_*)$, $F' = F(X,P')$, 
$F'_* = F(X,P'_*)$.


Finally, we have defined
$$
\forall R \in \mathbb{R}^4 \quad \chi(R) =  \delta_{R^2-m^2c^2}  
H(R^0)\eqno{(5.5)}
$$
where $H$ stands for the Heaviside function.
With these notations, if we denote $f(t,x,p) := F(X,P)$,  we have
$$
\langle P,\nabla _{X}\rangle  = {p^0 \over c}{\partial  \over \partial t}
+ p \cdot \nabla_x \,.\eqno{(5.6)}
$$
Therefore, equation (5.2) reads
$$
{\partial  \over \partial t} f + {cp\over p^0 }\cdot \nabla_x f = 
Q(F(t,x,.))(p^0,p)
:= {c \over p^0} \mathcal{Q} (F(t,x,.))(p^0,p).\eqno{(5.7)}
$$
 By Lemma \ref{lmA1.1}
$$
\chi(P) = {1 \over 2 p^0} \delta_{P^0 = p^0}
\eqno(5.8)
$$
we obtain, performing the integration in variables $P_*^0$, ${P'}^0$, ${P'}^0_*$,
$$
Q(F)(p^0,p) = \int_{\mathbb{R}^3} \!\! \int_{\mathbb{R}^3} \!\! 
\int_{\mathbb{R}^3} {c\mathcal{W} \over 8 p^0p_*^0 {p'}^0 {p'}^0_*} q(f) 
\delta_\mathcal{C} \, dp'dp'_*dp_*,\eqno{(5.9)}
$$
where $\mathcal{C}$ is defined by (1.1) with of course $\mathcal{E}(p) = p^0 / 
c$.
Gathering (5.7) and (5.9) we then recover (1.5), (1.6) with
$$
w = {c\,\mathcal{W} \over 8 p^0 p_*^0 {p'}^0 {p'}^0_*}.\eqno{(5.10)}
$$


The function $w$ is determined by the differential cross section as follows
$$
c{s \sigma(s,\theta) \over  2p^0 p_*^0 {p'}^0 {p'}^0_*} = w\eqno{(5.11)}
$$
where
$$
s=(P+P_*)^2, \quad \cos \theta={(P_*-P)\cdot (P'_*-P')\over (P_*-P)^2}.
\eqno(5.12)
$$
The parameter $s$ times $c^2$ is the square of the energy in the center of 
momentum system and  $\theta $ is the
scattering angle (see Remark \ref{rmk5.1}).
The differential cross section  $\sigma =\sigma (s, \theta)$ is a function of 
the energy and the scattering angle.
Since we consider identical particles it satisfies the symmetry relation:
$$
\sigma (s, \theta)=\sigma (s, \pi -\theta).
$$
Thus, the differential cross section determines the structure of the collision 
integral.
See Appendix \ref{A3} for more details about this function
and its physical meaning.
\smallskip
The 12-fold integral in (5.3) can be reduced to a 5-fold integral by carrying 
out the delta
function integrations. This can be done in two different ways. They correspond 
to the two different parametrizations
of the collision manifold which are well known for the classical Boltzmann 
equations.

\subsubsection{The center of mass parametrization}
We deduce this parametrization starting from the formulation (5.3) of the 
collision kernel.
To this end we first perform a change of variables in the $P'$ and $P'_*$ 
integrals. So,
given $P$ and $P_*$ fixed, consider the Lorentz transformation $\Lambda $ from 
$\mathbb{R}
^4$ into itself defined by:
$$
\Lambda \begin{pmatrix} \sqrt{s} \\ 0\end{pmatrix} = (P+P_*).
\eqno{(5.13)}
$$
It is given by the  $4\times 4$ matrix
$$
\Lambda = \begin{pmatrix}\gamma (v) & \gamma (v) v^{\top}\\ \gamma (v) v
&I+{\gamma (v)-1\over |v|^2} v v^{\top} \end{pmatrix}
\eqno{(5.14)}
$$
with
$$
v = {p + p_* \over \sqrt{|p + p_*|^2+s}} \in \mathbb{R}^3,
\quad \gamma (v)={1\over \sqrt{1-|v|^2}}\in \mathbb{R}
\eqno{(5.15)}
$$
and
$$
(v v^{\top})_{ik}=v_iv_k, \quad v v^{\top} \in \mathcal{M}_{3\times 3}.
\eqno{(5.16)}
$$
Define now,
$$
P'=\Lambda Q', \quad  P'_*=\Lambda Q'_*.
\eqno{(5.17)}
$$
For the sake of brevity we shall denote:
$$
P'=P'(Q'), \quad P_*'=P_*'(Q_*'),\quad p' \equiv p'(Q'),    \quad p_*' 
=p_*'(Q_*').
\eqno{(5.18)}
$$
By the definition of Lorentz transform we have that $s$ and $\theta$ are 
invariant
(see Appendix \ref{A2}) by the
change of variable.  Then:
$$\begin{aligned}
&Q(f)(p) \\
&= {4c\over p^0} 
\int_{\mathbb{R}^4}\!\!\int_{\mathbb{R}^4}\!\!\int_{\mathbb{R}^4}
s \sigma(s ,\theta) q(f) \delta_{\Lambda ^{-1}(P+P_*)-Q'-Q_*'}
\chi(P_*^0) \chi({{Q'}^0_*}) \chi({Q'}^0) \, {dQ'}{dQ'_*}dP_*,
\end{aligned}\eqno{(5.19)}
$$
with now
$$
\begin{aligned}
q(f)=& f(p'_*(Q_*'))f(p'(Q'))(1+ \tau  f(p))(1+\tau  f(p_*))\cdot\\
&-f(p)f(p_*)(1+\tau  f(p'(Q'))(1+\tau  f(p'_*(Q_*')), \end{aligned}
\eqno{(5.20)}
$$
where we have used that if
$(u_0, u_*, u_2, u_3)=\Lambda (v_0, v_*, v_2, v_3)$,
then $\mathop{\rm sign} u_0 =\mathop{\rm sign} v_0$,
and that
$$
\delta (P+P_*-\Lambda Q'-\Lambda Q_*') = \delta (\Lambda ^{-1}(P+P_*)-Q'-Q_*').
$$

We integrate with respect to  ${Q'}^0$, ${Q'_*}^0$, and ${Q_*'}^0$.
Thanks to $(5.8)$ we have
$$
Q(f)(p) = {c\over 2p^0} 
\int_{\mathbb{R}^3}\!\!\int_{\mathbb{R}^3}\!\!\int_{\mathbb{R}^3}
s \sigma(s ,\theta) q(f) \delta_{ \{(\sqrt{s},0) - Q' - Q'_*\}}
{dq'\over {q'}^0} {dq'_*\over {q'}_*^0} {dp_* \over p_*^0},
\eqno(5.21)
$$
with
$P = (p^0,p)$, $P_* = (p^0_*,p_*)$,  $Q' = ({q'}^0,q')$,
$Q'_* = ({q'_*}^0,q'_*)$. Observing that
$$
({q'}^0,q') + ({q'_*}^0,q'_*) = (\sqrt {s}, 0)
\ \hbox{ if and only if } \
q' = - q'_*, \quad {q'}^0 + {q'_*}^0 =\sqrt {s},
$$
we get
$$
{q'}^0 = {q'_*}^0 = {\sqrt{s}\over 2}.
\eqno{(5.22)}
$$
Therefore, integrating in $q'_*$ we get
$$
Q(f)(p) = {c\over p^0} \int_{\mathbb{R}^3}\!\!\int_{\mathbb{R}^3}
\sigma q(f) \delta_{{\sqrt{s}\over 2}-\sqrt{|q'|^2+m^2c^2}} \, dq'
{dp_*\over p_*^0}.
\eqno{(5.23)}
$$
We finally, change to spherical coordinates in the $q'$ integral:
$$
dq'=|q'|^2 \, d|q'| \, d\Omega\eqno{(5.24)}
$$
and use, again by Lemma \ref{lmA1.1},
$$
\delta({\sqrt{s}\over 2}-\sqrt{|q'|^2+m^2c^2})
= \sqrt{{s\over s-4m^2c^2}} \delta (|q'|-{1\over 2}\sqrt{s-4m^2c^2})
\eqno{(5.25)}
$$
to obtain
$$
Q(f)(p) = {c\over p^0}\int_{\mathbb{R}^3} \!\! \int_0^\infty \!\!\! \int_{S^2}
\sqrt{{s\over s-4m^2c^2}} \sigma q(f) \delta_{|q'|-{1\over 2}\sqrt{s-4m^2c^2}}
|q'|^2 \, d|q'| \, d\Omega {dp_*\over p_*^0}
$$
and finally
$$
Q(f)(p) = {c\over 4p^0} \int_{\mathbb{R}^3}\!\!\int_{S^2}
\sqrt{s(s-4m^2c^2)} \sigma q(f) \, d\Omega {dp_*\over p_*^0}.\eqno{(5.26)}
$$

Let us go back to the expression of $q(f)$ in order to give an explicit formula 
in terms of $p$, $p_*$
and $\Omega$.
Remember that
\begin{align*}q(f) =& f(p'_*(Q_*')) f(p'(Q')) (1+ \tau f(p))(1+ \tau f(p_*)) \\
& - f(p) f(p_*) (1+ \tau f(p'(Q')) (1+\tau f(p'_*(Q_*'))
\end{align*}
where
$P'=\Lambda Q'$, $P_*'=\Lambda Q_*'$, and
$\Lambda (\sqrt{s},  0)= (P+P_*)=(p^0+p_*^0, p+p_*)$,
so that we just have to express $p' = p'(Q')$ and $p'_* = p'_*(Q_*')$.
As we have said just before
$$
Q'=({\sqrt{s}\over 2 },|q'|\Omega), \quad Q_*'=({\sqrt{s}\over 2 },-|q'|\Omega).
$$
Therefore,
\begin{align*}
p' &= \Lambda ({\sqrt{s}\over 2 },q') \\
&= \Bigl( \gamma (v) {\sqrt{s}\over 2}
+ \gamma (v) |q'| v^{\top} \Omega \, , \gamma (v) v {\sqrt{s}\over 2}
+ |q'| \Omega + {\gamma (v) - 1 \over v^2} |q'| v v^{\top} \Omega \Bigr) \\
p'_* &= \Lambda ({\sqrt{s}\over 2 },-q')\\
& = \Bigl( \gamma (v) {\sqrt{s}\over 2}
- \gamma (v) |q'| v^{\top} \Omega \, , \gamma (v) v {\sqrt{s}\over 2}
- |q'| \Omega - {\gamma (v) - 1 \over v^2} |q'| v v^{\top} \Omega \Bigr)
\end{align*}
with
$$
v = {p+p_*\over \sqrt{|p+p_*|^2+s}} = {p+p_*\over p^0+p_*^0},  \quad
\gamma (v) = {1\over \sqrt{1-|v|^2}} = {p^0+p_*^0 \over
\sqrt{(p^0+p_*^0)^2-|p+p_*|^2}},
\eqno{(5.27)}
$$
and
$$
|q'| = {1\over 2} \sqrt{s-4m^2c^2} = {1\over 2} \sqrt{(p^0+p_*^0)^2
- |p+p_*|^2-4m^2c^2}.
\eqno{(5.28)}
$$
Finally we obtain
$$
\begin{gathered}
p' = {p+p_*\over 2} + {1\over 2} \sqrt {(p^0+p_*^0)^2 - |p+p_*|^2 - 4 m^2 c^2 } 
\bigl( \Omega + {\gamma (v) - 1 \over |v|^2} v v^{\top} \Omega \bigr),\\
p'_* = {p+p_*\over 2} - {1\over 2} \sqrt{(p^0+p_*^0)^2 -|p+p_*|^2-4m^2c^2} 
\bigl( \Omega + {\gamma (v) - 1 \over |v|^2} v v^{\top} \Omega\bigr).
\end{gathered}\eqno{(5.29)}
$$
Since the variable $\Omega$ describes the whole sphere $S^2$, it can then be
parametrized in spherical coordinates of polar axis $q$:
$$
\Omega={q\over |q|}\cos \theta+(\cos \phi \, \vec{\imath}  + \sin \phi \,
\vec{\jmath})\sin \theta \eqno{(5.30)}
$$
in such a way that
$$
d\Omega=\sin \theta d\theta d\phi .\eqno{(5.31)}
$$
To see this, we observe that
\begin{align}
\cos \theta&={(P_*-P)\cdot (P'_*-P')\over (P_*-P)^2}={(Q_*-Q)\cdot
(Q'_*-Q')\over (Q_*-Q)^2}\\
&={(q_*^0-q^0)({q_*'}^0-{q'}^0)-(q_*-q)\cdot(q_*'-q')\over
(q_*^0-q^0)^2-|q_*-q|^2}={q\cdot q' \over |q|^2},
\end{align}
where we use fact that $q_*^0=q^0={q_*'}^0={q'}^0=\sqrt {s}/2$ and
$q=-q_*$, $q'=-q_*'$. Finally since
$$
|q|=\sqrt{{q^0}^2-m^2c^2}=\sqrt{{{q'}^0}^2-m^2c^2}=|q'|,
$$
we get
$\cos \theta ={q q' \over |q| |q'|}$.

\begin{remark} \label{rmk5.1}
The reference frame of the variables $(Q, Q_*, Q', Q_*')$ in
the previous calculation is called the center of momentum system.
The geometry of the collision is
particularly simple in this frame since
$$ Q+Q_*=Q'+Q_*'=\begin{pmatrix}\sqrt{s}\\ 0\end{pmatrix}\eqno{(5.32)}
$$
and the scattering angle $\theta $ is
$$\cos \theta ={qq'\over |q\|q'|}.\eqno{(5.33)}
$$
It is easily seen that this is the angle formed by the trajectories before
and after the collision for each of the particles.
\end{remark}

Finally the M{\o}ller velocity is
$$
v={c\over2 p^0p_*^0}\sqrt{s(s-4m^2c^2)}\eqno{(5.34)}
$$
and the collision integral in (5.26) may be written as
$$
Q(f)(p)={1\over 2}\int_{\mathbb{R}^3}\int_{S^2}
v\sigma(s ,\theta)q(f)d\Omega dp_*.\eqno{(5.35)}
$$

\subsubsection{Another expression for the collision integral}
(We closely follow in this part the Appendix II of
Glassey and  Strauss [GS], see also \cite{Gl}). There is another way to reduce 
the 12-fold original integral to a 5-fold by using a
slightly different parametrization. Let us start again from the original 
equation,
perform directly the integration in the ${p'}^0$, ${p_*'}^0$ and $p_*^0$
variables to obtain:
$$
\begin{aligned}
Q(f)(p)={c\over 2\,p^0}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}
\int_{\mathbb{R}^3}&\delta
\big((P+P_*)-(\sqrt{|p'|^2+m^2c^2},p')\\
&-(\sqrt{|p'_*|^2+m^2c^2}, p_*')\big)
s\sigma q(f){dp'\over{p'}^0}
{dp'_*\over {p_*'}^0} {dp_*\over {p_*}^0} \,.
\end{aligned}\eqno{(5.36)}
$$
We integrate now in $ p_*'$ to obtain
$$Q(f)(p)={c\over 16p^0}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\delta
(p^0+p_*^0-{p'}^0-{p_*'}^0)s\sigma q(f){1\over {p_*'}^0}{dp'\over
{p'}^0}
{dp_*\over {p_*}^0}.\eqno{(5.37)}
$$
We perform now the change of variables
$$
p'=p+r\cdot \omega\,, \quad  p_*'=p_*-r\cdot \omega \eqno{(5.38)}
$$
with $\omega \in S^2$ and  $ r\in \mathbb{R}$ in such a way that
$p+p_*=p'+p'_*$.
for every $p$ and $p_*$ in $\mathbb{R} ^3$. We use now Lemma \ref{lmA1.1}
 (ii') to obtain
$$
\begin{aligned}
&\delta(p^0+p_*^0-{p'}^0-{p_*'}^0)\\
&=2(p^0+p_*^0)\delta (({p'}^0+{p'}^0_*)^2-(p^0+p^0_*)^2)\\
&=4(p^0+p_*^0){p'}^0{p'}^0_*\delta (4{{p'}^0}^2{{p'}^0_*}^2-[(p^0+p_*^0)^2-{{p'}^0}^2-
{{p'}^0_*}^2]^2).
\end{aligned}
\eqno{(5.39)}
$$
Observe that
$$\begin{aligned}\mathcal{P}
(r)&\equiv4{{p'}^0}^2{{p'}^0_*}^2-[(p^0+p_*^0)^2-{{p'}^0}^2-{{p'}^0_*}^2]^2\\
&=4{{p'}^0}^2{{p'}^0_*}^2-
[(p^0+p_*^0)^2-({{p'}^0}^2+{{p'}^0_*}^2)]^2\\
&=-(p^0+p_*^0)^4+2(p^0+p_*^0)^2({{p'}^0}^2+{{p'}^0_*}^2)-({{p'}^0}^2-{{p'}^0_*}^2)^2,
\end{aligned}\eqno{(5.40)}$$
where by the change of variables (5.38),
$$
{{p'}^0}^2-{{p'}^0_*}^2=|p|^2-|p_*|^2+2r(p+p_*)\cdot\omega
$$
It is now a simple matter to check that $\mathcal{P} $ is a polynomial of degree
two whose roots are $r=0$ and
$$
\begin{gathered}
a(p, p_*, \omega ) = {2(p^0+p_*^0)(\omega \cdot
(\hat p- \hat p_*)) p^0 {p_*}^0 \over (p^0+{p_*}^0)^2-(\omega
\cdot (p+p_*))^2}\equiv2{N\over D},\\
\hat p={p\over p^0}, \ \hat p_*={p_*\over p_*^0}.
\end{gathered}\eqno(5.41)
$$
Therefore, $\mathcal{P}(r)=Dr(r-a(p, p_*, \omega ))$.
We may then write
$$
\begin{aligned}
&\delta(p^0+{p_*}^0-{p'}^0-{p_*'}^0){1\over {p_*'}^0}{1\over {p'}^0}{dp'}\\
& = 4(p^0+p_*^0)\delta(4{{p'}^0}^2{{p'}^0_*}^2-[(p^0+p_*^0)^2-{{p'}^0}^2
-{{p'}^0_*}^2]^2){dp'}\\
& =4(p^0+p_*^0)\delta (Dr(r-a(p, p_*, \omega )))\,r^2\, dr\, d\omega.
\end{aligned}\eqno{(5.42)}
$$
Using Lemma \ref{lmA2.3}, we get
$$
\begin{aligned}
&\delta(p^0+p_*^0-{p'}^0-{p_*'}^0){1\over {p_*'}^0}{1\over{p'}^0}{dp'}\\
&=4(p^0+p_*^0)|Da|^{-1}[\delta (r)+\delta (r-a)]\,r^2\, dr\, d\omega.
\end{aligned}\eqno{(5.43)}
$$
The first delta function drops because of the factor $r^2$ and we are led them 
to:
$$
\begin{aligned}
&\delta(p^0+{p_*}^0-{p'}^0-{p_*'}^0){1\over {p_*'}^0}&{1\over {p'}^0}{dp'}\\
&=4(p^0+{p_*}^0)2|N|D^{-2}\delta (r-a)\, dr\, d\omega \\
&=8{(p^0+{p_*}^0)^2|\omega \cdot (\hat p- \hat p_*)|p^0{p_*}^0\over 
[(p^0+{p_*}^0)^2-(\omega \cdot
(p+p_*))^2]^2}\delta (r-a)\, dr\, d\omega.
\end{aligned}\eqno{(5.44)}
$$
Finally,
$$
\begin{gathered}
Q(f)(p)=\int_{\mathbb{R}^3}\int_{\mathcal{S}^2} \Gamma (p, p_*, \omega ) q(f)\, 
d\omega dp_*,\\
\Gamma (p, p_*, \omega )=4c\, s\sigma{(p^0+p_*^0)^2|\omega \cdot (\hat p-
\hat p_*)|\over [(p^0+p_*^0)^2-(\omega \cdot (p+p_*))^2]^2}.
\end{gathered}\eqno{(5.45)}
$$
Observe that
$\Gamma ( p, p_*, \omega )=\Gamma (p_*, p, \omega )$
and since $D\ge 2$,
$$0\le \Gamma ( p, p_*, \omega )\le c\,s\sigma (p^0+p_*^0)^2|\omega \cdot (\hat 
p- \hat p_*)|.$$

\subsection{Particles with different masses} %5.2

Similar calculations can be performed if we consider collisions of two quantum
relativistic particles with different masses $m_*$ ,$m_2$ and with momenta 
$P=(p^0, p)$ and $P_*=(p_*^0,
p_*)$  such that
$$
p^0=\sqrt{|p|^2+m_*^2c^2}, \quad p_*^0=\sqrt{|p_*|^2+m_2^2c^2}.\eqno{(5.46)}
$$
Let us denote by $f$ de density distribution of the particles $P$ of mass $m_*$ 
and $g$ that of the particles
$P_*$ of mass $m_2$. These functions satisfy the coupled system  (4.2).
Let us consider here only the integral $Q_{1,2}(f, g)\equiv Q(f, g)$
since $Q_{2,1}(g, f)$ is completely similar:
$$
\begin{aligned}
Q(f, g)(p)=&{8 c\over 
p^0}\int_{\mathbb{R}^4}\!\!\int_{\mathbb{R}^4}\!\!\int_{\mathbb{R}^4}
s \sigma q(f, g) \delta_{P+P_*-P'-P_*'=0}  \\
&\quad\times  \chi_2(P_*^0) \chi_1({P'}^0) \chi_2({P'_*}^0)
\, dP'dP'_*dP_*\\
q(f, g) =& g(p'_*) f(p')(1+\tau f(p))(1+\tau' g(p_*))\\
&\quad - f(p)g(p_*)(1+\tau f(p'))(1+\tau' g(p_*')),
\end{aligned}\eqno{(5.47)}
$$
with $\chi_i(P) = \delta_{P^2-m_i^2 c^2} \, H(P^0)$ for $i = 1,2$, and
$\tau , \tau ' \in \{-1, 0, 1\}$.


As before, the integral collision may be written as
$$
Q(f, g) = \int_{\mathbb{R}^3}\int_{S^2}
\nu \sigma q(f, g) \, d\Omega dp_*,\eqno{(5.48)}
$$
where the M{\o}ller velocity is in this case:
(where $\Omega$ and $\theta$ are defined as above)
$$
\nu = {c\over 2 p^0p_*^0}\sqrt{(s-(m_*-m_2)^2c^2)(s-(m_*+m_2)^2c^2)}.
\eqno{(5.49)}
$$
and the dependence of $p'$ and $p'_*$ in function of $p$, $p_*$ and
$\Omega$ in the expression (5.47) of $q(f,g)$ is
\begin{gather*}
p' = {p+p_*\over 2} + {\sqrt{(s-(m_*-m_2)^2c^2)(s-(m_*+m_2)^2c^2)}\over 2\sqrt 
s}
\bigl(\Omega+{\gamma (v)-1\over |v|^2} v v^{\top}\Omega\bigr), \\
p'_* = {p+p_*\over 2} - {\sqrt{(s-(m_*-m_2)^2c^2)(s-(m_*+m_2)^2c^2)}\over 2\sqrt 
s}
\bigl(\Omega+{\gamma (v)-1\over |v|^2} *, v v^{\top}\Omega\bigr).
\end{gather*}
An expression similar to (5.45) may also be obtained for the collision
integral $Q(f, g)$. From the expression in (5.47), we perform
an integration in the ${p'}^0$, ${p_*'}^0$ and $p_*^0$ variables to obtain
$$
\begin{aligned}
Q(f, g)(p)={c\over 
p^0}\int_{\mathbb{R}^3}\!\int_{\mathbb{R}^3}\!\int_{\mathbb{R}^3}\!
&\delta \big((P+P_*)-(\sqrt{|p'|^2+m_*^2c^2},p')\\
&-(\sqrt{|p'_*|^2+m_2^2c^2}, p_*')\big)
s\sigma q(f, f){dp'\over{p'}^0}
{dp'_*\over {p_*'}^0} {dp_*\over {p_*}^0}.
\end{aligned}
$$
We integrate now in $p_*'$ to obtain
$$Q(f, g)(p)={c\over p^0}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\delta
(p^0+p_*^0-{p'}^0-{p_*'}^0)s\sigma q(f, g){1\over {p_*'}^0}{dp'\over {p'}^0}
{dp_*\over {p_*}^0}.\eqno{(5.50)}
$$
We may then apply the calculations from (5.38) to (5.44) and obtain finally
$$
\begin{gathered}
Q(f, g)(p)=\int_{\mathbb{R}^3}\int_{\mathcal{S}^2} \Gamma (p, p_*, \omega ) q(f, 
g)\, d\omega
dp_*,\\
\Gamma (p, p_*, \omega )=8c\, s\sigma{(p^0+p_*^0)^2|\omega \cdot (\hat p-
\hat p_*)|\over [(p^0+p_*^0)^2-(\omega \cdot (p+p_*))^2]^2}.
\end{gathered}\eqno{(5.51)}
$$

\begin{remark} \label{rmk5.2} \rm
The expressions (5.34)-(5.35) and (5.48)-(5.49) correspond to the center of mass 
angular parametrizations of
the collisions in the classical and  non quantum Boltzmann equation.
\end{remark}


\subsection{Boltzmann-Compton equation for photon-electron \\
scattering} %5.3

In this section, we consider the system of Boltzmann equations for a dilute
 gas of low energy electrons and weakly dense
photons.  We assume that particles $P$ are photons and so are massless,
$m_*=0$ and particles $P_*$ are electrons of mass
$m_2>0$ that we take to be $m_2=m$. From the preceding sub Section 5.2,
if $f$ describes the density of  photons and $g$ the density of
electrons the collision integral writes
$$
Q(f, g)(p)={c\over 2 |p|}\int_{\mathbb{R}^3}\int_{S^2}
(s-m^2c^2) \sigma(s ,\theta)q(f, g)d\Omega
{dp_*\over p_*^0} \eqno{(5.52)}
$$
where,
$$\begin{aligned}s&=(P+P_*)^2=(|p|+\sqrt{|p_*|^2+m^2c^2})^2-|p+p_*|^2\\
&=c^2+2|p|\sqrt{|p_*|^2+m^2c^2}-2(p, p_*)
\end{aligned}\eqno{(5.53)}$$
and
$$
p'={p+p_*\over 2}+{s-m^2c^2\over 2\sqrt s}
\bigl(\Omega+{\gamma ( v)-1\over
v^2}vv^{\top}\Omega\bigr),\eqno{(5.54)}
$$
$$p'_*={p+p_*\over 2}-{s-m^2c^2\over 2\sqrt s}
\bigl(\Omega+{\gamma (v)-1\over v^2}vv^{\top}\Omega\bigr).\eqno{(5.55)}$$
In this context, the differential cross section $\sigma $ is given by the
Klein Nishina formula  given in Appendix \ref{A3}.

We briefly consider now the so called classical limit, which amounts to consider 
the higher order term in
(5.52) when $c\to \infty $. Notice first that
$$
s-m^2c^2\sim 2|p|mc,\quad \hbox{and}\quad v\sim 1 \quad
\hbox{as}\quad c\to \infty. \eqno{(5.56)}
$$
Since the photons have low energy we have $p^0 \sim {p'}^0$ from where we deduce
$$
\sigma (s, \theta)\sim {1\over 2}r_0^2\{1+\cos^2\theta\}, \quad
\hbox{as}\quad c\to \infty ,\eqno{(5.57)}
$$
(c.f. \cite[p. 163]{PS}) and finally
$$
\lim_{c\to \infty }{s-m^2c^2 \over 2\sqrt s }=|p|.\eqno{(5.58)}
$$
Then, in the classical limit we obtain
$$
\begin{aligned}Q(f, g)(p)=&{c\, r_0^2\over 2}\int_{\mathbb{R}^3}\int_{S^2}
 \{1+\cos^2\theta\}q(f, g)d\Omega \,dp_*
\end{aligned}\eqno{(5.59)}$$
with
$$
p'={p+p_*\over 2}+|p|\,\Omega, \quad \quad p_*'={p+p_*\over 2}-|p|\,\Omega.
\eqno{(5.60)}
$$

\subsubsection{Dilute and low energy electron gas at equilibrium}
We assume moreover that the dilute and low energy electron gas is at
equilibrium. Then, the density of electrons
is given by a Maxwell Boltzmann distribution. It is therefore  possible
to write the corresponding collision integral in a more explicit way as
it was already done in Section 4. To this end it is convenient to express
this collision integral in a
Carleman's type parametrization. It should be possible to obtain it
from (5.52) but nevertheless we proceed in a simpler way. We consider
then the classical limit, $c\to \infty $ directly in the integral
collision (5.47), where $\sigma $ is again the Klein Nishina differential
cross section.
Arguing as above we obtain that the collision integral reads
$$
Q(f)(p)={c\, r_0^2\over 2|p||p'|}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}
\int_{\mathbb{R}^3}(1+\cos^2 \theta )q(f) \delta _{\Sigma}dp_*dp'_*dp'
$$
where $\Sigma $ is now the manifold of 4-uplets $(p, p_*, p', p'_*)$
such that,
$$
p+p_*=p'+p'_*,\quad \hbox{and}\quad |p|+{|p_*|^2\over 2mc}=|p'|
+{|{p'_*}^2|\over 2mc}.\eqno{(5.61)}
$$

By the hypothesis on the electron gas, we may take
$g(p'_*)=e^{-\beta^0 {|p'_*|^2\over 2m}}e^\mu $
for some $\beta^0 >0$ and $\mu \in \mathbb{R}$ constants .
By (5.61) we deduce
$$
g(p'_*)=e^{-\beta ^0(|p|-|p'|)}e^{-\beta ^0{|p_*|^2\over 2m}}e^{\mu }, 
\eqno{(5.62)}
$$
and the integral collision reads,
$$ Q(f)(p)={r_0^2\over 2}e^{\mu }
\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}{\{1+\cos^2\theta\}\over |p||p'|}e^{-\beta
^0{|p_*|^2\over 2m}+\beta ^0|p'|} q(f)\delta _{\Sigma } dp' dp_*dp'_*,\eqno{(5.63)}
$$
with
$$
q(f)=e^{-\beta ^0|p|}f(p')(1+f(p))-e^{-\beta ^0|p'|}f(p)(1+ f(p')).
$$
Note that the integration in the $p'_*$ variable is straightforward.
Let us state the following auxiliary Lemma.


\begin{lemma} \label{lm5.3}
With $A=|p|-|p'|+{|p-p'|^2\over 2mc}$ and $w=p'-p$, we have
$$
S(p, p')=\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\delta _{\Sigma }
e^{-\beta ^0{|p_*|^2\over 2mc}}dp_*dp'_*={2\pi\, m^2\,c \over
\beta ^0|w|}e^{-\beta ^0{A^2mc^2\over 2|w|^2}}.
$$
\end{lemma}

\noindent {\bf Proof.}\quad
Since
$$
|p|+{|p_*|^2\over 2mc}-|p'|-{|p+p_*-p'|^2\over 2mc}=A- {1\over mc}(p_*, w),
$$
we have
$$
S(p, p')=\int_0^{\infty }\int_{\mathbb{S}^2}\delta _{A-{|p_*|\over mc}
(\Omega_*,w )}d\Omega_*e^{-\beta ^0{|p_*|^2\over 2m}}|p_*|^2d|p_*|.
$$
From
$$\int_{\mathbb{S}^2}\delta _{A-{|p_*|\over mc}(\Omega_*,w )}d\Omega_*={2\pi 
mc\over |p_*\|w|}H(1-{A^2m^2c^2\over |p_*|^2|w|^2})$$
(where $H$ stands for the Heaviside's function) we obtain
$$
S(p, p')={2\pi mc\over |w|}\int_{{Amc\over |w|}}^{\infty }|p_*
|e^{-\beta ^0{|p_*|^2\over 2m}}d|p_*|\eqno{(5.64)}
$$
which completes the proof.\hfill$\square$ \smallskip

The collision kernel reads now
$$Q(f)(p)={cr_0^2\over 2}e^{\mu }\int_{\mathbb{R}^3}{S(p, p')\over 
|p||p'|}\{1+\cos^2\theta\}e^{\beta ^0|p'|} q(f)dp'.
$$
Now, we assume that the photon distribution is radial. This is a further
simplification which allows to
write the equation in polar coordinates. We denote
$\varepsilon=|p|$, $\varepsilon'=|p'|$, and, with a slight abuse of notation,
 $S(p, p')=S(\varepsilon, \varepsilon', \theta )$,
$$
Q(f)(\varepsilon)={cr_0^2\over 2\varepsilon^2}e^{\mu }\int_0^{\infty }
B(\varepsilon,\varepsilon')q(f)e^{\beta _0 
\varepsilon'}d\varepsilon'\eqno{(5.65)}
$$
with
$$
B(\varepsilon, \varepsilon')=2\pi \,\varepsilon\varepsilon' \int_0^{\pi }(1+\cos 
^2 \theta ) S(\varepsilon, \varepsilon',
\theta )\sin \theta \, d\theta.\eqno{(5.66)}
$$
Finally, the Boltzmann-Compton equation reads:
$$
\partial _t f=\int_0^{\infty }b (\varepsilon, \varepsilon')\,(f'(1+f)e^{-\beta
^0\varepsilon}-f(1+f')e^{-\beta ^0\varepsilon'}){\varepsilon'}^{2}d\varepsilon\,
d\varepsilon', \eqno{(5.67)}
$$
where
\begin{align*}
b (\varepsilon, \varepsilon')
&={c\, r_0^2\over 2}e^{\mu }{B(\varepsilon, \varepsilon')\over \varepsilon ^2
{\varepsilon'}^2}e^{\beta^0 \varepsilon'}\\
&= 2{m^2c^2\, r_0^2\pi^2\over \beta ^0\varepsilon
\varepsilon'}e^{\mu }e^{\beta^0 \varepsilon'}
\int_0^{\pi }(1+\cos ^2 \theta ){1\over |w|}e^{-\beta ^0{A^2mc\over 2|w|^2}}
\sin \theta \, d\theta
\end{align*}
with $|w|^2=\varepsilon ^2+{\varepsilon'}^2-2\varepsilon\,\varepsilon'
\cos \theta$.

\begin{remark} \label{rmk5.3} \rm
Note that equation  (5.67) is formally
the same as (4.15)-(4.32) in Section 4 above and (1.1) of \cite{EM2}.
In particular, these equations have the same entropy function, given
by (4.52). Nevertheless,
in each of these cases, the functions $b(\varepsilon, \varepsilon')$ which 
appear
in the collision have different, although similar,
behaviour in the domain $\varepsilon>0, \varepsilon'>0$.
It is in particular easy to check that
the function $b$ in (5.67) does not satisfy any of the conditions
(2.1), (2.2) (2.3) imposed in the existence theorems obtained in \cite{EM2}.
The global existence of solutions in $L^1$ is then still an open question
for this equation.
\end{remark}

\subsection{The Kompaneets equation} %5.4

In this subsection we present the deduction of the
Kompaneets equation from the Boltzmann-Compton equation, and we mainly
follow \cite{K} and \cite[Vol. 3]{Le}. The Kompaneets limit is valid for
$$
\varepsilon, \varepsilon'\ll mc^2.
$$
We by start performing the change of variables
$\beta ^{0}\varepsilon=\overline{\varepsilon}$, $f(\varepsilon)=
g(\overline{\varepsilon})$ in (5.67) and, by abuse of notation, still
denote the variable as $\varepsilon$. This gives
$$
\varepsilon^2 {\partial g\over \partial t} =
{\beta ^{0}}^{-3}\int_0^\infty \bigl( g' (1+g) \, e^{-\varepsilon} -g
(1 + g') e^{-\varepsilon'} \bigr)\varepsilon^2{\varepsilon'}^2
b({\varepsilon\over \beta ^{0}}, {\varepsilon'\over \beta ^{0}})
\,d\varepsilon'.
$$

In a similar way as in the usual Fokker Planck approximation
(cf. \cite[vol. 10, Ch. 2]{LL}),
one notices that the main contribution to the
integral in (5.67) should come from the region where $|\varepsilon'-
\varepsilon|/|\varepsilon|<<1$.  This
is due to the fact that the kernel
$b(\varepsilon, \varepsilon')$ given in (5.67) is rather peaked around
$\varepsilon'=\varepsilon$. More precisely, as we shall see below (cf. (5.75)),
for any $\varepsilon>0$ fixed and
$\theta \in [0, \pi ] $:
$$
\lim_{m\to \infty }b(\varepsilon, \varepsilon'){\varepsilon'\over \varepsilon}
(\varepsilon'-\varepsilon)^2
=C(\theta )\delta _{\varepsilon'=\varepsilon}
$$
where $C(\theta )$ is a constant depending on $\theta$ and also on $m, c, \beta 
^{0}$. I.e., in the non
relativistic limit of Compton scattering, the dominant contribution to the 
collision integral comes from
the region where $|p|-|p'| (=
\varepsilon-\varepsilon')$ is small, and not from the region where  $p-p'$ is 
small. This is consistent
with the fact that the cross section for the non relativistic limit of Compton 
scattering, as described by
the Klein-Nishina formula (cf. Appendix \ref{A3},
Relativistic case), has no singular behaviour when the momentum
transfer $p'-p$ goes to zero.


We may then first expand
$g(\varepsilon')$ and $e^{-\varepsilon'}$ around $\varepsilon$ to obtain
$$
\begin{aligned}
&g({\varepsilon'}) (1+g(\varepsilon)) \, e^{-\varepsilon} - g(\varepsilon)
(1+ g({\varepsilon'})) \, e^{-{\varepsilon'}}\\
&\sim \Big\{({\varepsilon'}-\varepsilon){g'}(\varepsilon)+[({\varepsilon'}-
\varepsilon)-
{1\over 2}({\varepsilon'}-\varepsilon)^2]g(\varepsilon) \\
&\quad -[({\varepsilon'}-\varepsilon)-{1\over 2}({\varepsilon'}-\varepsilon)^2] g^2+
{1\over 2}({\varepsilon'}-\varepsilon)^{2g''}+ ({\varepsilon'}-\varepsilon)^2g{g'}
\Big\}e^{-\varepsilon}.
\end{aligned}
$$
From where,
$$
{\beta ^{0}}^3\varepsilon^2 {\partial g\over \partial t}\sim
a(\varepsilon)g(\varepsilon)+(a(\varepsilon)+\, b(\varepsilon))g'(\varepsilon)+
b(\varepsilon)g''(\varepsilon)+2\, b(\varepsilon)\, gg'+a(\varepsilon)g^2
$$
with,
\begin{gather*}
a(\varepsilon)=\varepsilon^2\int_0^{\infty }((\varepsilon'-\varepsilon)
-{1\over 2}(\varepsilon'-\varepsilon)^2)b({\varepsilon\over \beta ^{0}},
 {\varepsilon'\over \beta ^{0}})e^{-\varepsilon}{\varepsilon'}^2d\varepsilon'\\
b(\varepsilon)=\varepsilon^2{1\over 2}\int_0^{\infty}
(\varepsilon'-\varepsilon)^2b({\varepsilon\over \beta ^{0}},
{\varepsilon'\over \beta ^{0}})e^{-\varepsilon}{\varepsilon'}^2d\varepsilon'.
\end{gather*}
Therefore, if we set
$$
\alpha (\varepsilon)=\exp {\int_0^\varepsilon {a(\sigma )\over b(\sigma )}
d\sigma } \eqno{(5.68)}
$$
and
$$
d (\varepsilon)=b(\varepsilon)\exp ({-\int_0^\varepsilon {a(\sigma )\over 
b(\sigma )}d\sigma
})\eqno{(5.69)}
$$
we have
$${\beta ^{0}}^3\varepsilon^2{\partial g\over \partial t}\sim 
d(\varepsilon){\partial \over \partial
\varepsilon}[\alpha (\varepsilon)({\partial g\over
\partial \varepsilon }+g+g^2)].\eqno{(5.70)}
$$

The second step is to identify the two functions $d$ and $\alpha $. To this end 
one argues as follows.
We first assume that the equation (5.70) preserves the total number of 
particles, namely that
$${d\over dt}\int_0^{\infty }\varepsilon^2\, g(\varepsilon, t)\, 
d\varepsilon=0.\eqno{(5.71)}$$
This is reasonable as far as (5.70) is a good approximation of the Boltzmann-
Compton equation.
It requires that the following boundary conditions are satisfied:
$$
\lim_{\varepsilon\to 0}\alpha (\varepsilon)({\partial g\over
\partial \varepsilon }+g+g^2)=\lim_{\varepsilon\to \infty }\alpha 
(\varepsilon)({\partial g\over
\partial \varepsilon }+g+g^2)=0\eqno{(5.72)}
$$
i.e. that the particle flux is zero at the boundary of the domain.
Then, by a formal integration by parts on $\mathbb{R}^+$, $d(\varepsilon)$ has 
to be constant.
Let us denote it by $d$.


We have now to identify $\alpha $. By (5.68) and (5.69), $b(\varepsilon)=d\alpha 
(\varepsilon)$ where,
we recall
$$
b(\varepsilon)=\varepsilon^2{1\over 2}\int_0^{\infty}
({\varepsilon'}-\varepsilon)^2b({\varepsilon\over \beta ^{0}},
{\varepsilon'\over \beta ^{0}})e^{-\varepsilon}{\varepsilon'}^2d{\varepsilon'}
$$
and
$$
b({\varepsilon\over \beta ^{0}}, {\varepsilon'\over \beta ^{0}})=2\beta ^{0}
{m^2c^2\, r_0^2\pi^2\over \varepsilon
\varepsilon'}e^{\mu }e^{\varepsilon'}
\int_0^{\pi }(1+\cos ^2 \theta ){1\over |w|}e^{-\beta ^0{A^2mc\over 2|w|^2}}
\sin \theta  d\theta\,,
$$
with
$$
|\omega| ={1\over \beta ^{0}}(\varepsilon^2+{\varepsilon'}^2
-2\varepsilon\varepsilon'\cos \theta )^{1/2},\quad A={\varepsilon\over
\beta ^{0}}-{\varepsilon'\over \beta ^{0}}
+{|\omega |^2\over 2mc}.
$$
By the change of variables, $s =\cos \theta $ we slightly simplify this 
expression and obtain
$$b(\varepsilon)=m^2c^2r_0^2\pi ^2e^{\mu }\, e^{-\varepsilon}\varepsilon
\int_{-1}^{1}(1+s^2)\int_0^{\infty }(\varepsilon'-\varepsilon)^2
{e^{\varepsilon'}\, e^{-\beta ^{0}}{A^2mc\over 2|\omega |^2}\over |\omega
|}\varepsilon'd\varepsilon'ds.\eqno{(5.73)}
$$
To handle with the expression under the integral in (5.73) first notice
that by (5.61),
$$
{{\varepsilon'}^2\over 2mc}+\varepsilon'[1-{\varepsilon s\over mc}-({p_*\over 
mc}\cdot {p'\over |p'|})]
+{\varepsilon^2\over 2mc}-\varepsilon+\varepsilon({p_*\over mc}\cdot {p\over
|p|})=0.\eqno{(5.74)}
$$
A straightforward calculation gives
$$
\Delta =1+({\varepsilon\, s\over mc}+({p_*\over mc}\cdot {p'\over |p'|}))^2-
{\varepsilon^2\over m^2c^2}-{2\varepsilon\over mc}({p_*\over mc}
\cdot {p\over |p|})
-2({\varepsilon\, s\over mc}+({p_*\over mc}\cdot {p'\over |p'|}))
+{2\varepsilon\over mc}.
$$
Since all the terms in the right hand side, except $1$, are small when $m\gg 1$,
 $\varepsilon=\mathcal{O}(1)$,
$|p^{*}|=\mathcal{O}(1)$, we may use
$\sqrt{1+z}\sim 1+z/2-z^4/8+\cdots$ to obtain
$$
\sqrt{\Delta } \sim 1-{\varepsilon s\over mc}-({p_*\over mc}
\cdot {p'\over|p'|})+{\varepsilon\over mc}- {\varepsilon\over mc}
{p_*\over mc}\cdot \left( {p\over|p|}-{p'\over |p'|}\right)
-{\varepsilon^2\over m^2c^2}(1-s).
$$
Then, the two solutions of (5.74) are
\begin{align*}
\varepsilon'&=mc({\varepsilon\, s\over mc}+({p_*\over mc}\cdot
{p'\over |p'|})-1)\pm \, mc\sqrt {\Delta }\\
&=\varepsilon s+p_*\cdot {p'\over |p'|}-mc\pm mc\sqrt \Delta .
\end{align*}
The root with minus sign is negative, which is  inconsistent with the
fact that $\varepsilon'\ge 0$.  The root with the plus sign gives,
$$
\varepsilon'\sim \varepsilon-{\varepsilon^2\over mc}(1-s)
-{\varepsilon\over mc}p_*\cdot \big( {p\over |p|}-{p'\over |p'|}\big).
$$
Therefore,
\begin{align*}
|w|^2&={1\over {\beta ^{0}}^2}(\varepsilon^2+{\varepsilon'}^2-
2\varepsilon\varepsilon's)\sim
{1\over {\beta ^{0}}^2}(\varepsilon^2+(\varepsilon+{\varepsilon^2\over
2mc})^2-2\varepsilon(\varepsilon+{\varepsilon^2\over 2mc})\, s)
\\
&={1\over {\beta ^{0}}^2}(\varepsilon^2+\varepsilon^2(1+{\varepsilon\over
2mc})^2-2\varepsilon^2(1+{\varepsilon\over 2mc})\, s)\sim 2{\varepsilon^2\over 
{\beta ^{0}}^2}(1-s)
\end{align*}
Let us define
$$
\varepsilon_0=\varepsilon+{\varepsilon^2\over \beta ^{0}\, mc}(1-s).
$$
The quantity $\varepsilon-\varepsilon_0$ is the energy transfer of the photon as 
measured in the rest frame
of the initial electron (lab frame).
We have then
$$
|w|^2 \sim 2mc{(\varepsilon_0-\varepsilon)\over \beta ^{0}},
\quad \hbox{and}\quad A={\varepsilon\over \beta ^{0}}-
{\varepsilon'\over \beta ^{0}}+{|w|^2\over 2mc}
\sim {\varepsilon_0-\varepsilon'\over \beta ^{0}}.
$$
and therefore,
\begin{align*}
\int_0^{\infty }(\varepsilon'-\varepsilon)^2
{e^{-\beta ^0{A^2mc\over 2|w|^2}}\, e^{\varepsilon'}\over |w|}
\varepsilon'd\varepsilon'&\sim
{\beta ^{0}\over \sqrt {2(1-s)} \varepsilon}\int_0^{\infty }(\varepsilon'-
\varepsilon)^2
e^{-{|\varepsilon'-\varepsilon_0|^2\over 4|\varepsilon-\varepsilon_0|}}\,  
e^{\varepsilon'}\varepsilon'd\varepsilon'\\
&\sim \beta ^{0}{|\varepsilon-\varepsilon_0|\over \sqrt
{2(1-s)}\varepsilon}\int_0^{\infty }{(\varepsilon'-\varepsilon_0)^2\over 
|\varepsilon-\varepsilon_0|}
e^{-{|\varepsilon'-\varepsilon_0|^2\over 4|\varepsilon-\varepsilon_0|}}\,  
e^{\varepsilon'}\varepsilon'd\varepsilon'\\ &\sim {C\beta _0\, 
e^{\varepsilon}\over \sqrt
{2(1-s)}\varepsilon}|\varepsilon-\varepsilon_0|^{3/2}\varepsilon\\ &\sim   
C{\varepsilon^3\,
e^{\varepsilon}(1-s)\over  m^{3/2}c^{3/2}\sqrt {2\beta_0}},
\end{align*}
where we use the fact that $\varepsilon\sim \varepsilon_0$ and that
$$
\lim_{m\to 0}{1\over|\varepsilon-\varepsilon_0|^{1/2}}{(\varepsilon'
-\varepsilon_0)^2\over |\varepsilon-\varepsilon_0|}
e^{-{|\varepsilon'-\varepsilon_0|^2\over 4|\varepsilon-\varepsilon_0|}}
=C\delta _{\varepsilon'=\varepsilon}\,,\quad
C=\int_{-\infty }^{\infty }r^2\, e^{-r^2}\, dr.\eqno{(5.75)}
$$
Using this in (5.73) we obtain
\begin{align*}
b(\varepsilon)&=m^2c^2r_0^2\pi ^2e^{\mu }\, e^{-\varepsilon}\varepsilon
\int_{-1}^{1}(1+s^2)\int_0^{\infty }(\varepsilon'-\varepsilon)^2
{e^{\varepsilon'}\, e^{-\beta ^{0}}{A^2mc\over 2|\omega |^2}\over |\omega
|}\varepsilon'd\varepsilon'ds\\
&\sim C\sqrt{{m c\over 2\beta ^{0}}}\, r_0^2\pi ^2e^{\mu }\,
 \varepsilon^4\int_{-1}^{1}(1+s^2)(1-s)\,ds
\equiv \Lambda\varepsilon^4,
\end{align*}
where $\Lambda$ is independent of $\varepsilon$ and $s$.
The limiting equation reads,
$$
\varepsilon^2{\partial g\over \partial t}=
\Lambda{\beta ^{0}}^{-3\, }d^{-1} {\partial \over \partial
\varepsilon}[\varepsilon^4({\partial g\over
\partial \varepsilon }+g+g^2)].
$$
Finally, we perform the change of time variable
$t\to \Lambda t/d{\beta^{0}}^3$ and still denote by $t$
the new variable. The function $h(t, \varepsilon)$ of the new variables
satisfy then,
$$
\varepsilon^2{\partial h\over \partial t}=
{\partial \over \partial \varepsilon}[\varepsilon^4
({\partial h\over \partial \varepsilon }+h+h^2)].
$$
This is the so called Kompaneets equation such as it appears for example in
\cite{K, Le,Pe}. See also \cite{CL,EHV}.


\section{Appendix: A distributional lemma} \label{A1}

For a function $\Psi : \mathbb{R}^m \to \mathbb{R}$ we define $\delta_{\Psi}
= \delta_{\Psi = 0}$ by
\begin{equation}
\langle \delta_{\Psi = 0}, \varphi \rangle =  \lim_{\epsilon \to 0}
\int_{\mathbb{R}^m} \rho_\epsilon(\Psi(y)) \varphi(y) \, dy \quad
\forall \varphi \in C_c(\mathbb{R}^m) \quad \label{A1.1}
\end{equation}
where $(\rho_\epsilon)$ is any approximation of the 1-dimensional Dirac measure 
at the origin. Then, we have:


\begin{lemma} \label{lmA1.1}
For any $a,b \in \mathbb{R}$, $a \not= 0$
\begin{equation}
\delta_{a \, x - b} = {1 \over a} \delta_{x = b/a}.
\label{A1.2}
\end{equation}
For $a \not= b$,
\begin{equation}
\delta_{(x-a) (x-b)} = {1 \over |b-a|} ( \delta_{ x - a} + \delta_{ x - a} ).
\label{A1.3}
\end{equation}
\end{lemma}

\begin{remark} \label{rmkA1.2}\rm
As two particular cases we obtain
\begin{gather}
\forall a > 0 \quad \delta_{x^2-a^2} {\bf 1}_{x > 0}
= {1 \over 2 \, a} \delta_{x = a} ,\label{A1.4} \\
\forall a > 0 \quad \delta_{x (x-a)} {\bf 1}_{x > 0}
= {1 \over a} \delta_{x = a}.
\label{A1.5}
\end{gather}
\end{remark}

\paragraph{Proof of Lemma \ref{lmA1.1}}
Since \eqref{A1.2} is evident and \eqref{A1.4}, \eqref{A1.5} are immediate 
consequences of
\eqref{A1.3}, we just prove \eqref{A1.3}.
Let $(\rho_\epsilon)$ be a sequence of $L^1(\mathbb{R})$ such that
$\rho_\epsilon \rightharpoonup \delta$ in $\mathcal{D}'(\mathbb{R})$.
Then
\begin{align*}
\langle \delta_{(x-a) (x-b)},\phi\rangle
:=&  \lim_{\epsilon \to 0} \langle \rho_\epsilon((x-a) (x-b)),\phi\rangle  \\
=& \lim_{\epsilon \to 0} \int_\mathbb{R} \rho_\epsilon((x-a) (x-b)) \phi \, dx
= \lim_{\epsilon \to 0} I_\epsilon + J_\epsilon,\\
\end{align*}
with
$$
I_\epsilon = \int_{-\infty}^{a+b \over 2} \rho_\epsilon((x-a) (x-b)) \phi \, dx,
\quad
J_\epsilon = \int_{a+b \over 2}^\infty \rho_\epsilon((x-a) (x-b)) \phi \, dx.
$$
Without any loss of generality we may assume that $a<b$. Set $y = (x-a) (x-b) = 
x^2 - (a+b) \, x + a \, b$.
The function $x \mapsto y(x)$ is monotone for any $x \le (a+b)/2$ so that it is 
an allowed change
of variable. We compute $2 \, x = a + b \pm \sqrt{4 \, y + (a-b)^2}$ and
$dy = [2 \, x - (a+b) ]\, dx = \pm \sqrt{4 \, y + (a-b)^2} \, dx$, and hence
\begin{align*}
I_\epsilon &= \int_{- (b-a)^2 \over 4}^{\infty} \rho_\epsilon(y)
\phi \Bigl( {a + b \over 2} \pm \sqrt{ y + \bigl({a-b \over 2} \bigr)^2} \Bigr)
{dy \over \sqrt{4 \, y + (a-b)^2}} \\
&\rightarrow {1 \over |b-a|} \phi \bigl( {a + b - |a-b| \over 2} \bigr)
= {\phi(a) \over |b-a|}.
\end{align*}
Similarly, we prove $J_\epsilon \rightarrow \phi(a)/|b-a|$.
\hfill$\square$

\section{Appendix: Minkowsky space and Lorentz \\
transform} \label{A2}

We call $P = (P^0,p) \in \mathbb{R}^4$ with $P^0 \in \mathbb{R}$, $p \in 
\mathbb{R}^3$, or indifferently
$P = (P^\mu)$, the
{\it Lorentz metric} which is defined by
\begin{equation}
\langle P,Q \rangle = P^0 \, Q^0 - p \cdot q \quad \forall\, P,Q \in 
\mathbb{R}^4.
\label{A2.1}
\end{equation}
We also write
$$
\langle P,Q\rangle = P^\mu \, Q_\mu = P^\top \eta \, Q = \sum_{\mu,\nu=0}^4 
\eta_{\mu\nu} \, P^\mu \, Q^\nu,
\quad\hbox{with}\quad Q_\mu = \eta_{\mu\nu} \, Q^\mu,
$$
where
\begin{equation}
\eta = (\eta_{\mu\nu}) = \begin{pmatrix}1 &0^\top \\ 0 &-{\bf I}_3 \end{pmatrix}
\label{A2.2}
\end{equation}
is the Minkowsky matrix.
The inner product $\langle ,\rangle$ on $\mathbb{R}^4$ is symmetric, non
degenerated but not positive.

\paragraph{Definition} %\label{defA2.1} \rm
 A Lorentz transform is a linear operator $\Lambda : \mathbb{R}^4 \to 
\mathbb{R}^4$ such that
\begin{equation}
\langle \Lambda  P,\Lambda  Q \rangle = \langle P,Q\rangle  \quad
\forall \, P,Q \in \mathbb{R}^4.
\label{A2.3}
\end{equation}


\subsection{Examples of Lorentz transforms}
\paragraph{Rotations}
 For a rotation ${\bf R}$ of $\mathbb{R}^3$ (${\bf R} \in SO(3)$),
\begin{equation}
\Lambda = \begin{pmatrix}1 &0^\top \\ 0 &{\bf R} \end{pmatrix}
\label{A2.4}
\end{equation}
is a Lorentz transform.

\paragraph{Boosts} For $v \in \mathbb{R}^3$ such that $v := |v| < 1$,
\begin{equation}
\Lambda = \begin{pmatrix}\gamma &\gamma v^\top \\
\gamma v &{\bf I} + {\gamma - 1 \over v^2} v
v^\top\end{pmatrix},
\quad \gamma = {1 \over \sqrt{1 - v^2}}.
\label{A2.5}
\end{equation}
is a Lorentz transform.


\begin{remark} \label{rmkA2.2} \rm
Any Lorentz transform is the composition of a boost and a rotation, see
\cite{N}.
\end{remark}

For $(\beta^0,\beta) \in \mathbb{R}^4$ with $\beta^0 > |\beta|$ we define 
$\mathbf{b} > 0$ and $u \in \mathbb{R}^3$ by
\begin{equation}
\mathbf{b}^2 := (\beta^0)^2 - |\beta|^2, \quad {u \over c} := {\beta \over 
\beta^0}.
\label{A2.6}
\end{equation}
Then, setting
\begin{equation}
\gamma := {1 \over \sqrt{1 - (u/c)^2}},
\quad\hbox{we have}\quad
(\beta^\mu) = (\beta^0,\beta) = \bigr( \gamma \bar \beta, \gamma \bar \beta {u 
\over c} \bigl).
\label{A2.7}
\end{equation}
For such a 4-vector $(\beta^0,\beta)$ we define the boost transform $\Lambda = 
\Lambda_{(\beta^\mu)}$
associated to $v := u/c$. It satisfies
\begin{equation}
\Lambda \begin{pmatrix}\mathbf{b} \\ 0\end{pmatrix}
= \begin{pmatrix}\beta^0 \\ \beta\end{pmatrix}. \label{A2.8}
\end{equation}

\begin{lemma} \label{lmA2.3}
  Any Lorentz transform $\Lambda$ satisfies $\mathop{\rm det} \Lambda = \pm 1$.
\end{lemma}

\paragraph{Proof} For
$$\Lambda = \begin{pmatrix}a &b^\top \\ c & d\end{pmatrix}\quad
\mbox{define}\quad
\Lambda^\star = \begin{pmatrix}a &-c^\top \\ - b & d^\top\end{pmatrix}
$$
with $a \in \mathbb{R}$, $b,c \in \mathbb{R}^3$ and $d \in M(\mathbb{R}^3)$. We 
easily verify that
$\langle \Lambda  P, Q \rangle = \langle P,\Lambda^\star \, Q
\rangle$ for all $P,Q \in \mathbb{R}^4$.
In particular, by definition of a Lorentz transform, one has
$$
\langle \Lambda^\star \Lambda  P, Q \rangle = \langle \Lambda  P, \Lambda  Q
\rangle = \langle P,Q\rangle
\quad\hbox{for aall } P,Q \in \mathbb{R}^4,
$$
so that $\Lambda^\star \Lambda = Id_4$.
Since $\mathop{\rm det} \Lambda^\star = \mathop{\rm det} \Lambda$,
we get $(\mathop{\rm det} \Lambda)^2 = 1$. \hfill$\square$


\begin{definition} \label{defA2.4}\rm
For $s \in \mathbb{R}$ we define the set
\begin{equation}
M_+^s := \{ P \in \mathbb{R}^4, : \langle P,P\rangle = s, : P^0 > 0 \}
= \{ ( \sqrt{s + |p|^2},p), \quad p \in \mathbb{R}^3 \}.
\label{A2.9}
\end{equation}
We write $\Lambda \in\mathcal{L}_+^\uparrow$, if $\Lambda$ is a Lorentz 
transform
such that $\mathop{\rm det}\Lambda = +1$ and $(\Lambda \, P)^0 > 0$
for any $P \in M_+^s$, with $s > 0$.
\end{definition}

The boost $\Lambda$ associated to $(\beta^0,\beta)$ belongs to
$\mathcal{L}_+^\uparrow$ since $\mathop{\rm det}\Lambda = 1$ and
$$
(\Lambda \, P)^0 = \gamma p^0 + \gamma u \cdot p = {1 \over \mathbf{b}} [\beta^0 
p^0 + \beta \cdot p]
\ge {1 \over \mathbf{b}} [\beta^0 |p| - |\beta| | p| ] \ge 0
$$
for any $P \in M^s_+$ with $s > 0$.

\begin{lemma} \label{lmA2.5}
 Let $f : \mathbb{R} \to \mathbb{R}$ and $(\beta^\mu) = (\beta^0,\beta) \in 
\mathbb{R}^4$ such that
$\mathbf{b}^2 := \beta^{0 2} - |\beta|^2 > 0$. We define $F : \mathbb{R}^3 \to 
\mathbb{R}$ by
$F(p) = f(\beta^\mu p^\mu) = f(\beta^0 p^0 - \beta \cdot p)$ with $p^0 := 
\sqrt{s + |p|^2}$, $s > 0$.
Then, there are some constants $A_i = A_i(f,\mathbf{b})$ such that
\begin{gather}
\int_{\mathbb{R}^3} p^\mu F {dp \over p^0} = A_1 \,  \beta^\mu \label{A2.10}\\
\int_{\mathbb{R}^3} p^\mu p^\nu F {dp \over p^0} = A_2 \eta^{\mu\nu}
+ A_3 \beta^\mu \beta^\nu. \label{A2.11}
\end{gather}
In particular, one has
\begin{gather}
N(F) := \int_{\mathbb{R}^3} F \, dp = A_1 \beta^0  \label{A2.12}\\
P(F) := \int_{\mathbb{R}^3} F p \, dp = A_3 \beta^0 \beta \label{A2.13}\\
E(F) := \int_{\mathbb{R}^3} F p^0 \, dp = A_2  + A_3 (\beta^0)^2 \label{A2.14}\\
G(F) := s \int_{\mathbb{R}^3} F {dp \over p^0} = 4 \, A_2 + A_3 \mathbf{b}^2. 
\label{A2.15}
\end{gather}
\end{lemma}

\paragraph{Proof}
Using Lemma \ref{lmA2.3} we have
\begin{align*}
\delta_{P^2 = s} \, H(P^0) &=
\bigl( \delta_{(P^0 - \sqrt{s+|p|^2}) (P^0 + \sqrt{s+|p|^2})} \bigr) \, H(P^0) 
\\
&= {1 \over 2 \sqrt{s+|p|^2}} \bigl( \delta_{ P^0 - \sqrt{s+|p|^2}} +
\delta_{ P^0 + \sqrt{s+|p|^2} } \bigr) \, H(P^0) \\
&= {1 \over 2 \sqrt{s+|p|^2}} \delta_{ P^0 - \sqrt{s+|p|^2}}.
\end{align*}
where $H$  is the Heaviside function. Therefore, we get the fundamental
identity
\begin{align*}
\int_{\mathbb{R}^4} F(P) \delta_{P^2 = s} \, H(p^0) \, dP
&= \int_{\mathbb{R}^3} \Bigl\{ \int_\mathbb{R} F(P) {1 \over 2 \sqrt{s+|p|^2}} 
\delta_{ P^0 - \sqrt{s+|p|^2}}
\, P^0 \Bigr\} dp \\
&= \int_{\mathbb{R}^3} F(p^0,p) {dp \over p^0}.
\end{align*}
For $\Lambda \in \mathcal{L}_+^\uparrow$ we get
\begin{align*}
\int_{\mathbb{R}^3} F(\Lambda(p^0,p)) {dp \over p^0}
&= \int_{\mathbb{R}^4} F(\Lambda \, P) \delta_{P^2 = s} \, H(p^0) \, dP \\
&= \int_{\mathbb{R}^4} F(Q) \delta_{Q^2 = s} \, H(q^0) \, dQ
= \int_{\mathbb{R}^3} F(p^0,p) {dp \over p^0}.
\end{align*}
Now we choose as $\Lambda$ the boost associated to $(\beta^\mu)$ and using
\eqref{A2.8} we have, setting $P =\Lambda \, Q$, we get
\begin{align*}
\int_{\mathbb{R}^3} p^\mu f(\beta^\nu p^\nu) {dp \over p^0}
&= \int_{\mathbb{R}^3} p^\mu f(\mathbf{b} (\Lambda^{-1} P)^0) {dp \over p^0} \\
&= \Lambda^\mu \int_{\mathbb{R}^3} q f(\mathbf{b} q^0) {dq \over q^0}
= \Lambda^\nu \begin{pmatrix} \mathbf{b} \, A_1 \\ 0\end{pmatrix},
\end{align*}
with
$$
A_1 := \int_{\mathbb{R}^3} q^0 f(\mathbf{b} q^0) {dq \over q^0},
$$
since
$$
\int_{\mathbb{R}^3} q^i f(\mathbf{b} q^0) {dq \over q^0} = 0
$$
for $i = 1,2,3$ by rotation symmetry. This proves \eqref{A2.10}.
Similarly,
$$
\int_{\mathbb{R}^3} p^\mu p^\nu f(\beta^\sigma p^\sigma) {dp \over p^0}
= \Lambda^\mu_{\mu'} \Lambda^\nu_{\nu'} \int_{\mathbb{R}^3} q^{\mu'} q^{\nu'} 
f(\bar\beta q^0) {dq \over
q^0}
= \Lambda^\mu_{\mu'} \Lambda^\nu_{\nu'} \alpha_{\mu'} \delta_{\mu' \nu'}
$$
with
$$
\alpha_0 = \int_{\mathbb{R}^3} (q^0)^2 f(\bar\beta q^0) {dq \over q^0}
\quad
\alpha_1 = \alpha_2 = \alpha_3 = \int_{\mathbb{R}^3} (q^1)^2 f(\bar\beta q^0) 
{dq \over q^0}.
$$
By simple computations,
\begin{align*}
\Lambda^2 &= \begin{pmatrix}2\gamma^2 - 1 & 2\gamma^2 v^\top \\
2\gamma^2 v &{\bf I} + 2\gamma^2 v v^\top\end{pmatrix}
= - (\eta)^{\mu \nu} + 2 \begin{pmatrix}\gamma^2 &\gamma (\gamma u/c)^\top \\
\gamma (\gamma u/c)  &(\gamma u/c) (\gamma u/c)^\top \end{pmatrix} \\
&= - (\eta)^{\mu \nu} + {2 \over \mathbf{b}^2} \beta^\mu \beta^\nu,
\end{align*}
and
$\Lambda^\mu_0 = \begin{pmatrix}\gamma \\ \gamma u \end{pmatrix}
= {\beta^\mu \over \mathbf{b}}$.
Therefore
\begin{align*}
\int_{\mathbb{R}^3} p^\mu p^\nu f(\beta^\sigma p^\sigma) {dp \over p^0}
&= \alpha_1 \Lambda^\mu_{\mu'} \Lambda^\nu_{\mu'}
+ (\alpha_0 - \alpha_1) \Lambda^\mu_0 \Lambda^\nu_0 \\
&= \alpha_1 (\Lambda^2)^{\mu \nu}
+ (\alpha_0 - \alpha_1) \Lambda^\mu_0 \Lambda^\nu_0 \\
&= - \alpha_1 \eta^{\mu \nu} + {\alpha_0 + \alpha_1 \over \mathbf{b}^2} 
\beta^\mu
\beta^\nu,
\end{align*}
and \eqref{A2.11} is proved. The statements \eqref{A2.12}, \eqref{A2.13} and 
\eqref{A2.14} follow.
Finally, we compute, thanks to \eqref{A2.11},
$$
\int_{\mathbb{R}^3} \eta_{\mu\nu} p^\mu p^\nu f {dp \over p^0}
= A_1 \eta^{\mu\nu} \eta_{\mu\nu} + A_2 \beta^\mu \beta^\nu \eta_{\mu\nu},
$$
so that
$$
\int_{\mathbb{R}^3} ((p^0)^2 - |p|^2) f {dp \over p^0}
= 4 \, A_1 + A_2 [(\beta^0)^2 - |\beta|^2 ]
= 4 \, A_1 + A_2 \bar\beta^2,
$$
and \eqref{A2.15} follows, remarking that $(p^0)^2 - |p|^2 =s$. \hfill$\square$

\section{Appendix: Differential cross section}\label{A3}

We present in this Appendix some short notes about differential cross sections,
mainly taken from the volumes 3 and 10 of the Landau and Lifschitz.

Following Landau and Lifshitz vol.10, \S 2 let us consider collisions between
two molecules of a monatomic gas, one of which has momentum
$p$ in a given range $dp$ and the other in a range $dp_{*}$ and which
acquire in the collision values in the ranges $dp'$ and $dp_{*}'$
respectively. For brevity we refer simply to a collision of molecules
with $p$
and $p_{*}$, resulting in $p'$ and $p_{*}'$. The total number of such
collisions per unit time and unit volume of the gas may be written as a
product of the number of molecules per unit volume, $f(t, p)dp$, and the
probability that any of them has a collision of the type concerned. This
probability is always proportional to the number of molecules $p_{*}$ per
unit volume, $f(t, p_{*})dp_{*}$ and to the number of molecules $p'$ and
$p'_*$ per unit volume, $(1+\tau f(t, p'))dp'$ and $(1+\tau f(t, p'_*))dp'_*$.
Thus the number of collisions $p,p_{*}\to p', p_{*}'$ per unit time and
volume may be written as
$$W(p', p_{*}'; p, p_{*})ff_{*}(1+\tau f')(1+\tau f'_*)dpdp_{*} dp'dp_{*}',$$
where as usual, $\tau =1$ for Bose particles, $\tau =-1$ for Fermi particles and 
$\tau =0$
for non quantum particles.  The coefficient $W$ is a function of all its 
arguments. The
ratio of
$Wdp'dp_{*}'$ to the absolute value of the relative velocity $v-v_{*}$
of the colliding molecules has the dimensions of area, and is the
effective collision cross section:
$$d\sigma ={W(p', p_{*}'; p, p_{*})\over |p-p_{*}|}dp'dp_{*}'$$
The function $W$ can in principle be determined only by solving the
mechanical problem of collision of particles interacting according to some
given law. However, certain properties of this function can be elucidated
from general arguments.


The first property is called detailed balance and reads:
$$W(p', p_{*}';p, p_{*})=W(p, p_{*};p', p_{*}');
$$
i.e. each microscopic collision process is balanced by the reverse
process. It is a direct consequence of the following two symmetries:
\begin{enumerate}

\item The symmetry of the laws of the mechanic
(classical or quantum) under time reversal: according to this, the number
of collisions $p,p_{*}\to p', p_{*}'$ is equal, in equilibrium, to the
number $-p',-p_{*}'\to -p, -p_{*}$ from where one obtains:
$$W(p', p_{*}'; p, p_{*})=W(-p, -p_{*}; -p', -p_{*}').
$$
\item The symmetry of the molecules under spatial inversion i.e. change of
the signs of all coordinates. This symmetry implies
$$W(-p, -p_{*}; -p', -p_{*}')=W(p, p_{*}; p', p_{*}')$$
and the detailed balance follows.
\end{enumerate}

Moreover, we may also use the fact that, we do not integrate in the whole
momentum space but only along the manifold determined by the conservation
of energy and conservation  of the momentum. Let us assume for the sake of
brevity that the two particles have the same mass equal to one. Then, the
two conservation properties read:
\begin{gather*}
p'+p_{*}'=p+p_{*}\\
{|p|^2\over 2}+{|p_{*}|^2\over 2}={|p'|^2\over 2}+{|p'_{*}|^2\over 2}.
\end{gather*}
The expression of $W$ may therefore be written
\begin{align*}
&W(p', p_{*}'; p, p_{*})dp'dp'_{*}\\
&=H(p', p_{*}'; p,
p_{*})\delta (p'+p'_{*}-p-p_{*})
\delta({|p'|^2+|p_{*}'|^2-|p|^2-|p_{*}|^2\over 2})\,dp'\,dp'_{*}\\
&=H(p+p_{*}-p'_{*}, p_{*}'; p, p_{*})\delta
({|p'|^2+|p_{*}'|^2-|p|^2-|p_{*}|^2\over 2})\,dp'_{*}
\end{align*}
On this manifold we can write
$$
\begin{gathered}
p'=q(p, p_{*}, \omega )=p-(p-p_{*}, \omega )\omega\\
p_{*}'=q_{*}(p, p_{*}, \omega )=p_{*}+(p-p_{*}, \omega )\omega,
\end{gathered}
\quad \omega \in {\bf S}^2
$$
from where,
$$
W(p', p_{*}'; p, p_{*})dp'dp'_{*}=B(p, p_{*}, \omega )d\omega.
$$
Finally, since the two interacting particles constitute a closed physical
system, $W$  has to be Galilean invariant (or Lorentz invariant
in the relativistic case). Consider for the sake of
simplicity the classical case. This implies
$$
W(Tp', Tp_{*}'; Tp, Tp_{*})=W(p', p_{*}'; p, p_{*})
$$
for any rotation $T=R\in SO(3)$ and any translation $T(p)=a+p$ with $a\in
\mathbb{R}^3$. Also we have
$$(q(p+a, p_{*}+a,\omega), q_{*}(p+a, p_{*}+a, \omega ))=
(q(p, p_{*}, \omega )+a, q_{*}(p, p_{*}, \omega )+a)
$$
in such a way that
$$
B(p+a, p_{*}+a,\omega )=W(p+a, p_{*}+a, p'+a, p_{*}'+a)=W(p, p_{*},p',
p_{*}')=B(p, p_{*}, \omega )\,.
$$
Therefore, $B(p, p_{*}, \omega )=B(0, p_{*}-p,\omega )$
that we denote
$$B(0, p_{*}-p,\omega )\equiv B(p_{*}-p,\omega ).$$
On the other hand, we also have
$$
(q(Rp, Rp_{*}, R\omega ), q_{*}(Rp, Rp_{*}, R\omega ))=
(Rq(p, p_{*}, \omega ), Rq_{*}(p, p_{*}, \omega ))
$$
in such a way that
$$
\begin{aligned}B(R(p_{*}-p), R\omega )&=B(Rp, Rp_{*}, R\omega )\\
&=W(Rp, Rp_{*}, Rp',
Rp_{*}')=W(p, p_{*}, p', p_{*}')=B(p_{*}-p, \omega )
\end{aligned}
$$
for every rotation $R\in SO(3)$. Therefore
$$
B(p_{*}-p, \omega )=B(|p_{*}-p|, {p_{*}-p\over |p_{*}-p|}, \omega ).
$$
Note that  $B(p_{*}-p, \omega )f(p_{*})d\omega dp_{*}$ is the probability
per unit time and unit volume that any of the colliding particles $p$
has a collision of the type considered. The effective collision cross
section is then defined  by
$$
d\sigma={B(p_{*}-p, \omega )\over |p-p_{*}|}d\omega .
$$

\subsection{Scattering theory}
The differential cross section depends on a
crucial way on the kind of interaction between the two colliding particles
that one considers. If the interaction
between particles only depends on the distance between the two particles,
we may assume without any loss of generality that
$$
B(|p_{*}-p|, {p_{*}-p\over |p_{*}-p|}, \omega )
=B(|p_{*}-p|, {p_{*}-p\over |p_{*}-p|}\cdot \omega ),
$$
i.e. the function $B$ only depends on the modulus of the difference of
momentum of the two incident particles and of the angle $\alpha $ formed by the
two directions $p_{*}-p$ and $p_{*}'-p'$.


On the other hand, like any problem of
two bodies, the problem of elastic collision  amounts to a problem, of the
scattering of a single particle with the reduced mass in the field $U$ of a
fixed centered force. This simplification is effected by changing to a system
of coordinates in which the center of mass of the two particles is at rest.
We set
$$
\Omega ={p-p_{*}\over |p-p_{*}|}-2({p-p_{*}\over |p-p_{*}|},\omega )\omega
$$
in such a way that with this new variable,
\begin{align*}
p'={p+p_{*}\over 2}+{|p-p_{*}|\over 2}\Omega,\\
p_{*}'={p+p_{*}\over 2}-{|p-p_{*}|\over 2}\Omega.
\end{align*}
Then we have
$d\omega ={1\over 4\cos \alpha }d\Omega$, for
$\alpha =$ angle between $p-p_{*}$ and $\omega $.


Then, we use a spherical coordinate system with axis $p-p_{*}$:
$$
\Omega ={p-p_{*}\over |p-p_{*}|}\cos \theta +(\cos \phi h
+\sin \phi i)\sin \theta.
$$
With these new variables,
$$|(p-p_{*}, \omega )|=|p-p_{*}|\sin {\theta \over 2}\quad
\mbox{and}\quad
\cos \alpha = \sin {\theta \over 2}.
$$
Therefore,
$$
d\omega ={d\Omega \over 4\cos \alpha }={1 \sin \theta d\theta d\phi \over
4\cos \alpha}={\sin \theta \over 4 \sin (\theta /2)}d\theta
d\phi={1\over 2}\cos(\theta /2)d\theta d\phi
$$
from where,
$$
B(z, \omega )d\omega ={B(z, \omega  )\over 4\cos \alpha }d\Omega ={\cos
(\theta /2)\over 2}B(z, \omega  )d\theta d\phi.
$$
The differential cross section, $\sigma (z, \theta )$ is then such that
$$
B(z, \omega )d\omega =\sigma (z, \theta)\, d\Omega\equiv \sigma (z, \theta)\sin
\theta d\theta d\phi.
$$
Then write
$$
d\sigma (p, p_{*}, \theta )=\sigma {(|p-p_{*}|, \theta)\over |p-p_{*}|}d\Omega.
$$
The angle $\theta $ is the angle formed by the incident and scattered
trajectory of the particle interacting with the central potential.  In
classical mechanics, collisions of two particles are entirely determined by
their velocities and impact parameter. For a detailed study of the different
differential cross section depending on the potential
$U$ considered in the classical case the reader may consult the detailed work
by Cercignani \cite{C}. It is shown in particular that:


\noindent 1.- If $U$ is a power law potential:
$U(\rho )=|\rho |^{1-n}$, $n\not = 2,3$,
then
$$B(|p_{*}-p|, {p_{*}-p\over
|p_{*}-p|}\cdot \omega )=|p_{*}-p|^{\gamma }\beta ({p_{*}-p\over
|p_{*}-p|}\cdot \omega ),$$
where $p$ and $p_{*}$ are the velocities of the two particles.


\noindent 2.- \textit{Coulomb potential.}\quad
If $U(\rho )=\alpha |\rho |^{-1}$, then
$$\sigma (|p-p_{*}|, \theta )={\alpha ^2\over 16|p_{*}-p|^4\sin ^4{\theta
\over 2}}.$$
This is known as the Rutherford's formula (see Landau \& Lifschitz vol. 1 \S
19).

\noindent 3. \textit{Hard sphere potential.}\quad
For the so called Hard sphere potential $U$ defined as
$$
U(\rho )=\lim_{n\to \infty }U_n(\rho ),\quad U_n({\bf r})
=\begin{cases} n &\mbox{if }|\rho |<a\\
0 & \mbox{if }|\rho |>a \end{cases}
$$
we have $\sigma =a^2$.

\begin{remark} \label{rm1}\rm
In the general case, where the two interacting
particles $p$ and $p_{*}$ have masses $m_1$ and $m_2$ respectively , similar
arguments and calculations can be performed. In particular, the center of mass
parametrization may be written:
\begin{align*}
p'={m_1m_2\over m_1+m_2}(p+p_{*})+ {m_1\over m_1+m_2}|p-p_{*}|\Omega,\\
p_{*}'={m_1m_2\over m_1+m_2}(p+p_{*})- {m_2\over m_1+m_2}|p-p_{*}|\Omega.
\end{align*}
One defines again the angle $\theta$ by
$$\Omega ={p-p_{*}\over |p-p_{*}|}\cos \theta +(\cos \phi h+\sin \phi
i)\sin \theta.$$
As before,
$W(p', p_{*}'; p, p_{*})dp'dp'_{*}=\sigma (z, \theta)d\theta d\phi$
and
$$d\sigma (p, p_{*}, \theta )=\sigma {(|p-p_{*}|,
\theta )\over |p-p_{*}|}d\theta d\phi  .$$
\end{remark}

\begin{remark}[{\cite[Vol.10, \S 2.]{LL}}] \label{rmk2} \rm
 Although
the free motion of particles is assumed to be classical, this does not at all
mean that their collision cross section need not be determined  quantum
mechanically; in fact, it usually must be so determined.
\end{remark}

In quantum mechanics the very wording of the scattering problem must be
changed, since in motion with definite velocities the concept of path is
meaningless, and so is the impact parameter. The purpose of the theory is
then only to calculate the probability that, as a result of the collision,
the particles will scattered through any given angle. It is not our purpose
to present in detail the derivation and the properties of the differential
cross section from the scattering theory in detail. We only want to present
the general idea, the relevant results for our study and some precise
references for the interested readers.

We only give here a brief description of what M. Reed and B. Simon present
as ``na{\"\i}ve'' scattering theory, or a stationary picture of it (\cite{RS}, 
Vol.3, Notes on
\S X1.6). It is nevertheless the usual in the textbooks of quantum mechanics as 
for instance
vol.3, \S 123 of Landau Lifschitz.

For a fixed $\mathbb{R}^3$-vector $p$ and a positive real number $E$
let us consider the function of $\rho =(x, y, z)$ and $t$ called plane wave:
$$e^{-{i\over \hbar}Et}e^{{i\over \hbar}p\cdot\rho }.$$


This plane wave describes  a state in which the particle has a definite
energy $E$ and momentum $p$. The angular frequency of this wave is
$E/\hbar $ and its wave vector $k=p/\hbar$; the corresponding
wavelength $2\pi \hbar /|p|$ is the de Broglie wavelength of the
particle. The mass is $m=|p|^2/2E$ and the velocity is $v=p/m$.


Consider now a free particle, with mass $m$, total energy $E$, moving in the
direction of the
$z$-axis and  by abuse of notation ley us denote its plane wave as
$$\psi_1 (\rho )=e^{ikz},\quad \forall \rho =(x, y, z)\in \mathbb{R}^3.$$
Assume that it is scattered by a radially symmetric potential
$U(r)$ ($\rho =(r, \theta , \varphi )$ in polar
coordinates).


The basic ansatz of na{\"\i}ve scattering theory is that the scattering state
is the solution of the linear
Schr\"odinger equation
$$\Delta \psi (\rho ) +k^2\psi (\rho )
-{2m\over \hbar ^2}U(r)\psi (\rho )=0
$$ such that
$$\psi (\rho )\sim e^{ikz}+f(k, \theta ){e^{ikr}\over r}\quad
\hbox{as}\quad r\to \infty .$$
This is, at large $r$ we see the incident plane wave moving in the positive 
sense
of the $z$- axis and a spherical divergent wave, modulated by the function
$f\equiv f(k, \theta )$, called the scattering amplitude. This extra term 
describes the
``scattered particle''.


The scattered particle is described far from the center, as a
spherical divergent wave, i.e. a wave moving in the ``increasing'' sense of
the radial direction
$$ f(k, \theta )e^{ikr}/r. $$
Remember that the square of the modulus of the wave $\psi $ is the
density of probability to find the particle at the point $\rho $. The
presence of the factor $1/r$ is just to preserve that property since we are
in $\mathbb{R}^3$. The density of probability is not necessarily the same in
all the points but has to be independent of the
$\varphi $ and
$r$ variables due to the  spherical symmetry of the potential. This is taken
into account by the coefficient
$f(k, \theta )$ which is the amplitude diffusion and depends only on the angle
$\theta$ between the direction of the incoming particle, which is
$e_3=(0, 0, 1)$,
and the direction where we are looking for the scattered particle, i.e.
$\rho / r$.


As it is pointed out in \cite{RS}, at first sight, this ansatz looks
absurd, for, if
$\psi \sim e^{ikz}+f(\theta )r^{-1}e^{ikr}$ for $r\to \infty $, then, for all
the time $\psi $ has both a plane wave coming in and an outgoing spherical
wave. The point of the argument is to consider an initial state which is more
localized i.e. given by the function:
$$\psi (\rho  )=\int g(k)\left(e^{ikz}+f(\theta )r^{-1}e^{ikr}\right)dk$$
and $g$ peaked around $k=k_0$. Then following the same idea, for $r$ and $t$
large, the wave would be given by
$$
\psi (\rho ) \sim \int g(k)e^{ik(z-kt)}dk+r^{-1}f(\theta )
\int g(k)e^{ik(r-kt)}\,dk.
$$
Then, essentially by the Riemann-Lebesgue lemma, the first integral for $z$
and $t$ large has appreciable size only for $z\sim k_0t$ and the second
integral if $r  \sim k_0t.$ Therefore, if $t\to -\infty $, the second term is
negligible for all $r\ge 0$ and we recover, asymptotically only the incident
wave.


Therefore, the probability per unit time that the scattered particle will
pass through the surface element $dS=r^2d\Omega$  is
$(v/r^2)|f|^2dS=v|f|^2d\Omega$, where $v$ is the velocity of the particle. We
have then:
$$\sigma (k, \theta )d\Omega=|f(k, \theta )|^2d\Omega $$
and we recover the well known formula $d\sigma =|f(\theta )|^2d\Omega$.


We are then lead to see how do we get information about the function
$f(k,\theta )$. It is well known that the
solutions of the linear Schr\"odinger equation may be written as
$$\psi (r)=\sum _{l=0}^{\infty} A_{l}P_{l}(\cos \theta )R_{k, l}(r)$$
where $A_l$ are constants and the $R_{k, l}(r)$ are radial functions
satisfying the equation
$$
{1\over r^2}{d\over dr}(r^2{dR\over dr})+[k^2-{l(l+1)\over r^2}-{2m\over
\hbar }U(r)]R=0
$$
and the $P_l$ are the Legendre polynomials. The coefficients $A_l$ are
constants which have to be chosen so that the condition at $r\to
\infty $ is fulfilled. This implies that:
$$
A_{l}={1\over 2k}(2l+1)i^{l}\exp(i\delta _l)
$$
where for every $l$, $\delta _l$ is a constant called phase shift.


To see this observe that, the asymptotic form of each of the functions $R_{k,
l}$ is
$$
R_{k, l}(r)\sim {2\over r}\sin (kr-{l\pi \over 2}+\delta _{l})={1\over ir}
\{(-i)^{l}\exp [i(kr+\delta _l)]-i^{l}\exp [-i(kr+\delta _l)]\}
$$
Therefore, it is formally deduced that
$$
\psi (r)\sim \sum _{l=0}^{\infty }A_lP_l(\cos \theta
){i\over r }\Big\{ \exp [-i(kr-{l\pi \over
2}+\delta _{l})]-\exp [i(kr-{l\pi \over
2}+\delta _{l})]\Big\}.
$$
On the other hand, the plane wave expansion in spherical harmonics gives
$$
e^{ikz}=\sum _{l=0}^{\infty} (-i)^l(2l+1)P_l(\cos \theta )
({r\over k})^l({1\over r}{d\over dr})^l{\sin kr\over kr}\,.
$$
As $r\to \infty $ we have
\begin{align*}
e^{ikz}&\sim {1\over kr}\sum _{l=0}^{\infty }i^l(2l+1)P_l(\cos
\theta )\sin (kr-{l\pi \over 2})\\
&=\sum _{l=0}^{\infty }i^l(2l+1)P_l(\cos
\theta ){i\over 2kr }\big\{ \exp [-i(kr-{l\pi \over
2})]-\exp [i(kr-{l\pi \over 2})]\big\}
\end{align*}
For $A_{l}={1\over 2k}(2l+1)i^{l}\exp(i\delta _l)$, as indicated above,
$$\psi -e^{ikz}\sim {i\over 2kr}\sum _{l=0}^{\infty }(2l+1)P_{l}(\cos \theta
) [(-1)^{l}e^{-ikr}-S_le^{ikr}]$$
with $S_l=e^{2i\delta_l }$.
Then
$$
f(k, \theta )={1\over 2\, i\, k}\sum _{l=0}^{\infty }(2l+1)[S_l-1]P_{l}
(\cos \theta ).
$$
Integrating $d\sigma $ over all the values of the angles we obtain the total
cross section
$$\sigma =2\pi \int_0^{\pi } |f(k, \theta )|^2\sin\theta d\theta={4\pi \over
k^2}\sum _{l=0}^{\infty }(2l+1)\sin^2\delta _l$$
and since the Legendre polynomials are orthonormal,
$$\int_0^{\pi }P_{l}^2(\cos \theta )\sin \theta d\theta ={2\over 2l+1}.$$
The coefficients
$f_{l}(k, \theta )={1\over 2i\, k}(S_{l}-1)$
are called partial amplitude diffusion.


\subsection{Study of the general formula of $f(k, \theta )$}
This formula is valid for all radial potential $U(r)$ vanishing at infinity.
Its study reduces to that of the phases $\delta _l$.
(We quote \cite[Vol. 10, \S 124]{LL}).

Under the assumption that $U(r)\sim r^{-n}$ as $r\to \infty$, we have the
following two statements:


\noindent 1.- If $n>2$ then the total cross section is
finite and the differential cross section is integrable.


\noindent 2.- If $n\le 2$ the total cross section is infinite and the 
differential
cross section is not integrable. From a physical point of view this is due to
the fact that, since the field is slowly decreasing with the distance, the
probability of diffusion of very small angles becomes very large. Remember
that in classical mechanics, for every positive potential $U(r)$ vanishing
only at infinity, any particle with large but finite impact parameter is
deviated by a small but non zero angle and so the total cross section is
infinite, whatever is the decay of $U(r)$. (In that sense, it may be
considered that the quantum scattering is more regular, or less singular,
than the classical one).

Concerning the differential cross section itself
we have the following two statements:

\noindent 1.-  If $n\le 3$ the differential cross section becomes infinite
as $\theta \to 0$

\noindent 2.- If $n> 3$ the differential cross section is finite as
$\theta \to 0$ \smallskip

Finally, if $n\le 1$, then the total cross section is infinite, i.e. the
differential cross section is not integrable; the differential cross section
is singular as $\theta \to 0$ but it is well defined for $\theta \not=0$ and
is given by
$$f(k, \theta )={1\over 2\, i\, k}\sum _{l=0}^{\infty }(2l+1)P_l(\cos \theta
)(e^{2i\delta _l}-1).$$



\subsection{Non radial interaction}
Let us consider again the incident plane wave $e^{ikz}$ in the
radial potential considered above, moving in the
$(0,0,1)$ direction, which is reflected  and which at a large distance
point $\rho =(r, \theta , \phi )$ is seen as $e^{ikz}+f(\theta )e^{ikr}/r$.
Observe that, given the vector
$\rho =(x, y, z)=(r,\theta ,\phi )$, we have
$$
z=r\,e_{3}\cdot {\rho \over r}\,,\quad e_{3}=(0, 0, 1),
$$
i.e. $z$ may be seen as $r$ times the scalar product of the two unitary
vectors giving the directions of the incident and scattered particle's
velocities.  Consider now a general potential $U(\rho )$, and
$n$ a unitary vector of $\mathbb{R}^3$. Consider then an
incident particle in the direction $n$,
scattered by $U(\rho )$. The wave describing this particle  would then be
the solution of the Schr\"odinger equation
$$-\Delta \psi (\rho ) -k^2\psi (\rho ) +U(\rho )\psi (\rho )=0 $$
such that at the point $\rho =(r, \theta , \psi )$ with $|\rho |\to
\infty $,
$$\psi (\rho )\sim e^{ikrnn'}+{1\over r}f(n, n')e^{ikr}$$
where $ n'=r/\rho $.
The amplitude diffusion depends on the two directions
of the incident and scattered particles and not only on the angle that they
form.


\paragraph{Born's formula} (\cite[Vol. 3, \S 126]{LL})
This s formula gives an
explicit relation between the differential cross section and the potential
$U(\rho )$, non necessarily radial. As we have seen, in that case the
amplitude diffusion depends on the  incident and scattered directions and not
only on the angle they form. The explicit expression is:
$$f(q, q' )=-{m\over 2\pi \hbar ^2}\int_{\mathbb{R}^3}U({\bf
r})e^{-i(q'-q)\cdot \rho }dV(\rho )$$
$$
{d\sigma \over d\Omega}={m^2\over 4\pi ^2\hbar ^4}|\int_{\mathbb{R}^3}U(\rho 
)e^{-i(q'-q)\cdot
\rho }dV(\rho )|^2,\quad |q'-q|=2k\sin {\theta
\over 2} ,
$$
where $\theta $ is the angle between the two vectors $q$ and $q'$.


One may approximate the differential cross
section by the Born's formula whenever the perturbation field $U(\rho )$,
not necessarily spherically symmetric, may be considered as a perturbation.
[This corresponds to the case where all the phases $\delta _l$ are small].
This is possible when one of the following conditions are fulfilled:
$$
|U| \ll {\hbar ^2\over ma^2}\quad
\mbox{or}\quad
|U| \ll {\hbar v\over a}={\hbar ^2\over ma^2}ka
$$
where $a$ is the rayon of action of $U(\rho )$ and $U$ its order of magnitude
in the main region of its existence. In the first case, the Born's
approximation may be applied for all the velocities. In the second case, it
may be applied for particles with sufficiently large velocities.

Moreover, if the potential is spherically symmetric, $U=U(r)$, then we  obtain
$$
f(k, \theta )=-{m\over \hbar ^2}\int_0^{\infty }U(r){\sin [2rk\sin {\theta
\over 2}]\over k\sin {\theta
\over 2}}rdr.
$$
When $\theta =0$, this integral diverges provided $U(r)$ decreases
as $r^{-3}$ or slower when $r\to \infty $.


\subsection{Scattering of slow particles:} (\cite[Vol. 3, \S 132]{LL}).
We consider the limiting case where:

\noindent 1.- The potential $U(r)$ is radial and decreases at large distances
more rapidly than $1/r^3$.

\noindent 2.- The velocities of the particles
undergoing scattering are so small that their wavelength is large compared
with the radius of action $a$ of the field $U(r)$, i.e. $ka<<1$, and
their energy is small compared with the field within that radius,


Under these conditions, the total amplitude may be approximated as
$ f(\theta )\sim f_0$  which is the first partial amplitude.
Therefore, $d\sigma (k, \theta ) =|f_0|^2d\Omega$.
At low velocities, the scattering is isotropic, and the differential cross
section is independent of the particle energy.

If the potential decreases as $U(r)\sim r^{-n}$ with $n<3$ the above
approximation is not valid.


\subsection{Some examples of differential cross sections}
\noindent (i) \textit{Coulomb interaction}: (Rutherford formula
(\cite[Vol. 3 \S 135]{LL}.)
The scattering by central coulomb potential is particularly important with
respect to the applications in  physics. Moreover, the quantumechanic problem
of the collisions may be solved explicitly until the end.
So, if we assume the potential to be $U(\rho )= |\rho |^{-1}$, the
differential cross section is
$$
\sigma(k, \theta )= {1\over 4k^4 \sin ^4{\theta \over 2}}.
$$
Observe that it is the same as the differential cross section
obtained for the classical Coulomb interaction.


\noindent (ii) \textit{Hard sphere potential.}  (Slow particles;
\cite[Vol. 3 \S 132,  Problem 2]{LL}.)
For the Hard
sphere potential $U(r)=\lim_{n\to \infty }U_n(r)$, with
$$
U_n(r )=\begin{cases}n &\mbox{if } r<a\\
0 &\mbox{if } r>a, \end{cases}
$$
the differential cross section, under the conditions $ka\ll 1$ and for
small values of $k$ is $\sigma =a^2$.
Observe that this implies that the total cross section is $4\pi \,
a^2$, which is four times the result following classical mechanics.


\noindent (iii) \textit{Yukawa potential.} (Born approximation;
\cite[Vol. 3 \S 126,  Problem 3]{LL})
Assume that $U(r)=\alpha{{e^{-r/a}\over r}}$.
Then the Born approximation is
$$\sigma =({\alpha ma\over\hbar ^2 })^2{4a^2\over (q^2a^2+1)^2},
$$
with $q=2k\sin(\theta /2)$
This approximation is valid whenever $\alpha ma/\hbar ^2 \ll 1$ or $\alpha
/\hbar v\ll 1$. In the first case it is valid for all the velocities. In the
second, only for velocities sufficiently large.

One may find in \cite{LL}, Vol. 3,  \S 132, problem 1 and problem 2 the
Born approximations of the differential cross sections corresponding to the 
spherical well
and uniform potential  barrier and in \S 126 problem 2 to the Gaussian 
potential.

\paragraph{Collisions of identical particles} (\cite[Vol. 3 \S 137]{LL}.)
If the two particles are identical, then they are indiscernible and
$$W(p', p_{*}'; p, p_{*})=W(p_{*}', p'; p, p_{*}).
$$
The wave function of a system of two particles has to be symmetric or
antisymmetric with respect to the particles depending whether their total
spin is odd or even. Therefore, the wave function, describing the scattering,
obtained by solving the Schr\"odinger equation has to be symmetrized or
antisymmetrized. Their asymptotic expansion has then to be written as
$$
\psi \sim e^{ikz}\pm e^{-ikz}+{1\over r}e^{ikr}[f(\theta )\pm f(\pi -\theta)].
$$
In that way, if the total spin of the particles in the collision is even, the
differential section is
$$
d\sigma _s=|f(\theta )+f(\pi -\theta )|^2d\Omega.
$$
If the  total spin is odd, then the differential section is
$$
d\sigma _a=|f(\theta )-f(\pi -\theta )|^2d\Omega.
$$
We have assumed in these two formulas that the total spin of the particles in
the collision has a fixed value. But in general, we deal with collisions in
which the particles do not have their spins in a determined state. In order
to find the differential cross section one has then to take the mean over all
the possible states of the spin where we consider all of them
equiprobable. For an half integer $s$, the  probability for a system of
two particles of spin $s$ to have a spin $S$ even is
$s/(2s+1)$. The probability to have a spin $S$ odd is
$(s+1)/(2s+1)$. Then the differential cross section for interacting
identical particles of half integer spin $s$ is
$$
d\sigma ={s\over 2s+1}d\sigma _s+{s+1\over 2s+1}d\sigma _a.
$$
Similarly, for interacting
identical particles of integer spin $s$:
$$d\sigma ={s+1\over 2s+1}d\sigma _s+{s\over 2s+1}d\sigma _a.
$$


\subsection{Relativistic case}
The differential cross sections in the
relativistic case are calculated in a completely different way. Let us first 
mention here
that for short range interaction, one still has $s\sigma (s, \theta )\equiv
\hbox{constant}$ (cf. \cite{PS}). We conclude with the following example.


\paragraph{Photon-electron scattering}
Let $P=(p^0, p)$ and $P_{*}=(p_{*}^0, p_{*})$ be the 4-momenta of the photon
and electron before collision, and $P'=({p'}^0, p')$
and $P'_{*}=({p_{*}'}^0, p'_{*})$ their 4-momenta after the collision. Define
the center of mass coordinates:
$$s=(P+P_{*})^2,\quad t=(P-P')^2.
$$
The differential Compton cross section is given
by the Klein-Nishina formula:
$$
\sigma (s, \theta)={1\over 2}r_0^2(1-\xi )\{1+{1\over 4}{\xi ^2(1-x)^2\over 1-
{1\over 2}\xi (1-x)
}+[{1-(1-{1\over 2}\xi )(1-x)\over 1-{1\over 2}\xi (1-x)}]^2\}
$$
where $m$ is the mass of the electron,
$$
x=1+{2st\over (s-m^2)^2},\quad \xi
={((P+P_{*})^2-m^2c^2)\over (P+P_{*})^2},\quad r_0={e^2\over 4\pi mc^2}.
$$
See e.g. \cite{GLW}. If the energy of the photons is low,  the non relativistic
limit  of the Klein Nishina differential cross section is
$$
\xi \sim {2|p|mc\over m^2c^2}={2 |p|\over mc}
$$
and so it gives
$$
\sigma (s, \theta)\sim {1\over 2}r_0^2\{1+\cos^2\theta\}, \quad
\hbox{as}\quad c\to \infty.
$$
This is the Thomson formula for the efficient cross section of the diffusion
of an incident electromagnetic wave diffused by a single free charge at rest,
(in that case, $\theta $ is the angle formed by the direction of the
diffusion and the direction of the electric field
of the incident wave \cite[Vol. 2, \S 78.7]{LL}.


\paragraph{Acknowledgements}
M. Escobedo was supported by grant BFM2002-03345. M. Escobedo and S.
Mischler were supported by HPRN-CT-2002-00282, CNRS and UPV through a
PIC between the University of Pa{\'\i}s Vasco and the Ecole Normale
Sup\'erieure of Paris.
M. A. Valle was supported by the grants FPA2002-0203F and
 9/UPV00172.310-14497/2002.

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\noindent\textsc{Miguel. Escobedo }\\
Departamento de Matem\'aticas,\\
Universidad del Pa{\'\i}s Vasco\\
Apartado 644 Bilbao E 48080, Spain \\
e-mail:  mtpesmam@lg.ehu.es \smallskip

\noindent\textsc{Stephane Mischler }\\
DMA, Ecole Normale Sup\'erieure\\
Paris Cedex 05 F 75230, France\\
e-mail: Mischler@math.uvsq.fr \smallskip

\noindent\textsc{Manuel A. Valle} \\
Departamento de F{\'\i}sica Te\'orica e Historia de la Ciencia\\
Universidad del Pa{\'\i}s Vasco, \\
Apartado 644 Bilbao E48080, Spain\\
email: wtpvabam@lg.ehu.es

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