This article has been selected for the prize of the journal
''Rendiconti di Matematica e delle sue Applicazioni'' for the best PhD
Thesis of the Academic Year 2008/2009
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\begin{document}
\begin{center}
{\large\textbf{Mean-field Limit and Semiclassical Approximation for
Quantum Particle Systems }}
\end{center}
\vspace{0.1cm}
\begin{center}
F. Pezzotti\\{\footnotesize Departamento de Matem\'aticas, Universidad
del Pa\'is Vasco, Spain - \emph{e-mail:} federica.pezzotti@ehu.es}
\end{center}
\tableofcontents
\section{Introduction}
\setcounter{equation}{0}
\def\theequation{1.\arabic{equation}}
In this thesis we discuss the connection between the mean-field limit
and the semiclassical approximation for a system of $N$ identical
quantum particles. More precisely, we look at a system of $N$
identical particles (of mass $m=1$) interacting by means of the
mean-field potential:
\begin{equation}\label{eq: meanINT}
U\left(X_N\right)=\frac{1}{2N}\sum_{l\neq j}^{N}\phi(x_l-x_j),\ \ \ \
\phi:\R^3\rightarrow \R
\end{equation}
(where $X_N=\{x_1,\dots,x_N\},\ x_j\in\R^3,\ j=1,\dots, N $) in the
limit $N\to\infty$. It is well known that the effective (limiting)
dynamics of such a system is ruled by
the following nonlinear one-particle Schr$\ddot{\text{o}}$dinger equation:
\begin{equation}\label{eq: hartreeINTRO}
i\hbar\pa_t\psi_t=-\frac{\hbar^{2}}{2}\Delta\psi_t + \left(\phi \ast
\vert\psi_t\vert^{2}\right)\psi_t,
\end{equation}
where
\begin{equation}\label{eq: hartreeINT}
\left(\phi \ast \vert\psi_t\vert^{2}\right)(x)=\int_{\R^3}\ud y\
\phi(x-y)\vert \psi_t(y)\vert^2
\end{equation}
is the effective self-consistent interaction. Equation (\ref{eq:
hartreeINTRO}) is known as the Hartree equation. The rigorous
derivation of (\ref{eq: hartreeINTRO}) from the many-body evolution
can be formulated in terms of convergence of $j$-particle Reduced
Density Matrices (RDM).
In fact, by considering the $N$-particle wave function
$\Psi_{N,t}=\Psi_{N,t}\left(X_N\right)$ solution of the
Schr$\ddot{\text{o}}$dinger equation:
\begin{equation}\label{eq: mean}
i\hbar\pa_t\Psi_{N,t}=-\frac{\hbar^{2}}{2}\sum_{i=1}^{N}\Delta_{x_i}\Psi_{N,t}
+ U\Psi_{N,t},
\end{equation}
with $U$ given by (\ref{eq: meanINT}) and completely factorized
initial datum given by:
\begin{equation}\label{eq: meanINDAT}
\Psi_{N,0}(X_N)=\prod_{j=1}^{N}\psi_0(x_j),
\end{equation}
it can be proven that, for fixed $j$ (with $1\leq j\leq N$)
the $j$-particle reduced density matrix, defined as the trace class
operator with kernel
\begin{equation}\label{eq: redDENSmatrN}
\rho_{N,t}^{(j)}\left(X_j,Y_j\right)=\int_{\R^{3(N-j)}} \ud X_{N-j}
\overline{\Psi}_{N,t}\left(X_j, X_{N-j}\right)\Psi_{N,t}\left(Y_j,
X_{N-j}\right),
\end{equation}
converges, in the limit $N\to\infty$, to the factorized state:
\begin{equation}\label{eq: redDENSmatrJlim}
\rho_t^{(j)}\left(X_j,Y_j;t\right)=\prod_{k=1}^{j}\overline{\psi}_t\left(x_k\right)\psi_t\left(y_k\right),
\end{equation}
where $\psi_t(x)$ solves the one-particle
Hartree equation (\ref{eq: hartreeINTRO}) with initial datum $\psi_0$.
This feature is usually called ''propagation of chaos''.
The previous result was originally obtained for sufficiently smooth
potentials (see \cite{HEPP}, \cite{GV}, \cite{SPOHN});
then it has been
generalized to include Coulomb interactions (see \cite{BGM},
\cite{EY}, \cite{BEGMY}). Furthermore, some results concerning the
speed of convergence of the mean-field evolution to the Hartree
dynamics (for all fixed times), have been proven more recently (see
\cite{BENJIROD}, \cite{ErdBenji}).
The limit $N\to\infty$ for a classical system interacting by means of
the same mean-field interaction (\ref{eq: meanINT}), can be considered
as well (see
\cite{BH}, \cite{DOB}, \cite{Sp}, \cite{Neun} for the case of smooth
potential, and \cite{JABIN} for more singular interactions).
In fact, considering as initial state of the system a completely
factorized probability distribution
$F_{N,0}=F_{N,0}\left(X_N,V_N\right)\ud X_N \ud V_N$ in the
$N$-particle phase space $\R^{3N}\times \R^{3N}$, namely:
\begin{equation}\label{eq: classicINDAtN}
F_{N,0}\left(X_N,V_N\right)=
\prod_{j=1}^{N}f_0\left(x_j, v_j\right),\ \ \ \text{for some
one-particle density $f_0$,}
\end{equation}
it is known that its evolution $F_{N}\left(X_N,V_N;t\right)$ at time
$t>0$, is obtained by solving the Liouville equation:
\begin{equation}\label{eq: LiouvilleEQ}
\left(\pa_t +V_N\cdot \nabla_{X_N}\right)F_{N}(t)=\nabla_{X_N}U\cdot
\nabla_{V_N}F_{N}(t),
\end{equation}
with $U$ given by (\ref{eq: meanINT}).
Then, the $j$-particle marginal at time $t>0$, defined as
\begin{equation}\label{eq: redCLASS}
F_{N}^{(j)}\left(X_j,V_j;t\right)=\int_{\R^{3(N-j)}\times \R^{3(N-j)}}
\ud X_{N-j}\ud V_{N-j} F_N\left(X_j, X_{N-j},V_j, V_{N-j};t\right),
\end{equation}
converges, as $N\to\infty$, to the product state:
\begin{equation}\label{eq: prodCLASS}
F^{(j)}\left(X_j,V_j;t\right)=\prod_{k=1}^{j}f\left(x_k, v_k;t\right),
\end{equation}
where $f(x,v ;t)$ is the solution of the Vlasov equation:
\begin{equation}\label{eq: VlasovINTRO}
\left(\pa_t+v\cdot \nabla_x\right)f(t)=\left(\nabla_{x}\phi\ast f(t)
\right)\cdot\nabla_v f(t),
\end{equation}
(the convolution above is with respect to both the variables $x$ and
$v$) with initial datum $f_0$.\\
Equation (\ref{eq: Vlasov}) is the classical analogue of the Hartree
equation (\ref{eq: hartreeINTRO}).
Although the mean-field limit $N\to\infty$ is well understood for both
classical and quantum systems,
there is a question which is
still open, namely, does that limit hold for quantum systems uniformly
in $\hbar$, at least for systems having a reasonable classical analogue?
\\The proofs which are available up to now exhibit an error vanishing
when $N\to\infty$ but diverging as $\hbar\to 0$, although in
\cite{NS}, \cite{Graffi}, \cite{FGS}, \cite{FPK} some efforts in the
direction of a better control of the error term have been done.
If one wants to deal with the classical and quantum case
simultaneously, it is natural to work in the classical phase space by
using the Wigner formalism.
The one-particle Wigner function associated with the wave function
$\psi_t(x)$ is given by:
\begin{equation}\label{eq: WignerUNAINTRO}
f^{\hbar}\left(x, v;t\right)=(2\pi)^{-3}\int_{\R^{3}} \ud y\
e^{iy\cdot v}\overline{\psi}_t\left(x+\frac{\hbar
y}{2}\right)\psi_t\left(x-\frac{\hbar y}{2}\right),
\end{equation}
and, similarly, the $N$-particle Wigner function associated with the
wave function $\Psi_{N,t}(X_N)$ is defined as:
\begin{equation}\label{eq: WignerN}
W_N^{\hbar}\left(X_N, V_N;t\right)=(2\pi)^{-3N}\int_{\R^{3N}} \ud Y_N
\ e^{iY_N\cdot V_N}\overline{\Psi}_{N,t}\left(X_N+\frac{\hbar
Y_N}{2}\right)\Psi_{N,t}\left(X_N-\frac{\hbar Y_N}{2}\right).
\end{equation}
Then, by using that $\psi_t(x)$ and $\Psi_{N,t}(X_N)$ solve equations
(\ref{eq: hartreeINTRO}) and (\ref{eq: mean}) respectively, we find
the equations:
\begin{equation}\label{eq: WigLIOUhartree1}
\left(\pa_t+v\cdot \nabla_x\right)f^{\hbar}(t)=T^{\hbar}f^{\hbar}(t)
\end{equation}
and
\begin{equation}\label{eq: WigLIOUhartreeN}
\left(\pa_t+V_N\cdot
\nabla_{X_N}\right)W_N^{\hbar}(t)=T_N^{\hbar}W_N^{\hbar}(t),
\end{equation}
where $T^{\hbar}$ and $T_N^{\hbar}$ are suitable pseudodifferential operators.
The initial data for equations (\ref{eq: WigLIOUhartree1}) and
(\ref{eq: WigLIOUhartreeN}) are
\begin{equation}\label{eq: WignerUNAindat}
f_0^{\hbar}\left(x, v\right)=(2\pi)^{-3}\int_{\R^{3}} \ud y\
e^{iy\cdot v}\overline{\psi}_0\left(x+\frac{\hbar
y}{2}\right)\psi_0\left(x-\frac{\hbar y}{2}\right),
\end{equation}
and
\begin{eqnarray}\label{eq: WignerNindat}
W_{N,0}^{\hbar}\left(X_N, V_N\right)&&=(2\pi)^{-3N}\int_{\R^{3N}} \ud
Y_N \ e^{iY_N\cdot V_N}\overline{\Psi}_{N,0}\left(X_N+\frac{\hbar
Y_N}{2}\right)\Psi_{N,0}\left(X_N-\frac{\hbar Y_N}{2}\right)\nonumber\\
&& = \prod_{j=1}^{N}f_0^{\hbar}\left(x_j, v_j\right),
\end{eqnarray}
respectively.
One can easily rephrase the result of \cite{SPOHN} by showing that
the $j$-particle Wigner function
\begin{equation}\label{eq: WignerJN}
W_{N,j}^{\hbar}\left(X_j,
V_j;t\right)=\int_{\R^{3(N-j)}\times\R^{3(N-j)}} \ud X_{N-j} \ud
V_{N-j}W_N^{\hbar}\left(X_j, X_{N-j},V_j,V_{N-j};t\right)
\end{equation}
converges, in a suitable sense, to
\begin{equation}\label{eq: WignerJlim}
f_j^{\hbar}\left(X_j, V_j;t\right)=\prod_{k=1}^{j}f^{\hbar}\left(x_k,
v_k;t\right)\ \ \text{for any}\ t>0.
\end{equation}
However, the error in approximating the $N$-particle dynamics by the
limiting one is diverging when $\hbar\to 0$ (for example, for
sufficiently small times $t0.
\end{equation}
This is the main goal of the present research.
A complete proof of the uniformity in $\hbar$ of the limit
$N\to\infty$ would require a control of the remainder of the expansion
(\ref{eq: WignerJNexp}), but we are not able to provide it. However,
in proving (\ref{eq: convergence}) we characterize the quantum
corrections to the classical mean-field limit and we prove that they
are expressed in terms of classical quantities only.
The plan of the thesis is the following. \\
In Chapters 1 and 2 we discuss the mean-field model both in the
classical framework and in the quantum context. First we introduce
notation and technical tools that are needed to formulate the
mean-field results to which we referred previously. Then, we give an
outlook of the known results by discussing briefly the main approaches
in facing the problem both for smooth and singular interactions. Thus,
we focus on the case of sufficiently regular potential by showing
in detail the proof of the validity of ''propagation of chaos'' both
in the classical and in the quantum case, accenting the main
differences in the methods and, primarily, the inadequacy of the
''BBGKY hierarchy method'' in facing the classical mean-field limit
although in the quantum framework it plays a crucial role.
Furthermore, we highlight the non uniformity with respect to $\hbar$
of the error in the quantum mean-field approximation and we analyze in
detail which is the estimate that, providing a bound which diverges as
$\hbar\to 0$, is responsible for that.\\
In Chapter 3 we introduce the Wigner formalism. We accent first of all
why it is appropriate in looking at the semiclassical behavior of
quantum systems. Also, we point out the main difficulties of this
formalism with respect to the wave function (Schr\"odinger) and the
density matrix (Heisenberg) formulations introduced in Chapter 2.
Moreover, we rephrase the quantum mean-field result in the Wigner
formalism and we note that the error in the mean-field approximation
is still not uniform with respect to $\hbar$ and diverging when
$\hbar\to 0$. This ''bad'' behavior is due to the failure of the same
estimate we detected in Chapter 3, suitably rephrased in the Wigner
framework. Finally, we discuss some known results concerning the
connection between mean-field limit and semiclassical approximation.
\\
In Chapter 4 we prove our main result, namely, the convergence
(\ref{eq: convergence}). More precisely, we do the semiclassical
expansion both for the $N$-particle mean-field system (see (\ref{eq:
WignerJNexp})) and for the Hartree dynamics (\ref{eq: WignerJlimexp})
deriving explicitly what are the equations solved by the coefficients
at each order in $\hbar$. Then, we introduce the initial datum
as a suitable mixtures of coherent states. This choice guarantees that
the zeroth order coefficient of the $N$-particle expansion is a
factorized probability distribution. Finally, we identify the higher
order $N$-particle terms to be the expectation of certain derivatives
of the empirical measure. Such an expectation is with respect to the
probability distribution that we previously obtained from the
$N$-particle zeroth order coefficient.
By virtue of that, we obtain the limit $N\to\infty$ by using the
classical mean-field results presented in Chapter 1 and appropriate
properties of the derivatives of the classical trajectories associated
with the mean-field interaction.\\
In the last Chapter we present possible applications of our result
(presented in Chapter 4) in considering suitable mixtures of WKB
states (instead of coherent ones) and in dealing with other (related)
problems.\\
I wish to thank my advisor, Prof. Mario Pulvirenti, for suggesting me
a problem which I considered challenging from the very first moment,
for offering me his precious experience and considering me not only as
a student but indeed as a co-worker.
\section{Classical Mean-Field Limit}
\setcounter{equation}{0}
\def\theequation{2.\arabic{equation}}
In this chapter we analyze the mean-field limit for a many-body
classical system. More precisely, we are looking at a system
constituted by $N$ identical particles interacting by the potential
\begin{equation}\label{eq: class1}
U^{cl}(X_N)=\frac{1}{2N}\sum_{k\neq l}^{N}\phi(x_k-x_l),
\end{equation}
where we used the notation $X_N=(x_1,\dots, x_N)\in \R^{3N}$ for the
positions of the $N$ particles
(here and henceforth we put the superscripts ''$cl$'' and ''$Q$'' to
distinguish between classical and quantum quantities denoted by the
same symbol).
We note that $U^{cl}$ is given by a sum over all interactions among
pair of particles; the two-body interactions are governed by the
potential $\phi$ which we assume to be spherically symmetric (as it is
in all reasonable physical situations), namely $\phi(x)=\phi(\vert
x\vert)\ \forall\ x\in \R^3$. We set the dimension to be equal to $3$
but the results we are going to discuss hold in any dimension.
Sometimes we will refer to the system under consideration as
''mean-field system''.
We want to characterize the dynamics when the number of particles $N$
is very big. In this sense we speak about ''macroscopic'' or
''effective'' dynamics and from a mathematical point of view that
purpose is realized by taking the limit $N\to\infty$.
As a second step, it is also important to describe the dynamics of the
fluctuations of the $N$-particle evolution around the limiting one,
but here we do not discuss in detail this topic which is analyzed, for
example, in \cite{BH}. \\
\subsection{Setting of the problem: general features and known results}
\setcounter{equation}{0}
\def\theequation{2.1.\arabic{equation}}
A mean-field system is described by an $N$-body Hamiltonian of the form
\begin{equation}\label{eq: class2V}
H_N^{cl,V}(X_N,V_N)=\sum_{k=1}^{N}\(\frac{v_k^2}{2}+V^{cl}(x_k)\)+U^{cl}(X_N),
\end{equation}
where we used the notation $V_N=(v_1,\dots,v_N)\in \R^{3N}$ to
indicate the velocities of the $N$ particles and, for the sake
simplicity, the mass of the (identical) particles is chosen equal to
one. The first part of the Hamiltonian is simply the sum of the
one-body Hamiltonians associated with the motion of each particle (the
function $V^{cl}$ describes
an external potential which acts in the same way on all $N$
particles), while the remaining term involving $U^{cl}$ describes the
interaction among the particles. For the sake of simplicity we assume
that the force experienced by each particle is only that arising from
the many-body interaction, namely, the one-particle potential $V^{cl}$
is assumed to be equal to zero. We can do that without loss of
generality because the results we are going to discuss can be
generalized easily to the case $V^{cl}\neq 0$. \\Thus the Hamiltonian
we consider is
\begin{equation}\label{eq: class2}
H_N^{cl}(X_N,V_N)=\sum_{k=1}^{N}\frac{v_k^2}{2}+U^{cl}(X_N),
\end{equation}
and we note that $H_N^{cl}$ is symmetric with respect to any
permutation of the labeling.
The factor $1/N$ in the potential $U^{cl}$ (see (\ref{eq: class1}))
forces the energy per particle to remain finite in the limit
$N\to\infty$ and this is the crucial feature in order to obtain a
well-defined but non-trivial limiting dynamics. Moreover we observe
that $U^{cl}$ is such that the interaction among the particles is
quite weak when $N$ is very big (the strength of the pair interaction
is of the order $1/N$)
but it is long range (because the pair interaction potential $\phi$ is
unscaled, thus its support remains of order one in the limit
$N\to\infty$). Therefore when $N$ becomes large (for example, in the
applications related to the gas dynamics we have $N\approx 10^{23}$)
the mutual interaction turns to be weaker and weaker but the total
effect of such an interaction is not negligible (the force experienced
by a fixed particle because of the presence of all the others is
proportional to $(N-1)/N\approx O(1)$) . We will see that these two
features are responsible for the validity of ''propagation of chaos''
and of the nonlinearity of the macroscopic equation we find in the
limit (see Sections 1.3 and 1.4).\\
The dynamics of an $N$-particle system associated with the Hamiltonian
(\ref{eq: class2}) is governed by the Newton equations
\begin{equation}\label{eq: class3}
\left\{
\begin{aligned}
& \dot{x}_i = v_i,\\
& \dot{v}_i= - \frac{1}{N}\sum_{k\neq i}^N\nabla_{x_i}
\phi(x_i-x_k),\ \ i=1,\dots,N
\end{aligned}
\right.
\end{equation}
Thus
we know that given an initial configuration $Z_N:=(X_N,V_N)\in
\R^{3N}\times \R^{3N}$ in the $N$-particle phase-space, the
time-evolved configuration $Z_N(t)$ up to some $t>0$ is obtained
solving (\ref{eq: class3}) with initial datum $Z_N$. \\
As we have already noticed, in many interesting physical situations
the number $N$ is very big thus it is not possible (and even not
particularly relevant for the applications) to know which are the
positions and the velocities of all particles at a certain time. In
other words, it is quite difficult to determine a unique initial
$N$-particle configuration for the time-evolution defined by (\ref{eq:
class3}) and, even if one was able to provide that, it would be
impossible to solve such a huge number of equations, even by using
numerical methods.
Nevertheless, one can provide collective and more useful informations
such as the probability to find $N_1$ particles ($N_1\leq N$) in a
region $\Lambda_1\subset \R^{3N}$, the probability that $N_2$
particles ($N_2\leq N$) have velocities belonging to some
$\Lambda_2\subset \R^{3N}$ or the probability to find $N_3$ particles
($N_3\leq N$) with positions belonging to some $\Lambda_3^x\subset
\R^{3N}$ and velocities belonging to some $\Lambda_3^v\subset
\R^{3N}$. In other words, one can give the $N$-particle probability
distribution in the phase-space $\R^{3N}\times \R^{3N}$.
Therefore, denoting by $F_{N,0}(Z_N)\ud Z_N$ such a distribution, we
have that $F_{N,0}$ is symmetric with respect to any permutation of
the variables, $F_{N,0}\geq 0$ and:
\begin{equation}\label{eq: class3bis}
\int \ud Z_N F_{N,0}(Z_N)=1.
\end{equation}
Moreover, by computing the marginals of $F_{N,0}(Z_N)$ with respect to
the velocities $V_N$ and to the positions $X_N$ one obtains
respectively the spatial and the velocity probability density.\\
The time-evolved probability density $F_N(t):=F_{N}(Z_N;t)$ is
obtained by solving the Liouville equation
\begin{equation}\label{eq: class4}
\(\pa_t +V_N\cdot\nabla_{X_N}\)F_{N}(t)=\nabla_{X_N}U\cdot\nabla_{V_N}F_N(t),
\end{equation}
with initial condition $F_{N,0}$, where $U$ is the potential defined
in (\ref{eq: class1}). By denoting as $\Phi^t\(X_N, V_N\)$ the
Hamiltonian flow associated with equations (\ref{eq: class3}), it is
easy to verify that the solution of equation (\ref{eq: class4}) is
obtained by propagating the initial datum $F_{N,0}$ through the
characteristic curves of $\Phi^t\(X_N, V_N\)$, namely
\begin{equation}\label{eq: class5}
F_N(t)=F_{N,0}\left(\Phi^{-t}\left(X_N,V_N\right)\right).
\end{equation}
Thus we are guaranteed that starting from an $N$-particle probability
density at time $t=0$, we have a probability density for each time
$t>0$ and the evolution preserves also the symmetry with respect to
permutations of the variables (because the Hamiltonian $H_N^{cl}$ is
symmetric with respect to permutations).\\
In the classical framework observables of the $N$-particle system are
represented by real functions defined on the phase-space
$\R^{3N}\times \R^{3N}$. Then, if we know that the configuration of
the system at a certain time $\tau$ is $Z_N(\tau)$, the value of the
observable associated with a certain function $u_N$ at time $\tau$, is
given by $u_N(Z_N(\tau))$. On the other side, if what we have is the
$N$-particle probability distribution at a certain time $\tau_1$,
namely $F_N(\tau_1)$, we are able to give probabilistic predictions
about the value of the observables at time $\tau_1$. More precisely,
the expectation of the observable associated with a certain function
$u_N$ at time $\tau_1$, is given by
\begin{equation}\label{eq: class3tris}
\left\langle u_N\right\rangle_{F_{N}(\tau_1)}:=\int \ud Z_N
u_N(Z_N)F_{N}(Z_N;\tau_1).
\end{equation}
By (\ref{eq: class3}) it is clear that to guarantee existence and
uniqueness of the flow $\Phi^t\(X_N, V_N\)$ for each $t$ we need to
assume $\phi\in \mathcal{C}_b^2\(\R^3\)$\footnote{Here and henceforth
we denote by $\mathcal{C}_b^k\(\R^d\)$ the space of functions on
$\R^d$ with continuous and uniformly bounded derivatives up to the
order $k$}. Therefore the first rigorous results concerning the
analysis of the limit $N\to\infty$ for the $N$-particles mean-field
system have been proven under suitable smoothness assumption on the
pair interaction potential $\phi$ (see for example \cite{BH} and
\cite{Neun}).
Nevertheless, several systems of physical interest are described by
more singular potential.
For example, a system of gravitating particles can be described by the
potential (\ref{eq: class1}) where $\phi$ is the Coulomb interaction
among the particles and,
in that case, the factor of $1/N$ in front of the potential energy can
be justified
by the smallness of the gravitational
constant. \\Mean-field systems with singular interactions are clearly
hard to face because one has to deal with a system of ODE (namely,
(\ref{eq: class3})) with non regular fields. Quite recently some
progress have been done in \cite{JABIN} where the mean-field limit is
realized
by only assuming $\nabla_x \phi\approx 1/\vert x\vert^\alpha,\ \
\alpha<1$ for the pair interaction $\phi$. On the other side, the
assumptions on the initial datum are very strong and they are quite
good for numerical purposes but not satisfying from a statistical
physics point of view (for example, ''chaotic'' initial data are not
admissible, namely it is not possible to consider initially factorized
$N$-particle distribution). The problem involving the Coulomb
potential is still open. \\
Here we will not discuss the ''singular case'' because for our
purposes we need to deal with a smooth interaction potential and with
a classical mean-field result involving ''chaotic'' initial data (see
Chapter 3 and 4), thus from now on we will focus on the mean-field
limit in the ''smooth case''.\\
In \cite{BH} and \cite{Neun} it is proved that the effective single
particle dynamics of a mean-field system with smooth interaction
potential ($\phi\in\mathcal{C}_b^2\(\R^3\)\ $) in the limit
$N\to\infty$ is governed by the Vlasov equation:
\begin{equation}\label{eq: Vlasov}
\(\pa_t +v\cdot \nabla_x\)f(t)=\(\nabla_{x}\phi\ast f(t)\)\cdot\nabla_v f(t),
\end{equation}
where $f(t)=f(x,v;t)$ for each time $t\geq0$ is a one-particle
probability density and here and in the rest of the chapter we denote
by $\ast$ the convolution with respect to both position and velocity.
By computing the marginals of $f(x,v;t)$ with respect to the velocity
$v$ and the position $x$ one finds respectively the spatial and the
velocity probability density.
It is remarkable that the results proven in \cite{BH} describe both
the continuum limit of the point particle dynamics associated with the
mean-field interaction, as we specified previously, and the so called
''propagation of chaos'' for the many-body mean-field system.
Moreover, in \cite{BH} it has been proven that the fluctuations of a
certain class of observables (called ''intensive observables'')
converge to a gaussian stochastic process. We will not discuss this
last feature, on the contrary we will show in detail the emergence of
the Vlasov dynamics as the limit of the $N$-particle evolution and the
proof of propagation of chaos.
\subsection{The Vlasov equation}
\setcounter{equation}{0}
\def\theequation{2.2.\arabic{equation}}
Let us consider a one-particle density $f_0\in C^1(\R^6)$ and let us
look at the solution $f(t)$ of the Vlasov equation (\ref{eq: Vlasov})
with initial datum $f_0$.
Denoting by $\Phi^t_V(x,v)$ the flow associated with the system:
\begin{equation}\label{eq: NewtonVlasov}
\left\{
\begin{aligned}
& \dot{x} = v,\\
& \dot{v}= - \nabla_{x} \phi\ast f(t),
\end{aligned}
\right.
\end{equation}
one can easily verify that $f(t)$ is obtained by propagating
$f_0(x,v)$ through the characteristic curves of the flow
$\Phi^t_V(x,v)$, namely
\begin{equation}\label{eq: class6}
f(t)=f(x,v;t)=f_0\left(\Phi^{-t}_V\left(x,v\right)\right).
\end{equation}
Therefore in proving existence and uniqueness of the solution of
(\ref{eq: Vlasov}) one has to deal with a system of ODE with a
self-consistent field (see (\ref{eq: NewtonVlasov})) and the
smoothness of the potential $\phi$ is not sufficient to make a
standard fixed point argument to be successful. One needs a more
involved analysis and it has been done
by R.L. Dobrushin in \cite{DOB}. It is remarkable that the Vlasov
equation (\ref{eq: Vlasov}) makes sense even for a generic probability
measure $\nu$ because $\nabla\phi \ \ast\ \nu$ is sufficiently smooth
(thanks to the regularity of $\phi$) then the proof presented in
\cite{DOB} ensures existence and uniqueness of the solution in this
framework. In particular, if the initial datum is an absolutely
continuous measure with respect to the Lebesgue measure in
$\R^3\times\R^3$ with a smooth density $f_0$
(which is the case we discussed previously), the solution $f(t)$ is a
strong solution whose regularity depends on that of $f_0$ and $\phi$ (
$f_0\in C^1\(\R^6\)$ and $\phi\in C^2_b\(\R^3\)$, at least).
Furthermore, introducing the Wasserstein distance $\mathcal{W}$,
in \cite{DOB} it has been proven the following stability result for
solutions of the Vlasov equation:
\begin{eqnarray}\label{eq: DOBineq}
&&\mathcal{W}(\nu^t_1,\nu^t_2)\leq e^{Ct}\mathcal{W}(\nu^0_1,\nu^0_2)
\end{eqnarray}
where $\nu^0_1$ and $\nu^0_2$ are two probability measures and
$\nu^t_1$ and $\nu^t_2$ are the weak solutions of the Vlasov equation
with initial data given by $\nu^0_1$ and $\nu^0_2$ respectively. The
metric induced by $\mathcal{W}$ on the space of probability measures
on $\R^3\times\R^3$ is equivalent to the weak topology of probability
measures, namely we have to look at measures tested versus functions
in $\mathcal{C}_b^{0}(\R^3\times\R^3)$ (the space of continuous and
uniformly bounded functions). Thus by (\ref{eq: DOBineq}) it follows
that
\begin{eqnarray}\label{eq: class7}
&&\int u(x,v)\nu^0_n(\ud x\ud v)\to\int u(x,v)\nu^0(\ud x\ud v)\ \
\text{as}\ n\to\infty,\ \forall\ u\in \mathcal{C}_b^{0}(\R^3\times\R^3)
\end{eqnarray}
implies
\begin{eqnarray}\label{eq: class8}
&&\int u(x,v)\nu^t_n(\ud x\ud v)\to\int u(x,v)\nu^t(\ud x\ud v)\ \
\text{as}\ n\to\infty,\ \forall\ u\in \mathcal{C}_b^{0}(\R^3\times\R^3),
\end{eqnarray}
where $\{\nu^0_n\}_{n\geq 0}$ is a sequence of probability measures
converging to some $\nu^0$ when the parameter $n$ goes to infinity and
$\nu^t_n$ and $\nu^t$ are the weak solutions of the Vlasov equation
with initial data given by $\nu^0_n$ and $\nu^0$ respectively. In the
sequel we will denote the weak convergence of probability measures by
the symbol $\stackrel{M}{\rightarrow}$.
\subsection{The Vlasov dynamics as the continuum limit of the
$N$-particle Mean-Field dynamics}
\setcounter{equation}{0}
\def\theequation{2.3.\arabic{equation}}
Let us introduce the empirical measure associated with an $N$-particle
configuration $Z'_N$
\begin{eqnarray}\label{eq: class9}
&&\mu_N(z\vert Z'_N)=\frac{1}{N}\sum_{i=1}^{N}\delta\left(z-z'_i\right),
\end{eqnarray}
where $z:=(x,v)$ is the generic point in the one-particle phase-space
$\R^3\times\R^3$ and $Z'_N=(z'_1,\dots, z'_N)\in \R^{3N}\times\R^{3N}$.
By definition $\mu_N$ is a measure on the one-particle phase-space
but, as it is clear by (\ref{eq: class9}), it depends on all the
configuration $Z'_N$. Then let us consider an initial $N$-particle
configuration $Z_N$ for equations (\ref{eq: class3}) distributed
according to a factorized (smooth) $N$-particle measure $F_{N,0} \ud
Z_N$, namely
\begin{eqnarray}\label{eq: class10}
&&F_{N,0}(Z_N)=\prod_{i=1}^N f_0(z_i)=f_0^{\otimes N},\ \ \ \ \ \
f_0\in C^1(\R^6).
\end{eqnarray}
We denote by $\mu_N^0$ the empirical measure associated with $Z_N$ and
by considering the empirical measure $\mu_N(t)=\mu_N(z\vert Z_N(t))$
associated with the time-evolved configuration $Z_N(t)$ (solution of
equations (\ref{eq: class3}) with initial datum $Z_N$), it is easy to
verify that $\mu_N(t)$ is the unique (weak) solution of the Vlasov
equation (\ref{eq: Vlasov}) with initial datum $\mu_N^0$. In fact, by
integrating versus $\mu_N(t)$ versus a smooth test function
$u=u(x,v)$, we find:
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAK}
&&\(u,\mu_N(t)\)=\int\ud z\ \mu_N(z\vert
Z_N(t))=\frac{1}{N}\sum_{i=1}^N u(z_i(t))=\frac{1}{N}\sum_{i=1}^N
u(x_i(t),v_i(t)).
\end{eqnarray}
Then, we obtain
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKI}
\frac{d}{dt}\(u,\mu_N(t)\)&&=\frac{1}{N}\sum_{i=1}^N
\frac{d}{dt}u(x_i(t),v_i(t))=\nonumber\\
&&=\frac{1}{N}\sum_{i=1}^N \[\nabla_x u(x_i(t),v_i(t))\dot{x}_i(t)+
\nabla_v u(x_i(t),v_i(t))\dot{v}_i(t)\],\nonumber\\
&&
\end{eqnarray}
that, by virtue of (\ref{eq: class3}), implies
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKII}
\frac{d}{dt}\(u,\mu_N(t)\)=\frac{1}{N}\sum_{i=1}^N \nabla_x
u(x_i(t),v_i(t))v_i(t)- \frac{1}{N}\sum_{i=1}^N \nabla_v
u(x_i(t),v_i(t))\(\frac{1}{N}\sum_{k\neq i}^{N}\nabla_{x_i}\phi(x_i
-x_k)\).\nonumber\\
&&
\end{eqnarray}
Following (\ref{eq: muSOLVESvlasovWEAK}), the equation (\ref{eq:
muSOLVESvlasovWEAKII}) becomes
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKIV}
\frac{d}{dt}\(u,\mu_N(t)\)=\(v\cdot\nabla_x u,\mu_N(t)\)-
\(\(\nabla\phi\ast \mu_N(t)\) \cdot\nabla_v u,\mu_N(t)\),
\end{eqnarray}
where
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKV}
&&\(\nabla\phi\ast \mu_N(t)\)(x)=\int \ud y \ud
w\nabla_x\phi(x-y)\(\frac{1}{N}\sum_{k=1}^N
\delta(y-x_k(t))\delta(w-v_k(t))\)=\nonumber\\
&&=\frac{1}{N}\sum_{k=1}^N \int \ud y \nabla_x\phi(x-y)
\delta(y-x_k(t))=\frac{1}{N}\sum_{k=1}^N\nabla_{x}\phi(x-x_k(t)).
\end{eqnarray}
Therefore, $\mu_N(t)$ verifies (\ref{eq: muSOLVESvlasovWEAKIV}) for
any function $u$ sufficiently smooth and
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKVI}
&&\(u,\mu_N(t)\)\vert_{t=0}=\(u,\mu_N^0\)=\frac{1}{N}\sum_{i=1}^N u(x_i, v_i).
\end{eqnarray}
In other words, $\mu_N(t)$ satisfies the following weak equation
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKVII}
\pa_t \mu_N(t) + v\cdot\nabla_x\mu_N(t)= \(\nabla_{x}\phi\ast
\mu_N(t)\) \cdot\nabla_v \mu_N(t),
\end{eqnarray}
with initial datum $\mu_N^0$, and we note that (\ref{eq:
muSOLVESvlasovWEAKVII}) is precisely the Vlasov equation (\ref{eq:
Vlasov}).\\
By the Strong Law of Large Numbers (SLLN) we know that
\begin{eqnarray}\label{eq: class11}
&&\mu^0_N\stackrel{M}{\rightarrow}f_0\ \ \text{as}\ N\to\infty,\
\text{a.e with respect to the product measure }\ f_0^{\otimes \infty},
\end{eqnarray}
therefore, by (\ref{eq: class11}), by knowing that $\mu_N(t)$ solves
the Vlasov (weak) equation (\ref{eq: muSOLVESvlasovWEAKVII})
and by (\ref{eq: DOBineq}), it follows that
\begin{eqnarray}\label{eq: class12}
&&\mu_N(t)\stackrel{M}{\rightarrow}f(t)\ \ \text{as}\ N\to\infty,\
\text{a.e with respect to the product measure }\ f_0^{\otimes \infty},
\end{eqnarray}
where $f(t)$ is the (strong) solution of the Vlasov equation (\ref{eq:
Vlasov}) with initial datum $f_0$. \\
From now on, we will say that a configuration $Z_N$ is ''typical''
with respect to the measure $f_0$ if the empirical measure $\mu_N^0$
associated with $Z_N$ verifies
\begin{eqnarray}\label{eq: class11NUOVA}
&&\mu^0_N\stackrel{M}{\rightarrow}f_0\ \ \text{as}\ N\to\infty.
\end{eqnarray}
\subsection{Hierarchies and Propagation of Chaos}
\setcounter{equation}{0}
\def\theequation{2.4.\arabic{equation}}
In the previous paragraph we proved that the Vlasov equation arises
from the continuum limit of a system of $N$ particles interacting by
the mean-field potential (\ref{eq: class1}). This is precisely what
convergence (\ref{eq: class12}) tells us and it can be seen as a
one-particle effect, namely (\ref{eq: class12}) provides the equation
governing the single-particle dynamics in the limit.\\
Now we want to show how (\ref{eq: class12}) works in order to
characterize the effective dynamics of a subsystem
made by a fixed number $j$ of particles.
This is a natural approach in looking at the macroscopic behavior of
many-body systems because we want to look at the limit $N\to\infty$
and we need to deal with quantities depending on a number of variables
which remains finite in the limit. \\
In this perspective, for any $j=1,\dots,N$ we introduce the
''$j$-particle marginal density'' (or simply ''$j$-particle marginal''
) associated with an $N$-particle density $F_N(X_N,V_N)$ as
\begin{eqnarray}\label{eq: class13}
&&F_N^{(j)}(X_j,V_j)=\int_{\R^{3(N-j)}\times\R^{3(N-j)}}\ud X_{N-j}\ud
V_{N-j}F_N(X_N,V_N),
\end{eqnarray}
where we used the notation $X_{j}=(x_{1},\dots, x_j),
V_{j}=(v_{1},\dots, v_j) \in \R^{3j}$ and $X_{N-j}=(x_{j+1},\dots,
x_N),$ $V_{N-j}=(v_{j+1},\dots, v_N)\in \R^{3(N-j)}$. Indeed, the
marginal $F_N^{(j)}$ is obtained by integrating $F_N$ with respect to
the ''last'' $N-j$ variables thus it is a $j$-particle probability
density (we remind that all quantities under consideration are
symmetric with respect to permutations of the variables then, without
loss of generality, in order to refer to any subsystem made by $N-j$
particles we can consider the last $N-j$). Clearly if $j=N$ we have
$F_N^{(N)}=F_N$.
For fixed $j < N$, the $j$-particle
marginal does not contain the full information about the $N$-particle
configuration described by $F_N$. Knowledge
of the $j$-particle marginal $F_N^{(j)}$, however, is sufficient to
compute the expected value of every $j$-particle
observable in the configuration described by the probability
distribution $F_N\ud Z_N$. In fact, if $u_j$ denotes an arbitrary
continuous and uniformly bounded function on $\R^{3j}$, and if
$u_j\otimes 1_{N-j}$ denotes the function on $\R^{3N}$ which is
associated with the $N$-particle observable corresponding to $u_j$ for
the first $j$ particles and to $1_{N-j}$ for the last $(N-j)$ particles,
we have
\begin{eqnarray}\label{eq: class24}
&&\left\langle u_j\otimes 1_{N-j}\right\rangle_{F_N}=\int \ud Z_N
u_j(Z_j)F_N(Z_N)=\int \ud Z_j u_j(Z_j)F_N^{(j)}(Z_j)=\left\langle
u_j\right\rangle_{F_N^{(j)}},\nonumber\\
&&
\end{eqnarray}
where we denoted by $\left\langle u_j\otimes
1_{N-j}\right\rangle_{F_N}$ the expected value of the $N$-particle
observable corresponding to $ u_j\otimes 1_{N-j}$ with respect to the
distribution $F_N\ud Z_N$ and with $\left\langle
u_j\right\rangle_{F_N^{(j)}}$ the expected value of the $j$-particle
observable corresponding to $u_j$ with respect to $F_N^{(j)}\ud Z_j$.
Thus, $F_N^{(j)}$ is sufficient to compute the expectation of
arbitrary observables which depend non-trivially
on at most $j$ particles (because of the permutation symmetry, it is
not important on which particles
it acts, just that it acts at most on $j$ particles).\\
We are interested in characterizing the time-evolution of the
marginals $F_{N}^{(j)}(t):=F_{N}^{(j)}(Z_j;t)$ associated with the
solution $F_N(t)$ of the Liouville equation (\ref{eq: class4}).
By integrating the Liouville equation versus the variables
$Z_{N-j}=(X_{N-j},V_{N-j})$ we find
the following family of equations (one for each $j=1,\dots,N$)
\begin{eqnarray}\label{eq: class14}
&&\(\pa_t+V_j\cdot
\nabla_{X_j}\)F_{N}^{(j)}(t)=T_{N,j}^{cl}F_{N}^{(j)}(t)+\frac{N-j}{N}C_{j,j+1}^{cl}F_{N}^{(j+1)}(t),
\end{eqnarray}
where $T_{N,j}^{cl}$ is precisely the $j$-particle Liouville operator,
namely $T_{N,j}^{cl}=\nabla_{X_j}U^{cl}(X_j)\cdot \nabla_{V_j}$, while
the operator $C_{j,j+1}^{cl}$ maps $j+1$-particle densities in
$j$-particle ones (if $j=N$ we find $C_{N,N+1}^{cl}\equiv 0$).
The family of equations (\ref{eq: class14}) is known as BBGKY
hierarchy (in honor of the authors who independently derived it: Born,
Bogoliubov, Green, Kirkwood, Yvon)
and it is called ''hierarchy'' because we can see that the equation
for the $j$-particle marginal is linked to the subsequent one by the
term $C_{j,j+1}^{cl}F_{N}^{(j+1)}(t)$. The physical meaning is clear:
the variation in time of $F_{N}^{(j+1)}(t)$ is due to the free motion
of the $j$ particles, which is encoded in the free-transport term
$V_j\cdot \nabla_{X_j}F_{N}^{(j)}(t)$, to the interaction among
themselves, which is modeled by the term $T_{N,j}^{cl}F_{N}^{(j)}(t)$,
and to the interaction among the $j$-particle subsystem and the
remaining $N-j$ particles, which is encoded in the term
$\frac{N-j}{N}C_{j,j+1}^{cl}F_{N}^{(j+1)}(t)$ (the factor $1/N$ is
precisely the factor appearing in the potential $U^{cl}$ (see
(\ref{eq: class1})) while the interaction with the last $N-j$
particles can be modeled by $N-j$ times the interaction with the
$j+1$-th because of the symmetry with respect to permutations of the
labeling (which follows from the fact that we are dealing with $N$
identical particles).
Writing explicitly the action of the operators $T_{N,j}^{cl}$ and
$C_{j,j+1}^{cl}$, we find:
\begin{eqnarray}\label{eq: class15}
&&\left(T_{N,j}^{cl}F_{N}^{(j)}\right)(X_j, V_j)=\frac{1}{N}\sum_{
k\neq l}^{ j}\nabla_{x_k}\phi(x_k-x_l)\cdot
\nabla_{v_k}F_{N}^{(j)}(X_j,V_{j}),
\end{eqnarray}
and
\begin{eqnarray}\label{eq: class16}
&&\left(C_{j,j+1}^{cl}F_{N}^{(j+1)}\right)(X_j, V_j)=\nonumber\\
&&=\sum_{k=1}^{j}\int_{\R^3\times\R^3}\ud x_{j+1}\ud
v_{j+1}\nabla_{x_k}\phi(x_k-x_{j+1})\cdot
\nabla_{v_k}F_{N}^{(j+1)}(X_j,x_{j+1},V_j,v_{j+1}).
\end{eqnarray}
By these expressions we can argue that the operator $T_{N,j}^{cl}$
gives a vanishing contribution in the limit because it is of size
$j^2/N$, while the operator $C_{j,j+1}^{cl}$ is of order one in the
limit and the factor $(N-j)/N$ appearing in (\ref{eq: class14}) is
also of order one. Therefore denoting by $F^{(j)}(t)$ the expected
limit of $F_N^{(j)}(t)$ when $N\to\infty$, the formal limit of the
BBGKY hierarchy (\ref{eq: class14}) is
\begin{eqnarray}\label{eq: class17}
&&\(\pa_t+V_j\cdot \nabla_{X_j}\)F^{(j)}(t)=C_{j,j+1}^{cl}F^{(j+1)}(t),
\end{eqnarray}
which in the case $j=1$ is equal to:
\begin{eqnarray}\label{eq: class18}
&&\(\pa_t+v_1\cdot \nabla_{x_1}\)F^{(1)}(t)=\int\ud x_{2}\ud
v_{2}\nabla_{x_1}\phi(x_1-x_{2})\cdot
\nabla_{v_1}F^{(2)}(x_1,x_{2},v_1,v_{2};t).
\end{eqnarray}
We observe that the Vlasov equation (\ref{eq: Vlasov}) can be rewritten as
\begin{eqnarray}\label{eq: class19}
&&\(\pa_t+v\cdot \nabla_{x}\)f^t=\int\ud x_{2}\ud
v_{2}\nabla_{x}\phi(x-x_{2})\cdot \nabla_{v}f^t(x,v)f^t(x_{2},v_{2}).
\end{eqnarray}
Replacing $(x,v)$ by $(x_1,v_1)$, $f^t$ by $F^{(1)}(t)$ and the
product $f^t\ f^t$ by $F^{(2)}$ we realize that (\ref{eq: class19}) is
precisely the same of (\ref{eq: class18}). Thus the equation of the
hierarchy (\ref{eq: class17}) corresponding to $j=1$ is properly the
Vlasov equation, provided that the the two-particle distribution
$F^{(2)}(t)$ is factorized, and for this reason (\ref{eq: class17})
(which is an infinite hierarchy because $j$ can be equal to any
positive number
) is usually called ''Vlasov hierarchy''. More precisely, by
considering (\ref{eq: class17})
and by assuming the marginals $\{F^{(j)}(t)\}_{j\geq 1}$ to be
factorized, namely
\begin{eqnarray}\label{eq: class20}
&&F^{(j)}(t)=f(t)^{\otimes j}\ \ \forall\ j,
\end{eqnarray}
it is easy to verify that $f(t)$ has to solve the Vlasov equation.
Conversely, if we consider a time dependent one-particle density
$f(t)$ solving the Vlasov equation and we take the $j$-particle
densities $F^{(j)}(t)=f(t)^{\otimes j}$, for $j=1,2\dots$, we find
that the sequence $\{F^{(j)}(t)\}_{j\geq 1}$ solves the hierarchy
(\ref{eq: class17}). \\
An interesting problem is that of the uniqueness of the solution of
the Vlasov hierarchy which plays an important role in facing the mean
field limit when a generic (namely, non factorized) initial datum is
considered for the many-body dynamics (such a case is also studied in
\cite{BH}). This topic has been discussed in \cite{NS}, under strong
smoothness assumptions on the interaction potential, and in \cite{Sp}
by assuming $\phi\in \mathcal{C}_b^2(\R^3)$.
Here we will not enter into details because we are going to show that
in the present context there is no need to prove the uniqueness of the
solution of the Vlasov hierarchy in order to establish the validity of
propagation of chaos. (In the next chapter we will see that the
situation in the quantum case can be very different).\\
First of all let us explain what we mean by ''propagation of chaos''.
As we have already specified, we consider as initial datum for the
Liouville equation (\ref{eq: class4}) the factorized $N$-particle
probability density (\ref{eq: class10}). This choice means that we are
assuming that the particles are identically and independently
distributed at time $t=0$, or equivalently, the particles are
initially uncorrelated. This is quite reasonable from the physical
point of view and this is what is usually called ''hypotheses of
molecular chaos''.
Because of the interaction among the particles, the factorization
(\ref{eq: class10}) is not preserved by the
time evolution because some correlations are introduced by the
dynamics; in other words, the evolved $N$-particle density $F_N(t)$ is
not given by the product of one-particle
densities, if $t\neq 0$. However, due to the mean-field character of
the interaction each
particle interacts very weakly (we remind that the strength of the
interaction is of the order $1/N$) with all other
$(N - 1)$ particles.
For this reason, we may expect that, in
the limit of large $N$, the total interaction force experienced by a
typical particle in the system
can be effectively replaced by an averaged, mean-field, force, and
therefore that factorization is
approximately, and in an appropriate sense, preserved by the time
evolution. In other words, we
may expect that, in a sense to be made precise,
\begin{eqnarray}\label{eq: class21}
&&F_N(t)\approx f(t)^{\otimes N} \ \ \text{as}\ N\to\infty
\end{eqnarray}
for an evolved one-particle density $f(t)=f(x,v;t)$. This asymptotic
factorization is precisely what is called ''propagation of chaos''
Assuming (\ref{eq: class21}), it is simple to derive a self-consistent
equation
for the time-evolution of the one-particle density $f(t)$. In fact,
(\ref{eq: class21}) states that, for every fixed time $t$,
the $N$ particles are independently distributed in space according to
the density $\rho(x;t)=\int \ud v f(x,v;t)$. If this is true, the total
force experienced, for example, by the first particle can be approximated by
\begin{eqnarray}\label{eq: class22}
&&\frac{1}{N}\sum_{k\geq 2}
\nabla_{x_1}
\phi(x_1 - x_k)\approx\frac{1}{N}\sum_{k\geq 2}\int \ud y\nabla_{x_1}
\phi(x_1 - y)\rho(y;t)=\nonumber\\
&&=\frac{N-1}{N}\int \ud y\ud w\nabla_{x_1} \phi(x_1 -
y)f(y,w;t)=\frac{N-1}{N}\(\nabla_{x_1}\phi\ast
f(t)\)\approx\(\nabla_{x_1}\phi\ast f(t)\),\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. It follows that, if (\ref{eq: class21}) holds true,
the one-particle density $f(t)$ must satisfy the self-consistent
equation
\begin{eqnarray}\label{eq: class23}
&&\(\pa_t +v\cdot \nabla_x\)f(t)=\(\nabla_{x}\phi\ast f(t)\)\cdot
\nabla_v f(t)
\end{eqnarray}
with initial data $f(t)\vert_{t=0}=f_0$ given by (\ref{eq: class10}).
Equation (\ref{eq: class23}) is precisely the Vlasov equation
and we have just presented an heuristic argument to explain
how it is related to the propagation of chaos. We observe that the
Vlasov equation is a nonlinear Liouville equation on $\R^3\times
\R^3$. Therefore starting from the linear Liouville equation
(\ref{eq: class4}) on $\R^{3N}\times \R^{3N}$, we obtain, for the
evolution of factorized densities, a nonlinear Liouville equation on
$\R^3\times \R^3$; the nonlinearity in the Vlasov equation is a
consequence of the many-body effects
in the linear dynamics. \\
The validity of propagation of chaos (namely, the precise statement
concerning the asymptotic factorization (\ref{eq: class21})) is
expressed in terms of convergence of the $j$-particle marginal
densities associated with the solution of the Liouville equation
(\ref{eq: class4}) to the $j$-fold product of solutions of the Vlasov
equation when $N\to\infty$. We are going to show that it is a
straightforward consequence of the convergence (\ref{eq: class12})
(e.g. \cite{BH}). \\
Let us consider the $j$-particle marginal $F_N^{(j)}(t)$ associated
with the solution $F_N(t)$ of the Liouville equation with factorized
initial datum $F_{N,0}$ given by (\ref{eq: class10}).
We want to look at the behavior of $F_N^{(j)}(t)$ when $N\to\infty$.
Denoting by $\mathbb{E}_N$ the expectation with respect to the initial
$N$-particle distribution $F_{N,0}(Z_N)$, after straightforward
computations, we obtain:
\begin{eqnarray}\label{eq: EprodEMP}
\mathbb{E}_N\left[\mu_N\(z'_1\vert Z_N(t)\)\dots\mu_N\(z'_j\vert
Z_N(t)\)\right]&&=\frac{N(N-1)\dots(N-j+1)}{N^j}\
F_N^{(j)}(Z'_j;t)+\nonumber\\
&&+\ O\left(\frac{1}{N}\right),\nonumber\\
&&
\end{eqnarray}
where
$F_N^{(j)}(Z'_j;t)=F_N^{(j)}(\Phi^{-t}(Z'_j)=F_N^{(j)}(Z'_j(-t))$ (see
(\ref{eq: class5})).
Consider now a typical sequence $Z_N$ with respect to $f_0$, namely
such that (\ref{eq: class11NUOVA}) holds. By the Strong Law of Large
Numbers (\ref{eq: class11}) we know that (\ref{eq: class11NUOVA})
holds a.e. with respect to the product measure $f_0^{\otimes \infty}$.
Then,
by (\ref{eq: class12}) and (\ref{eq: EprodEMP}) we have:
\begin{eqnarray}\label{eq: prodEMP}
&&\lim_{N\to\infty}\mathbb{E}_N\left[\mu_N\(z'_1\vert Z_N(t)\)\dots
\mu_N\(z'_j\vert Z_N(t)\)\right]=\lim_{N\to\infty}
F_N^{(j)}(Z'_j;t)=\prod_{k=1}^{j}f(z'_k;t),\nonumber\\
&&
\end{eqnarray}
in the weak topology of probability measures, where $f(z'_k;t)=f(t)$
solves the Vlasov equation with initial datum $f_0$. Thus propagation
of chaos holds. In the end, we found that starting from an initial
uncorrelated state (\ref{eq: class10}) for the $N$-particle system,
for times $t>0$ we loose the factorization, but it is recovered in the
limit because the correlations created by the dynamics are smaller and
smaller when $N\to\infty$. On the other side, the effect of the
many-body interaction is ''translated'' into the self-consistent force
appearing in the Vlasov equation.\\
The convergence (\ref{eq: prodEMP}) of $F_N^{(j)}(t)$ to
$f(t)^{\otimes j}$ implies that:
\begin{eqnarray}\label{eq: convOBSj}
&&\left\langle u_j\otimes 1_{N-j}\right\rangle_{ F_N(t)}=\left\langle
u_j\right\rangle_{ F_N^{(j)}(t)}\rightarrow \left\langle
u_j\right\rangle_{f(t)^{\otimes j}}\ \ \text{as}\ N\to\infty,\nonumber\\
&&
\end{eqnarray}
for each $u_j\in \mathcal{C}^{0}_{b}\(\R^{3j}\times\R^{3j}\)$. In
other words, we are able to compute the ''macroscopic'' expected value
of $j$-particle observables.\\
A remarkable fact is that the validity of propagation of chaos has
been proven without using the hierarchies and this is really a big
advantage because to deal with the hierarchies (\ref{eq: class14}),
(\ref{eq: class17}) seems to be quite difficult. A priori, one could
think to prove the convergence (\ref{eq: convOBSj}) of the marginals
$F_N^{(j)}(t)$ to the products $f(t)^{\otimes j}$, by using that the
first ones solve the BBGKY hierarchy (\ref{eq: class14}) and the
second ones solve the Vlasov hierarchy (\ref{eq: class17}). Thus, if
one would be able to prove the convergence of solutions of the
$N$-dependent hierarchy to the Vlasov one, by knowing that the
limiting hierarchy has factorized solutions arising from the Vlasov
equation (as we previously discussed), the final step for proving the
propagation of chaos would be to show the uniqueness of the solution
of the Vlasov hierarchy over the class in which one is able to prove
convergence. As regard to the ''convergence problem'' the difficulty
is that the BBGKY hierarchy involves operators which are unbounded, at
least in reasonable spaces, thus it does not seem possible to apply
any compactness argument to ensure the convergence of the solution. On
the other side, concerning the ''uniqueness problem'' for the limiting
hierarchy (\ref{eq: class17}), the crucial point is the connection
between the space in which one could show convergence and those in
which it would be possible to prove uniqueness. Therefore, the problem
of realizing the classical mean-field limit by dealing with the
hierarchies is quite hard. On the other side, we have just seen that
it can be faced more naturally by using two crucial tools: the Law of
Large Numbers (\ref{eq: class11}) and the continuity of solutions of
the Vlasov equations with respect to the weak convergence of measures
(\ref{eq: DOBineq}).\\
In the next chapter we will see that in the quantum case to deal with
hierarchies is not so difficult and a possible approach to realize the
limit (indeed the one that has been used more in the last years) is
properly the one we have just described (convergence + uniqueness),
particularly to deal with singular pair interaction potentials.
\\
\section{Quantum Mean-Field Limit}
\setcounter{equation}{0}
\def\theequation{3.\arabic{equation}}
This chapter is devoted to the analysis of the macroscopic properties
of the dynamics of a quantum system constituted by $N$ identical
particles interacting by a mean-field potential in the limit
$N\to\infty$. As in the previous chapter, we set the dimension of the
system equal to $3$ but the main results we are going to discuss hold
in any dimension.
\\
The mean-field interaction potential is represented by the (right hand
side) multiplication operator
\begin{equation}\label{eq: quant1}
U^{Q}(X_N)=\frac{1}{2N}\sum_{k\neq l}^N\phi(x_k-x_l),
\end{equation}
where, as in the previous chapter, we denote by
$X_N=\(x_1,\dots,x_N\)\in \R^{3N}$ the positions of the $N$ particles
and we assume $\phi$ to be spherically symmetric. We want to
characterize the effective dynamics of such a system for large $N$.
The problem of investigating the error in the approximation of the
many-body evolution with the limiting macroscopic dynamics, which we
do not discuss here, has been studied in \cite{BENJIROD} and
\cite{ErdBenji}.
\subsection{Setting of the problem: general features and known results}
\setcounter{equation}{0}
\def\theequation{3.1.\arabic{equation}}
The state of an $N$-particle quantum mechanical system in $\R^3$
can be described by a complex valued wave function $\Psi_N\in L^2\(\R^{3N}\)$.
Physically the absolute value squared of $\Psi_N(x_1,\dots,x_N)$ is
interpreted as the
probability density for finding particle one at $x_1$, particle two at
$x_2$, and so on. Moreover the absolute value squared of the Fourier
transform $\hat{\Psi}_N(v_1,\dots,v_N)$ is interpreted as the
probability density for having particle one with velocity $v_1$,
particle two with velocity $v_2$, and so on (for the sake simplicity
we always consider identical particles with mass $m=1$ thus velocities
are always equal to momenta). Because of this
probabilistic interpretation, we will always consider wave functions
$\Psi_N$ with $L^2$-norm equal to one.\\
In nature there exist two different types of particles; bosons and
fermions. Bosonic systems are
described by wave functions which are symmetric with respect to
permutations, in the sense that
\begin{equation}\label{eq: quant3}
\Psi_N(x_{\pi(1)},\dots,x_{\pi(N)})=\Psi_N(x_1,\dots,x_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$. Fermionic systems,
on the other hand, are described by antisymmetric
wave functions satisfying
\begin{equation}\label{eq: quant4}
\Psi_N(x_{\pi(1)},\dots,x_{\pi(N)})=(-1)^{\sigma(\pi)}\Psi_N(x_1,\dots,x_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$
where $\sigma(\pi)=0$ if $\pi$ is even (in the sense that it can be written
as the composition of an even number of transpositions) and
$\sigma(\pi)=1$ if it is odd. In the sequel we will denote by
$L_s^2\(\R^{3N}\)$ the space of bosonic wave functions (namely the
subspace of $L^2\(\R^{3N}\)$ consisting of all functions satisfying
(\ref{eq: quant3})).
Equations (\ref{eq: quant3}) and (\ref{eq: quant4}) are responsible
for substantial differences between an $N$-particle bosonic system and
a fermionic one. Actually these features determine a different way to
look at the limit $N\to\infty$ in the mean-field context, the use of
different techniques leading to (a bit) different effective dynamics
(see paragraph ''Joint limit $N\to \infty$ and $\var\to 0$'' in
Section 3.4). Furthermore,
the different nature of bosons and fermions is crucial in the
perspective of looking at the connection between mean-field limit and
semiclassical approximation (
as we will observe in Section 3.4) and, at least from this point of
view, bosonic systems seem to be more difficult to treat.
Anyway, here and henceforth we consider undistinguishable quantum
particles by neglecting the statistics. In particular, in some cases
the states we consider are indeed admissible for bosons.\\
We know that the observables of an $N$-particle system are represented
by self adjoint operators $A$ on $L^2\(\R^{3N}\)$, then the expectation
\begin{equation}\label{eq: EXP}
\left\langle A\right\rangle_{\Psi_N}=<\Psi_N,A\Psi_N>=\int
\bar{\Psi}_N(X_N)\(A\Psi_N\)(X_N)\ud X_N
\end{equation}
gives the value of the observable represented by $A$ in the state
described by $\Psi_N$. \\
The Hamiltonian of an $N$-particle system interacting by (\ref{eq:
quant1}), assuming the mass of the particles to be equal to one, is
the standard quantization of (\ref{eq: class2V}), namely
\begin{equation}\label{eq: quant2V}
H_N^{Q,V}=-\sum_{k=1}^{N}\(\frac{\var^2 \Delta_k}{2}+V^Q(x_k)\)+U^{Q}(X_N),
\end{equation}
where we denoted by $\Delta_k$ the Laplace operator acting on the
variable $x_k$, $k=1,2,\dots,N$ and here and henceforth we denote the
Planck constant by $\var$. The potentials $V^Q$ and $\phi$ (appearing
in (\ref{eq: quant1}) are such that the Hamiltonian $H_N^{Q,V}$ is
guaranteed to be a
self-adjoint operator acting on the Hilbert space $L^2\(\R^{3N}\)$ and
it is invariant with respect to any permutation of the labeling
(namely, the Hamiltonian is symmetric in the exchange of particle
names).
The first part of $H_N^{Q,V}$ is a sum of one-body operators
(operators acting on one particle
only); the sum of the Laplacians is the kinetic part of the
Hamiltonian. The function $V^Q$ describes
an external potential which acts in the same way on all $N$ particles.
The second part of the Hamiltonian describes the interaction among the
particles.
As in the classical case, we can assume without loss of generality
that the potential experienced by each particle is only that arising
from the many-body interaction, namely, the one-particle potential
$V^Q$ is assumed to be equal to zero. Thus the Hamiltonian we consider
is
\begin{equation}\label{eq: quant2}
H_N^{Q}=-\sum_{k=1}^{N}\frac{\var^2\Delta_k}{2}+U^{Q}(X_N).
\end{equation}
The Hamiltonian (\ref{eq: quant2}) is the observable associated with
the energy of the
$N$-particle system interacting by the mean-field potential (\ref{eq:
quant1}), thus the expectation
\begin{eqnarray}\label{eq: quant7}
&&\left\langle H_N^Q\right\rangle_{\Psi_N}=<\Psi_N,H_N^Q\Psi_N>=\int
\ud X_N \bar{\Psi}_N(X_N)\(H_N^Q\Psi_N\)(X_N)
\end{eqnarray}
gives the energy of the system in the state described by the wave
function $\Psi_N$.\\
The considerations we did in the previous chapter as regard to the
scaling of the potential hold also in the quantum context. Therefore
we are guaranteed that the energy per particle is of order one for
large $N$, as it is crucial in looking for a non-trivial and
well-defined limiting dynamics, and we realize that the basic features
of the model are that the mutual interaction among the particles is
weak (again of size $1/N$) and of long range type (unscaled support of
$\phi$). Again, as a consequence of such two effects we will have
propagation of chaos and nonlinearity of the equation governing the
limiting one-particle dynamics respectively (see Section 2.2).\\
The time evolution of a wave function $\Psi_{N}\in L^2\(\R^{3N}\)$
associated with the $N$-particle system whose Hamiltonian is (\ref{eq:
quant2}) is governed by the linear Schr\"odinger
equation
\begin{equation}\label{eq: quant5}
i\var\pa_t\Psi_{N,t}=H_N^Q\Psi_{N,t},
\end{equation}
and, since $H_N^Q$ is a self-adjoint operator, the time-evolution
associated with the equation (\ref{eq: quant5}) preserves the
$L^2$-norm of the wave function.
The solution to
(\ref{eq: quant5}), with initial condition $\Psi_{N,t}\vert_{t=0} =
\Psi_{N,0}\in L^2\(\R^{3N}\)$, can be written by means of the unitary
group generated
by $H_N^Q$ as
\begin{equation}\label{eq: quant6}
\Psi_{N,t}=e^{-i\frac{t}{\var}H_N^Q}\Psi_{N,0}\ \ \text{for all}\ t\in \R.
\end{equation}
The global well-posedness of (\ref{eq: quant5}) is not an issue here.
The study of (\ref{eq: quant5}) is focused, therefore, on
other questions concerning the qualitative and quantitative behavior
of the solution $\Psi_{N,t}$. Despite
the linearity of the equation, these questions are usually quite hard
to answer, because in physically
interesting situation the number of particles $N$ is very large; for
example, in applications related to the study of boson stars we have
$N \approx 10^{30}$. For such huge values of $N$,
it is of course impossible to compute the solution (\ref{eq: quant5})
explicitly; numerical methods are completely
useless as well (unless the interaction among the particles is switched off).
Fortunately, also from the point of view of physics, it is not so
important to know the precise solution
to (\ref{eq: quant5}); it is much more important, for physicists
performing experiments, to have information
about the macroscopic properties of the system, which describe the
typical behavior of the particles,
and result from averaging over a large number of particles.
Restricting the attention to macroscopic
quantities simplifies the study of the solution $\Psi_{N,t}$, but it
still does not make it accessible to mathematical
analysis. To further simplify matters, we are going to let the number
of particles $N$ tend
to infinity. The macroscopic properties of the system, computed in the
limiting regime $N\to\infty$, are
then expected to be a good approximation for the macroscopic
properties observed in experiments,
where the number of particles $N$ is very large, but finite (explicit
bounds on the difference between the limiting behavior as $N\to\infty$
and the behavior for large but
finite $N$ are obtained in \cite{BENJIROD} and \cite{ErdBenji}).
\subsubsection{The density matrix formalism}
To consider the limit of large $N$, we are going to make use of the
Reduced (or Marginal) Density
Matrices (RDM) associated with an $N$-particle wave function
$\Psi_N\in L^2\(\R^{3N}\)$. First of all, we define the
density matrix $\hat{\rho}_N=\vert \Psi_N><\Psi_N\vert$ associated
with $\Psi_N$ as the orthogonal projection onto $\Psi_N$; we use
here and henceforth the notation $\vert \psi><\psi\vert$ to indicate
the orthogonal projection onto $\psi$ (Dirac bracket
notation).
Therefore $\hat{\rho}_N$ is a non-negative integral operator acting
from $L^2\(\R^{3N}\)$ to $L^2\(\R^{3N}\)$ with kernel given by
\begin{equation}\label{eq: BenjiPRELdensMATRIX1}
\rho_N(X_N;Y_N)=\bar{\Psi}_N(X_N)\Psi_N(Y_N),
\end{equation}
where $Y_N=(y_1,\dots,y_N)\in \R^{3N}$. Note that, by virtue of the $L^2$-
normalization of $\Psi_N$, we have
\begin{equation}\label{eq: BenjiPRELdensMATRIX1bis}
\texttt{Tr}\hat{\rho}_N=\int \ud X_N \rho_N(X_N;X_N)=\int \ud X_N
\bar{\Psi}_N(X_N)\Psi_N(X_N)=\left\|\Psi_N\right\|_{L^2\(\R^{3N}\)}^2=1.
\end{equation}
Thus $\hat{\rho}_N\in\mathscr{L}^1\(L^2\(\R^{3N}\)\)$, where
$\mathscr{L}^1\(L^2\(\R^{3N}\)\)$ is the Banach space (with respect to
the norm
$\left\|\cdot\right\|_{\mathscr{L}^1\(L^2\(\R^{3N}\)\)}=\texttt{Tr}\left\vert
\cdot \right\vert$) of the trace class operators acting on
$L^2\(\R^{3N}\)$. Moreover,
the positivity of $\hat{\rho}_N$ implies $\left\|\hat{\rho}_N
\right\|_{\mathscr{L}^1\(L^2\(\R^{3N}\)\)} =\texttt{Tr}\hat{\rho}_N
=1$.\\
It turns out that the state of a quantum mechanical system can be
equivalently represented in the wave function (Schr\"odinger) picture
and in the density matrix (Heisenberg) formalism and the expectation
$\left\langle A\right\rangle_{\Psi_N}=<\Psi_N,A\Psi_N>$ of an
observable $A$ in the state described by $\Psi_N$, expressed through
the density matrix $\hat{\rho}_N$,
can be written as \texttt{Tr}$ A\hat{\rho}_N$. For example, the energy
of the mean-field system in the state described by $\hat{\rho}_N$ is
\begin{eqnarray}\label{eq: quant7conRHO}
&&\left\langle
H_N^Q\right\rangle_{\Psi_N}=<\Psi_N,H_N^Q\Psi_N>=\texttt{Tr}
H_N^Q\hat{\rho}_N,\nonumber\\
&&
\end{eqnarray}
$H_N^Q$ defined in (\ref{eq: quant2}).\\
The time evolution of a density matrix
describing the state of the $N$-particle mean-field system
is governed by the linear equation
\begin{equation}\label{eq: quant8}
i\var\pa_t\hat{\rho}_{N,t}=\[H_N^Q,\hat{\rho}_{N,t}\],
\end{equation}
where
$\[H_N^Q,\hat{\rho}_{N,t}\]$ denotes the commutator between $H_N^Q$
and $\hat{\rho}_{N,t}$, namely
$\[H_N^Q,\hat{\rho}_{N,t}\]=H_N^Q\hat{\rho}_{N,t}-\hat{\rho}_{N,t}H_N^Q$.
Equation (\ref{eq: quant8}) is usually called Heisenberg equation and,
by knowing that $\hat{\rho}_{N,t}=\vert\Psi_{N,t}><\Psi_{N,t}\vert $,
it can be derived easily by the Schr\"odinger equation (\ref{eq:
quant5}) solved by $\Psi_{N,t}$. The self-adjointness of the
Hamiltonian $H_N^Q$, responsible for conservation of the $L^2$-norm of
the wave function, implies that positivity and trace of the density
matrix are also preserved in time. \\
We remind that we are looking at systems constituted by
undistinguishable particles. Then we consider density matrices
$\hat{\rho}_N$ such that their kernel $\rho_N(x_1, . . . , x_N; y_1 .
. . , y_N)$ is symmetric in the exchange of particle names, namely
\begin{equation}\label{eq: quant3NEW}
\rho_N(x_{\pi(1)},\dots,x_{\pi(N)};y_{\pi(1)},\dots,y_{\pi(N)})=\rho_N(x_1,\dots,x_N;y_1,\dots,y_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$. By the definition
of the time-evolution (\ref{eq: quant8}) it is easy to verify that
this property is preserved in time. \\
The solution to
(\ref{eq: quant8}), with initial condition
$\hat{\rho}_{N,t}\vert_{t=0} =\hat{\rho}_{N,0}$,
can be written by means of the unitary group generated
by $H_N^Q$ as
\begin{equation}\label{eq: quant9}
\hat{\rho}_{N,t}=
e^{-i\frac{t}{\var}H_N^Q}\hat{\rho}_{N,0}e^{i\frac{t}{\var}H_N^Q}\ \
\text{for all}\ t\in \R.
\end{equation}
The main advantage in describing the state and the dynamics of an
$N$-particle system by using the density matrix formalism is that it
gives the possibility to investigate the properties of subsystems made
by a fixed number of variables.
The way to do that is to introduce the Reduced Density Matrices (RDM).
For $j = 1,\dots,N$, we define the $j$-particle marginal
density $\hat{\rho}_N^{(j)}$ associated with $\hat{\rho}_N$ as the
partial trace of $\hat{\rho}_N$ over the degrees of freedom of the last
$(N - j)$ particles:
\begin{equation}\label{eq: BenjiPRELdensMATRIX2}
\hat{\rho}_N^{(j)}=\texttt{Tr}_{j+1}\hat{\rho}_N
\end{equation}
where $\texttt{Tr}_{j+1}$ denotes the partial trace over the particles
$j + 1, j + 2, \dots ,N$. In other words, $\hat{\rho}_N^{(j)}$ is
defined as the non-negative trace class operator on $L^2\(\R^{3j}\)$
with kernel given by
\begin{equation}\label{eq: BenjiPRELdensMATRIX3}
\rho_N^{(j)}(X_j;Y_j)=\int \ud X_{N-j}\rho_N(X_j,X_{N-j};Y_j, X_{N-j}).
\end{equation}
The last equation can be considered as the definition of partial
trace. As in the previous chapter, we used the notation
$X_j = (x_1,\dots, x_j), Y_j= (y_1,\dots,y_j)\in \R^{3j}$ and $X_{N-j}
= (x_{j+1},\dots, x_N) \in \R^{3(N-j)}$. By definition,
$\texttt{Tr}\hat{\rho}_N^{(j)} = 1$ for all $N$ and for all $j = 1,
\dots,N$ (clearly, if $j=N$ we find $\hat{\rho}_N^{(N)}=\hat{\rho}_N$)
thus $\hat{\rho}_N^{(j)}\in \mathscr{L}^1\(L^2\(\R^{3j}\)\)$ for all
$N$ and for all $j$.
\begin{rem}\label{REMARKsuNORMALIZZAZIONE}
\textnormal{Note that, in the physics literature, one normally
uses a different normalization for the reduced density matrices. If
the statistics are taken into account, the reduced density matrices
are defined as expectation of bosonic and fermionic fields in the
framework of the ''second quantization formalism''. }
\end{rem}
For fixed $j < N$, the $j$-particle
density matrix does not contain the full information about the state
described by $\hat{\rho}_N$. Knowledge
of the $j$-particle marginal $\hat{\rho}_N^{(j)}$, however, is
sufficient to compute the expectation of every $j$-particle
observable in the state described by the density matrix
$\hat{\rho}_N$. In fact, if $A^{(j)}$ denotes an arbitrary
bounded operator on $L^2\(\R^{3j}\)$, and if $A^{(j)} \otimes
1^{(N-j)}$ denotes the operator on $L^2(\R^{3N})$ which acts as
$A^{(j)}$ on the first $j$ particles, and as the identity on the last
$(N -j)$ particles, we have
\begin{equation}\label{eq: BenjiPRELdensMATRIX4}
\texttt{Tr}\(A^{(j)} \otimes
1^{(N-j)}\)\hat{\rho}_N=\texttt{Tr}A^{(j)}\hat{\rho}_N^{(j)}.
\end{equation}
Thus, $\hat{\rho}_N^{(j)}$ is sufficient to compute the expectation of
arbitrary observables which depend non-trivially
on at most $j$ particles (because of the permutation symmetry, it is
not important on which particles
it acts, just that it acts at most on $j$ particles).\\
Marginal densities play an important role in the analysis of the
$N\to\infty$ limit because, in contrast
to the wave function $\Psi_N$ and to the density matrix
$\hat{\rho}_N$, the $j$-particle marginal $\hat{\rho}_N^{(j)}$ can
have, for
every fixed $j\in\mathbb{N}$, a well-defined limit as $N\to\infty$
(because, if we fix $j\in \mathbb{N}$, $\{\hat{\rho}_N^{(j)}\}_N$
defines a sequence
of operators all acting on the same space $L^2(\R^{3j})$). In other
words, $\hat{\rho}_N^{(j)}$ is a function of a fixed number of
variables (which remains finite in the limit $N\to\infty$), while
$\Psi_N$ and $\hat{\rho}_N$ are functions of $N$ variables thus in the
limit we would have to deal with functions of an infinite number of
variables and clearly it prevents the possibility to find a
well-defined limit for them.
\subsubsection{Mixed states}
In the previous analysis we have always considered systems whose state
is described by a density matrix
$\hat{\rho}\in\mathscr{L}^1\(L^2\(\R^{3d}\)\)$ which is the orthogonal
projection onto a wave function $\Psi\in L^2\(\R^{3d}\) $ with
$\texttt{Tr}\hat{\rho}=\left\|\Psi\right\|_{L^2\(\R^{3d}\)}=1$ (we had
$d=N$). Such kind of states are called ''pure'' states. Indeed, we say
that a system is in a pure state whenever we know that it is described
by a uniquely determined wave function with probability equal to one.
As a consequence, the density matrix describing a ''pure'' state is a
rank-one projection on $L^2\(\R^{3d}\)$, namely
$\left\vert\Psi><\Psi\right\vert$. Nevertheless, in some cases it is
not possible to know precisely (namely, with probability equal to one)
which is the wave function describing the state of a system but one
only has probabilistic predictions about that.
For example, one can have a certain number (possibly infinite) $k$ of
known wave functions $\Psi^{1},\dots,\Psi^{k}\in L^2\(\R^{3d}\)$, with
$\left\|\Psi^{1}\right\|_{L^2\(\R^{3d}\)}=\dots=
\left\|\Psi^{k}\right\|_{L^2\(\R^{3d}\)}=1$, and a sequence of
non-negative numbers $\lambda_1,\dots, \lambda_k$ such that it is
known that the state can be described by $\Psi^{1}$ with probability
equal to $\lambda_1$, by $\Psi^{2}$ with probability equal to
$\lambda_2$ and so on..., where $\lambda_s\leq 1$ for $s=1,\dots,k$
and $\sum_s \lambda_s=1$. These kind of states are called ''mixed''
states (or equivalently, one can say that the state ''associated
with'' the sequence $\Psi^{1},\dots,\Psi^{k}$ is a ''mixture'' of the
pure states $\Psi^{1},\dots,\Psi^{k}$). \\
It turns out that one of the advantages of the density matrix
formalism is that it encodes both the case of pure states and the case
of mixtures. In fact, denoting by $\hat{\rho}_{s}$ the orthogonal
projection onto $\Psi^s$, for $s=1,\dots,k$, the density matrix
$\hat{\rho}_{mix}$ associated with the system under consideration is
given by
\begin{equation}\label{eq: mixture}
\hat{\rho}_{mix}=\sum_{s=1}^{k}\lambda_s
\hat{\rho}_{s}=\sum_{s=1}^{k}\lambda_s
\left\vert\Psi^{s}><\Psi^{s}\right\vert.
\end{equation}
Then, since $\left\|\Psi^{s}\right\|_{L^2\(\R^{3d}\)}=1$ for any $s$, we have
\begin{equation}\label{eq: CORREZIONE1}
\texttt{Tr}\hat{\rho}_{mix}=\sum_{s=1}^{k}\lambda_s
\texttt{Tr}\hat{\rho}_{s}=\sum_{s=1}^{k}\lambda_s
\left\|\Psi^{s}\right\|_{L^2\(\R^{3d}\)}= \sum_{s=1}^{k}\lambda_s=1,
\end{equation}
namely, $\hat{\rho}_{mix}\in \mathscr{L}^1\(L^2\(\R^{3d}\)\)$ and
$\left\|\hat{\rho}_{mix}\right\|_{\mathscr{L}^1\(L^2\(\R^{3d}\)\)}=1$.
\\
Clearly, a mixed state reduces to a pure state if
$\lambda_{\bar{s}}=1$ for some $\bar{s}$ and $\lambda_{s}=0$ for
$s\neq \bar{s}$.
By virtue of (\ref{eq: CORREZIONE1}), it turns out that the analysis
done previously by starting from a pure state can be generalized
straightforward to the case of mixtures.
Furthermore, there are also states that are made by a ''continuum''
mixture of pure states. Indeed, let us take the parameter $s$ in
(\ref{eq: mixture}) as a continuum variable varying in a certain set
$\Lambda\subset \R^n$, for some $n$ (for example, in the initial state
considered in Section 4.5 we have $s=(x_0,v_0)\in \Lambda=\R^6$), and
let us consider a function $g=g(s)$ such that $g\ud s$ is a
probability distribution on $\Lambda$. If we have a family of
$L^2$-normalized wave functions $\left\{\Psi^s\right\}_{s\in \Lambda}$
on $\R^d$ (for example, in Section 4.5 we considered the family of
coherent states ''centered'' in $(x_0, v_0)$), we can costruct a mixed
state which is the ''continuum'' mixture of the pure states
$\left\{\Psi^s\right\}_{s\in \Lambda}$ through the probability
distribution $g\ud s$. It turns out that the kernel $\rho_{mix}$ of
the density matrix $\hat{\rho}_{mix}$ describing the mixed state under
consideration is
\begin{equation}\label{eq: mixtureDENScontinue}
\rho_{mix}(X;Y)=\int_\Lambda \ud s\ g(s)\rho^s(X;Y)=\int_\Lambda \ud
s\ g(s)\overline{\Psi}^{s}(X) \Psi^{s}(Y),
\end{equation}
where $X\in \R^d,\ Y\in \R^d$. Clearly, all considerations we did for
''discrete'' mixtures hold also in that case.
\subsubsection{The limit $N\to\infty$}
We will discuss several known results about the study of the limiting
dynamics when $N\to\infty$ for a mean-field
system and we will see that what has been established, by using
different techniques and various formalisms, is that the effective
single-particle dynamics is governed by a cubic nonlinear
Schr\"odinger equation
\begin{equation}\label{eq: Hartree}
i\var\pa_t \psi_t=-\frac{\var^2}{2}\Delta \psi_t+\(\phi\ast \vert
\psi_t\vert^2\)\psi_t
\end{equation}
which is known as Hartree equation. Clearly, in that case the symbol
''$\ast$'' denotes the convolution with respect to the spatial
variable, namely
\begin{equation}\label{eq: HartreePOT}
\(\phi\ast \vert \psi_t\vert^2\)(x)=\int \ud y\ \phi(x-y)\vert
\psi_t(y)\vert^2.
\end{equation}
The first rigorous results establishing a relation between the many
body Schr\"odinger evolution
and the nonlinear Hartree dynamics were obtained by K. Hepp in
\cite{HEPP}(for smooth interaction potentials)
and then generalized by J. Ginibre and G. Velo to singular potentials
in \cite{GV}. These works were inspired
by techniques used in quantum field theory. We will not discuss this
method because we want to focus on other techniques which are more
related to the topic we are going to face in the next chapters (the
connection between mean-field limit and semiclassical approximation).
The first proof of the emergence of the Hartree dynamics by using the
RDM formalism (''RDM-convergence'') was obtained by H. Spohn in
\cite{SPOHN}, for bounded potentials (see Theorem \ref{teoSPOHN} in
Section 2.3).
The method introduced by Spohn was then extended to singular
potentials: in \cite{EY}, L. Erd{\H{o}}s and H. T. Yau faced the RDM-
convergence for a Coulomb potential $\phi (x) = ±1/\vert x\vert$;
partial results for this kind of interaction were
also obtained by C. Bardos, F. Golse and N. Mauser in \cite{BGM} (note
that recently a new proof in the
case of a Coulomb interaction has been proposed by J. Fr\"ohlich, A.
Knowles, and S. Schwarz in \cite{AnntiNEW}).
A different approach to the proof of the rigorous derivation of the
Hartree equation from a mean-field bosonic system has been proposed by
Fr\"ohlich, Schwarz and
Graffi in \cite{FGS}. By using the Wigner formalism (see Chapter 3)
they can consider the mean-field limit uniformly in the Planck
constant $\var$ (up to an exponential error depending on time); this
allows them to combine the semiclassical
limit and the mean field limit by assuming restrictive assumptions on
the pair interaction potential (we will come back on this result in
Chapter 3). It is also interesting to remark that the mean-field limit
can be
interpreted as a Egorov-type theorem; this was observed
in \cite{FPK} for sufficiently smooth potentials and in
\cite{AnntiNEW} for the Coulomb interaction.
\subsection{Quantum BBGKY hierarchy and its formal limit as $N\to\infty$}
\setcounter{equation}{0}
\def\theequation{3.2.\arabic{equation}}
We have already remarked that, for any $j=1,\dots,N$, the marginal
densities $\hat{\rho}_{N,t}^{(j)}$ associated with the solution
$\hat{\rho}_{N,t}$ of the equation (\ref{eq: quant8}), are crucial
tools in studying the mean-field limit because they can have, for
every fixed $j$, a well-defined limit as $N\to\infty$. Thus, we are
interested in their time-evolution as $N\to\infty$.
By taking the partial trace over the degrees of freedom of the last
$N-j$ particles in the Heisenberg equation (\ref{eq: quant8}) we find
the following family of equations (one for each $j=1,\dots,N$)
\begin{eqnarray}\label{eq: quant10}
&&i\var \pa_t
\hat{\rho}_{N,t}^{(j)}=\sum_{k=1}^{j}\[-\frac{\var^2}{2}\Delta_k,\hat{\rho}_{N,t}^{(j)}\] +
T_{N,j}^{Q}\hat{\rho}_{N,t}^{(j)}+\frac{N-j}{N}C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)},
\end{eqnarray}
where the operator $T_{N,j}^{Q}$ acts on $\mathscr{L}^1\(L^2_s\(\R^{3j}\)\)$
while the operator $C_{j,j+1}^Q$ maps $j+1$-particle densities in
$j$-particle ones (if $j=N$ we find $C_{N,N+1}^Q\equiv 0$).
The family of equations (\ref{eq: quant10}) is called BBGKY hierarchy
in analogy to the classical case and, again, it is called
''hierarchy'' because we can see that the equation for the
$j$-particle marginal density is linked to the subsequent one by the
term $C_{j,j+1}^Q \hat{\rho}_{N,t}^{(j)}$. The physical meaning is the
same we discussed in the classical case: the variation in time of
$\hat{\rho}_{N,t}^{(j)}$ is due to the free motion of the $j$
particles, to their interaction among themselves and to the
interaction among the $j$-particle subsystem and the remaining $N-j$
particles. The first effect is modeled by the l.h.s and by the first
term in the r.h.s of (\ref{eq: quant10}), the second one is encoded in
$T_{N,j}^{Q}\hat{\rho}_{N,t}^{(j)}$), while the interaction between
the $j$-particle subsystem and the remaining $N-j$ particles is
modeled by $\frac{N-j}{N}C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)}$. The
factor $1/N$ in front of $C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)}$ arises
from the scaling of the potential $U^{Q}$ (see (\ref{eq: quant1})
while the factor $N-j$ is due to the symmetry with respect to
permutations of the labeling (we remind that we are dealing with $N$
identical particles): indeed, the interaction of the $j$ particles
under consideration with the last $N-j$ can be modeled by $N-j$ times
the interaction with the $j+1$-th particle.
Writing explicitly the action of the operators $T_{N,j}^{Q}$ and
$C_{j,j+1}^{Q}$, we find:
\begin{eqnarray}\label{eq: quant11}
&&T_{N,j}^{Q}\hat{\rho}_{N,t}^{(j)}=\frac{1}{2N}\sum_{k\neq
l}^{j}\[\phi(x_k-x_l),\hat{\rho}_{N,t}^{(j)}\],
\end{eqnarray}
and
\begin{eqnarray}\label{eq: quant12}
&&C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)}=\sum_{k=1}^{j}\texttt{Tr}_{j+1}\left\{\[\phi(x_k-x_{j+1})
,\hat{\rho}_{N,t}^{(j+1)}\]\right\}.
\end{eqnarray}
By (\ref{eq: quant11})
we can argue that the operator $T_{N,j}^{Q}$ gives a vanishing
contribution in the limit because it is of size $j^2/N$, while the
operator $C_{j,j+1}^Q$ is of order one in the limit and the factor
$(N-j)/N$ appearing in (\ref{eq: quant10}) is also of order one.
Therefore denoting by $\hat{\rho}_{t}^{(j)}$ the expected limit of
$\hat{\rho}_{N,t}^{(j)}$ when $N\to\infty$, the formal limit of the
BBGKY hierarchy (\ref{eq: quant10}) is
\begin{eqnarray}\label{eq: quant13}
&&i\var\pa_t\hat{\rho}_{t}^{(j)}=\sum_{k=1}^{j}\[-\frac{\var^2}{2}
\Delta_k,\hat{\rho}_{t}^{(j)}\]+ C_{j,j+1}^Q\hat{\rho}_{t}^{(j+1)},
\end{eqnarray}
which in the case $j=1$ is equal to:
\begin{eqnarray}\label{eq: quant14}
&&i\var\pa_t\hat{\rho}_{t}^{(1)}=\[-\frac{\var^2}{2}
\Delta_1,\hat{\rho}_{t}^{(1)}\]+\texttt{Tr}_{2}\left\{\[\phi(x_1-x_{2})
,\hat{\rho}_{t}^{(2)}\]\right\}.
\end{eqnarray}
We observe that the Hartree equation (\ref{eq: Hartree}) in the
density matrix formalism (''Heisenberg form'') is
\begin{eqnarray}\label{eq: quant15}
&&i\var\pa_t\hat{\rho}_{t}=\[-\frac{\var^2}{2}
\Delta,\hat{\rho}_{t}\]+\texttt{Tr}_{2}\left\{\[\phi(x-x_{2})
,\hat{\rho}_{t}\otimes\hat{\rho}_{t}\]\right\}.
\end{eqnarray}
Replacing $x$ by $x_1$, $\hat{\rho}_{t}$ by $\hat{\rho}_{t}^{(1)}$ and
the product $\hat{\rho}_{t}\otimes\hat{\rho}_{t}$ by
$\hat{\rho}_{t}^{(2)}$ we realize that (\ref{eq: quant15}) is
precisely the same of (\ref{eq: quant14}). Thus the equation of the
hierarchy (\ref{eq: quant13}) corresponding to $j=1$ is properly the
Hartree equation, provided that the the two-particle density
$\hat{\rho}_{t}^{(2)}$ is factorized, and for this reason (\ref{eq:
quant13})
is usually called ''Hartree hierarchy''. More precisely, by
considering (\ref{eq: quant13})
and by assuming the reduced density matrices $\hat{\rho}_{t}^{(j)}$,
$j=1,2,\dots$, to be factorized, namely
\begin{eqnarray}\label{eq: quant16}
&&\hat{\rho}_{t}^{(j)}=\hat{\rho}_{t}^{\otimes j}\ \ \forall\ j,
\end{eqnarray}
it is easy to verify that $\hat{\rho}_{t}$ has to solve the Hartree
equation. Conversely, if we consider a time dependent one-particle
density $\hat{\rho}_{t}$ solving the Hartree equation (\ref{eq:
quant15}) and we take the $j$-particles densities
$\hat{\rho}_{t}^{(j)}=\hat{\rho}_{t}^{\otimes j}$, $j=1,2\dots$, we
find that the sequence $\{\hat{\rho}_{t}^{(j)}\}_{j\geq 1}$ solves the
hierarchy (\ref{eq: quant13}). \\
An interesting problem is that of the uniqueness of the solution of
the Hartree hierarchy. The situation in the quantum case is quite
different from that of the classical one. In fact, as we will see in
the next section, the Hartree hierarchy is much more controllable than
the Vlasov one because the operators involved are bounded with respect
to the norms appropriate to study the convergence of the sequence of
reduced density matrices to the solution of the limiting hierarchy.
Thus, it is possible to follow the approach we described briefly at
the end of the previous chapter (convergence + uniqueness) in order to
prove ''propagation of chaos'' in the quantum context, namely,
asymptotic factorization of the dynamics (in the sense specified in
the forthcoming paragraph). Nonetheless, the proof of uniqueness of
the solution of the quantum limiting hierarchy is very far to be a
trivial stuff. Indeed, in the case of bounded interaction the problem
is quite easy to face and, in particular, by following the strategy of
\cite{SPOHN} (originally introduced by O. Lanford for the derivation
(for short times) of the Boltzmann equation from the hard-sphere
dynamics (see \cite{LANFORD})) it is possible to prove ''at the same
time'' convergence and uniqueness (see Theorem \ref{teoSPOHN}). On the
contrary, in case of more singular interactions (as the Coulomb one),
the proof of propagation of chaos consists really of two steps:
proving the convergence of solutions of the BBGKY hierarchy to the
Hartree hierarchy and showing the uniqueness of the solution of such a
hierarchy (which implies factorization of the limiting $j$-particle
density matrices because, as we have already remarked, the Hartree
hierarchy admits factorized solutions as (\ref{eq: quant16})). In the
Coulomb case, the ''uniqueness' problem is quite hard to deal with
because of the singularity of the interaction (see \cite{EY}).
Anyway, we will come back later on the rigorous proof of propagation
of chaos, analyzing in detail the case of bounded potentials. Moreover
we will discuss briefly the Coulomb case, accenting which are the main
new tools with respect to the bounded case, why there is need of them
and in which way they make the proof harder requiring a more refined
analysis of the limiting hierarchy. \\
Now let us clarify what we mean by ''propagation of chaos'' in the
quantum framework.
Let us consider as initial datum for the Schr\"odinger equation
(\ref{eq: quant5}) a factorized $N$-particle wave function
\begin{equation}\label{eq: quant17bis}
\Psi_{N,0}=\psi_0^{\otimes N},\ \ \ \ \ \ \text{for some}\ \
\psi_0\in L^2\(\R^3\).
\end{equation}
This assumption, rephrased in the density matrix formalism, leads to
consider the following factorized $N$-particle density matrix
\begin{equation}\label{eq: quant17}
\hat{\rho}_{N,0}=\vert
\Psi_{N,0}><\Psi_{N,0}\vert=\hat{\rho}_0^{\otimes N},\ \ \ \text{with}
\ \hat{\rho}_0=\vert \psi_0><\psi_0\vert,
\end{equation}
as initial datum for the Heisenberg equation (\ref{eq: quant8}).
As in the classical context, (\ref{eq: quant17bis}) (or equivalently
(\ref{eq: quant17})) is called ''hypotheses of molecular chaos''
because we are assuming that the particles are initially uncorrelated.
Furthermore, they are all in the same (one-particle) state at time
$t=0$ and clearly $\Psi_{N,0}\in L^2_s\(\R^{3N}\)$. Thus, (\ref{eq:
quant17bis}) is an admissible state for bosons (while for fermions it
is prevented by the Pauli exclusion principle).
The physical motivation for studying the evolution of factorized wave
functions
is that states close to the ground state of $H_N^Q$ (the eigenvector
associated with the lowest eigenvalue),
which are the most accessible and thus the most interesting states,
can be approximately described
by wave functions like (\ref{eq: quant17bis}) (some of the results
which we are going to discuss in the following sections do not require
strict factorization as in (\ref{eq: quant17bis}); instead asymptotic
factorization of the initial wave function in the sense of
$\mathscr{L}^1$-convergence of the RDM to the $j$-fold product of
one-particle densities
would be sufficient (see Theorem \ref{teoSPOHN} and the discussion below)).\\
Because of the interaction among the particles, the factorization
(\ref{eq: quant17bis}) (or equivalently (\ref{eq: quant17})) is not
preserved by the
time evolution; in other words, the evolved $N$-particle wave function
$\Psi_{N,t}$ is not given by the product of one-particle
wave functions, if $t\neq 0$. All considerations done in the classical
case concerning the mean-field (weak) character of the interaction
hold, then we may expect that, in
the limit of large $N$, the total interaction potential experienced by
a typical particle in the system
can be effectively replaced by an averaged, mean-field, potential, and
therefore that factorization is
approximately, and in an appropriate sense, preserved by the time
evolution. In other words, we
may expect that, in a sense to be made precise,
\begin{eqnarray}\label{eq: quant18}
&&\Psi_{N,t}\approx \psi_t^{\otimes N} \ \ \text{as}\ N\to\infty
\end{eqnarray}
or
\begin{eqnarray}\label{eq: quant18tris}
&&\hat{\rho}_{N,t}\approx \hat{\rho}_t^{\otimes N} \ \ \text{as}\
N\to\infty, \ \ \text{with} \ \ \hat{\rho}_{N,t}=\vert
\Psi_{N,t}><\Psi_{N,t}\vert,\ \ \ \hat{\rho}_t=\vert \psi_t><\psi_t\vert
\end{eqnarray}
for an evolved one-particle wave function $\psi_t$. This asymptotic
factorization
is precisely what is called ''propagation of chaos''. Assuming
(\ref{eq: quant18}), it is simple to derive a self-consistent equation
for the time-evolution of the wave function $\psi_t$. In fact,
(\ref{eq: quant18}) states that, for every fixed time $t$,
the $N$ bosons are independently distributed in space according to the
density $\vert \psi_t(x)\vert^2$. If this is true, the total
potential experienced, for example, by the first particle can be
approximated by
\begin{eqnarray}\label{eq: quant18bis}
&&\frac{1}{N}\sum_{k\geq 2}
\phi(x_1 - x_k)\approx\frac{1}{N}\sum_{k\geq 2}\int \ud y \phi(x_1 -
y)\vert\psi_t(y)\vert^2=\frac{N-1}{N}\( \phi\ast
\vert\psi_t\vert^2\)\approx\(\phi\ast \vert\psi_t\vert^2 \),\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. It follows that, if (\ref{eq: quant18}) holds true,
the one-particle wave function $\psi_t$ must satisfy the self-consistent
equation
\begin{eqnarray}\label{eq: quant19}
&& i\var \pa_t \psi_t = -\frac{\var^2}{2}\Delta \psi_t +\(\phi\ast
\vert\psi_t\vert^2 \)\psi_t
\end{eqnarray}
with initial datum $\psi_0$ given by (\ref{eq: quant17bis}). Equation
(\ref{eq: quant19}) is precisely the Hartree equation
and we have just presented an heuristic argument to explain
how it is related to the propagation of chaos. We observe that the
Hartree equation is a nonlinear Schr\"odinger equation on $\R^3\times
\R^3$. Therefore starting from the linear Schr\"odinger equation
(\ref{eq: quant5}) on $\R^{3N}\times \R^{3N}$, we obtain, for the
evolution of factorized densities, a nonlinear Schr\"odinger equation
on $\R^3\times \R^3$; the nonlinearity in the Hartree equation is a
consequence of the many-body effects
in the linear dynamics.\\
The validity of propagation of chaos (namely, the precise statement
concerning the asymptotic factorization (\ref{eq: quant18}) or
(\ref{eq: quant18tris})) is expressed in terms convergence in
$\mathscr{L}^1\(L^2(\R^{3j})\)$ of the $j$-particle marginal densities
associated with the solution of the Heisenberg equation (\ref{eq:
quant8}) to the $j$-fold product of solutions of the Hartree equation
when $N\to\infty$, namely
\begin{equation}\label{eq: spohn2PRIMA}
\left\| \hat{\rho}_{N,t}^{(j)}-\hat{\rho}_t^{\otimes
j}\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\rightarrow 0,\ \ \text{as}\ N\to\infty,
\end{equation}
$\hat{\rho}_t\in\mathscr{L}^1\(L^2\(\R^{3}\)\)$ solving the Hartree
equation (in the ''Heisenberg form'') (\ref{eq: quant15})
with initial datum $\hat{\rho}_0$ given by (\ref{eq: quant18tris}).
Clearly, $\hat{\rho}_t=\vert \psi_t><\psi_t\vert$, $\psi_t$ solving
the Hartree equation (\ref{eq: quant19}) with initial datum $\psi_0$
given by (\ref{eq: quant18}). \\
We have already remarked that, for fixed $j < N$, the $j$-particle RDM
$\hat{\rho}_{N,t}^{(j)}$
does not contain the full information about the $N$-particle system
described by $\hat{\rho}_{N,t}$. Nonetheless,
$\hat{\rho}_{N,t}^{(j)}$ is sufficient to compute the expectation of
arbitrary observables of the form $A_j\otimes 1_{N-j}$ which depend
non-trivially
on at most $j$ particles (because of the permutation symmetry, it is
not important on which particles
it acts, just that it acts at most on $j$ particles).
Therefore the convergence (\ref{eq: spohn2PRIMA}) implies that:
\begin{eqnarray}\label{eq: convOBSjQUANTUM}
&&\left\langle A_j\otimes 1_{N-j}\right\rangle_{
\Psi_{N,t}}=\texttt{Tr}\(A_j\otimes
1_{N-j}\)\hat{\rho}_{N,t}=\texttt{Tr} A_j\hat{\rho}_{N,t}^{(j)}\
\rightarrow \ \texttt{Tr} A_j\hat{\rho}_{t}^{\otimes j}= \left\langle
A_j\right\rangle_{ \psi_{t}^{\otimes j}}\ \text{as}\
N\to\infty,\nonumber\\
&&
\end{eqnarray}
for each bounded operator $A_j$ acting on $L^2\(\R^{3j}\)$. In other
words, (\ref{eq: spohn2PRIMA}) allows to know the ''macroscopic''
expected value of $j$-particle observables for an $N$-particle
system interacting by a men-field potential.
\subsection{Mean-Field limit for bounded potentials}
\setcounter{equation}{0}
\def\theequation{3.3.\arabic{equation}}
We consider, in this section, the dynamics generated by the mean field
Hamiltonian (\ref{eq: quant2}) under the
assumption that the interaction potential is a bounded operator. We
will assume, in other words,
that $\phi\in L^\infty\(\R^3\)$ (recall that the operator norm of the
multiplication operator $\phi (x_k - x_l)$ is given
by the $L^\infty$-norm of the function $\phi$ ).
In the sequel we will use the notation $\phi_{kl}:=\phi (x_k - x_l)$.
\begin{teo} \label{teoSPOHN}\textnormal{{ \bf [Spohn 1980]}}\ \
Let the pair interaction potential $\phi$ be in $L^\infty\(\R^3\)$ and
the initial state of the system be described by a factorized
$N$-particle wave function $\Psi_{N,0}\in L^2_s\(\R^{3N}\)$, namely
\begin{equation}\label{eq: teoSPOHNwave}
\Psi_{N,0}=\psi_0^{\otimes N},\ \ \ \ \ \ \text{for some}\ \
\psi_0\in L^2\(\R^3\):\ \ \left\| \psi_{0}\right\|_{L^2\(\R^{3}\)}=1.
\end{equation}
This implies that the initial $N$-particle density matrix
$\hat{\rho}_{N,0}\in \mathscr{L}^1\(L^2\(\R^{3N}\)\)$ is given by
\begin{equation}\label{eq: spohn1}
\hat{\rho}_{N,0}=\vert
\Psi_{N,0}><\Psi_{N,0}\vert=\hat{\rho}_0^{\otimes N},\ \ \
\hat{\rho}_0=\vert\psi_0><\psi_0\vert.
\end{equation}
Then, for any fixed $j$,
\begin{equation}\label{eq: spohn2}
\left\| \hat{\rho}_{N,t}^{(j)}-\hat{\rho}_t^{\otimes
j}\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\longrightarrow 0,\ \ as\ N\to\infty,
\end{equation}
where $\hat{\rho}_{N,t}^{(j)}$ solves the BBGKY hierarchy (\ref{eq:
quant10}) with initial datum $\hat{\rho}_{0}^{\otimes j}$
and $\hat{\rho}_t\in\mathscr{L}^1\(L^2\(\R^{3}\)\)$ is the solution of
the Hartree equation (in the ''Heisenberg form'')
\begin{eqnarray}\label{eq: spohn3}
i\var \pa_t
\hat{\rho}_t=\[-\frac{\var^2}{2}\Delta,\hat{\rho}_t\]+\textnormal{\texttt{Tr}}_2\{\[\phi(x-x_2)
,\hat{\rho}_t\otimes\hat{\rho}_t\]\},
\end{eqnarray}
with initial datum $\hat{\rho}_0$. In terms of wave functions, we find
that $\hat{\rho}_t=\vert\psi_t><\psi_t\vert$, $\psi_t$ solving the
Hartree equation (\ref{eq: Hartree}) with initial datum $\psi_0$.
\end{teo}
\begin{dem}
Let $\hat{\rho}^{(N)}
\in \mathscr{L}^1\(L^2\(\R^{3N}\)\) $ be a trace class operator with
kernel $\rho^{(N)}$ invariant under permutations of the labeling.
For fixed $j$, let $\hat{\rho}_{j}^{(N)}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\)$ be
\begin{equation}\label{eq: spohn4}
\hat{\rho}_{j}^{(N)}=\texttt{Tr}_{j+1}\hat{\rho}^{(N)}.
\end{equation}
Then, by considering the time-evolution
$\hat{\rho}^{(N)}(t)=e^{-\frac{i}{\var}H_N^Q
t}\hat{\rho}^{(N)}e^{\frac{i}{\var}H_N^Q t}$, $H_N^Q$ defined in
(\ref{eq: quant2}), it is also invariant under permutations of the
labeling and
the $j$-particle trace class operator
$\hat{\rho}_{j}^{(N)}(t)=\texttt{Tr}_{j+1}\hat{\rho}^{(N)}(t)$
satisfies the differential equation
\begin{eqnarray}\label{eq: spohn5}
i\var \pa_t
\hat{\rho}_j^{(N)}(t)&&=\[\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\)+\frac{1}{2N}\sum_{k\neq l}^j\phi_{k
l},\hat{\rho}_j^{(N)}(t)\]+\nonumber\\
&&+\(\frac{N-j}{N}\)\sum_{k=1}^{j}\texttt{Tr}_{j+1}\left\{\[\phi_{k j+1}
,\hat{\rho}_{j+1}^{(N)}(t)\]\right\}.
\end{eqnarray}
This is what we previously called BBGKY hierarchy (see (\ref{eq:
quant10})) as it can be seen by using the ''compact'' notation
\begin{eqnarray}\label{eq: spohn6}
\pa_t
\hat{\rho}_j^{(N)}(t)=-\frac{i}{\var}\[\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\),\hat{\rho}_j^{(N)}(t)\]-\frac{i}{\var}T_{N,j}^Q\hat{\rho}_j^{(N)}(t)-\frac{i}{\var}\(\frac{N-j}{N}\)C_{j,j+1}^Q\hat{\rho}_{j+1}^{(N)}(t),\nonumber\\
&&
\end{eqnarray}
$T_{N,j}^Q$ and
$C_{j,j+1}^Q$ as in (\ref{eq: quant11}) and (\ref{eq: quant12}) respectively.
Let $S_j^{(N)}(t)$ is the flow associated with the equation:
\begin{eqnarray}\label{eq: spohn6bis}
\pa_t
\hat{\rho}_j^{(N)}(t)=-\frac{i}{\var}\[H_{N,j}^Q,\hat{\rho}_j^{(N)}(t)\],
\end{eqnarray}
with
\begin{eqnarray}\label{eq: HAMaJ}
H_{N,j}^Q:=\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\)+T_{N,j}^Q.
\end{eqnarray}
Thus, $S_{j}^{(N)}(t)\hat{\rho}_{j}=e^{-\frac{i}{\var}H_{N,j}^Q
t}\hat{\rho}_{j}e^{\frac{i}{\var}H_{N,j}^Q t}$, for any
$\hat{\rho}_{j}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\) $. By the Duhamel
formula, the solution of (\ref{eq: spohn6}) can be written as
\begin{eqnarray}\label{eq: spohn7prima}
\hat{\rho}_j^{(N)}(t)=S_{j}^{(N)}(t)\hat{\rho}_j^{(N)}+
\(\frac{N-j}{N}\)\(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\
S_j^{(N)}(t-t_1)C_{j,j+1}^Q \hat{\rho}_{j+1}^{(N)}(t_1).
\end{eqnarray}
Iterating the integral equation (\ref{eq: spohn7prima}),
we obtain the
series
\begin{eqnarray}\label{eq: spohn7}
\hat{\rho}_j^{(N)}(t)&&=S_{j}^{(N)}(t)\hat{\rho}_j^{(N)}+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud
t_n\dots\ud t_1\
S_j^{(N)}(t-t_1)\(\frac{N-j}{N}\)\(-\frac{i}{\var}\)C_{j,j+1}^Q\dots\nonumber\\
&&\qquad \quad\dots
\(\frac{N-j-n+1}{N}\)\(-\frac{i}{\var}\)C_{j+n-1,j+n}^Q\
S_{j+n}^{(N)}(t_n) \hat{\rho}_{j+n}^{(N)}.
\end{eqnarray}
Let $\left\|\cdot \right\|_j$ denote the trace norm in
$\mathscr{L}^1\(L^2\(\R^{3j}\)\)$. Since $S_j^{(N)}(t)$ preserves the
$\left\|\cdot \right\|_j$ norm (because $H_{N,j}^Q$ is a self-adjoint
operator on $L^2\(\R^{3j}\)$), by the expression (\ref{eq: quant12})
for $C_{j,j+1}^Q$, it is easy to verify that the $n$-th term of the
series (\ref{eq: spohn7}) is bounded by
\begin{eqnarray}\label{eq: spohn8}
\frac{t^n}{n!}\ j(j+1)\dots
(j+n-1)\(\frac{2\left\|\phi\right\|_{L^\infty}}{\var}\)^n
\left\|\hat{\rho}_{j+n}^{(N)}\right\|_{j+n}.
\end{eqnarray}
If one assumes
\begin{eqnarray}\label{eq: spohn9}
\mathbf{P1)}\ \left\|\hat{\rho}_{j}^{(N)}\right\|_{j}\leq a^j\ \ \
\text{for any}\ j,
\end{eqnarray}
then the series (\ref{eq: spohn7}) converges in trace norm for $\vert
t\vert\leq t_0$ with
$t_0<\frac{\var}{4\left\|\phi\right\|_{L^\infty}a}$.
For any
$\hat{\rho}_{j}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\) $, let
$S_{j}(t)\hat{\rho}_{j}=e^{-\frac{i}{\var}H_{j}t}\hat{\rho}_{j}e^{\frac{i}{\var}H_j
t}$,
where $H_{j}=\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\)$ is the
$j$-particle free Hamiltonian. We note that
\begin{eqnarray}\label{eq: spohn8opTbound}
\left\|T_{N,j}^Q\hat{\rho}_{j}^{(N)}\right\|\leq
\frac{j(j-1)}{2N}\|[\phi, \hat{\rho}_{j}^{(N)}]\|_{j}\leq
\frac{j(j-1)}{N}\left\|\phi\right\|_{L^\infty}
\left\|\hat{\rho}_{j}^{(N)}\right\|_{j},\nonumber\\
&&
\end{eqnarray}
then by Property $\mathbf{P1)}$ we find
\begin{eqnarray}\label{eq: spohn8opTboundII}
\left\|T_{N,j}^Q
\right\|\leq \frac{j(j-1)}{N}\left\|\phi\right\|_{L^\infty}a^j
\ \to 0\ \ \ \text{as}\ \ N\to\infty,
\end{eqnarray}
where $\left\|\cdot\right\|$ is the operator norm on
$\mathscr{L}^1\(L^2\(\R^{3j}\)\)$.
We note that, for any $\hat{\rho}_{j}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\)$,
\begin{eqnarray}\label{eq: nuova}
\left\|S_j^{(N)}(t)\hat{\rho}_{j}-S_{j}(t)\hat{\rho}_{j}\right\|_j&&\leq\frac{1}{\var}\int_0^t\ud \tau \left\|S_{j}(t-\tau)T_{N,j}^Q\hat{\rho}_{j}(\tau)\right\|_j\leq
\nonumber\\
&&\leq\frac{1}{\var}\int_0^t\ud \tau
\left\|T_{N,j}^Q\hat{\rho}_{j}(\tau)\right\|_j,
\end{eqnarray}
where we use that $S_j(t)$ preserves the trace norm. Then, from
(\ref{eq: spohn8opTboundII}), it follows that
\begin{eqnarray}\label{eq: spohn10ante}
\lim_{N\to\infty}\left\|S_j^{(N)}(t)-S_{j}(t)\right\|=0.
\end{eqnarray}
If one assumes
\begin{eqnarray}\label{eq: spohn10}
\mathbf{P2)}\
\lim_{N\to\infty}\left\|\hat{\rho}_{j}^{(N)}-\hat{\rho}_{j}\right\|_{j}=0,
\end{eqnarray}
for some $\hat{\rho}_{j}\in \mathscr{L}^1\(L^2\(\R^{3j}\)\)$,
then by (\ref{eq: spohn7}) and (\ref{eq: spohn10ante}), it follows
that $\hat{\rho}_{j}^{(N)}(t)$ converges as $N\to\infty$ in trace norm
to
\begin{eqnarray}\label{eq: spohn11}
\hat{\rho}_{j}(t)&&=\sum_{n=0}^{+\infty}\int_{0\leq t_n\leq \dots \leq
t_1\leq t}\ud t_n\dots\ud t_1\
S_{j}(t-t_1)\(-\frac{i}{\var}\)C_{j,j+1}^Q\dots\nonumber\\
&&\qquad \qquad\dots \(-\frac{i}{\var}\)C_{j+n-1,j+n}^Q\ S_{j+n}(t_n)
\hat{\rho}_{j+n},
\end{eqnarray}
for $\vert t\vert\leq t_0$. We note that the $n$-th term of the above
series in bounded in trace norm by (\ref{eq: spohn8}), then for short
times $\vert t\vert\leq t_0$ we are ensures that (\ref{eq: spohn11})
converges in trace norm.\\
Let $\hat{\rho}^{(N)}$ be a density matrix. Then
$\left\|\hat{\rho}_{j}^{(N)}(t)\right\|_{j}=\left\|\hat{\rho}_{j}^{(N)}\right\|_{j}$ by preservation of positivity and trace. Therefore, if for the initial state the bound $\mathbf{P1)}$ is satisfied, it remains valid for all times, and the argument just given can be iterated to prove convergence of $\hat{\rho}_{j}^{(N)}(t)$ to $\hat{\rho}_{j}(t)$ as $N\to\infty$ for all times. Furthermore, $\hat{\rho}_{j}(t)$ is uniquely determined for all times because by iteration we prove that (\ref{eq: spohn11}) converges in trace norm for all times.
\\
One checks that for the particular initial state $\hat{\rho}_{N,0}$ in
(\ref{eq: spohn1}) the conditions $\mathbf{P1)}$ and $\mathbf{P2)}$
are satisfied with $a=1$ and $\hat{\rho}_j=\hat{\rho}_0^{\otimes j}$.
Therefore, we can claim that the solution $\rho_{N,t}^{(j)}$ of the
BBGKY hierarchy (\ref{eq: spohn6}) with initial datum
$\hat{\rho}_0^{\otimes j}$ converges in trace norm to the unique
$j$-particle density matrix $\hat{\rho}_{j}(t)$ identified by the
series (\ref{eq: spohn11}) with $
\hat{\rho}_{j+n}=\hat{\rho}_0^{\otimes j+n}$ \ $\forall\ n$.
Differentiating (\ref{eq: spohn11}) with respect to $t$, one obtains
the limiting hierarchy of equations
\begin{eqnarray}\label{eq: spohn12}
i\var \pa_t
\hat{\rho}_j(t)&&=\[\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\),\hat{\rho}_j(t)\]+
C_{j,j+1}^Q\ \hat{\rho}_{j+1}(t),
\end{eqnarray}
whose unique trace class solution is $\hat{\rho}_{j}(t)$
with initial datum $\hat{\rho}_0^{\otimes j}$.
Moreover, (\ref{eq: spohn12}) preserves the factorization property for
all $t$ according to the Hartree equation (in the Heisenberg form)
(\ref{eq: spohn3}). This ensures the validity of propagation of chaos,
namely $\hat{\rho}_j(t)=\hat{\rho}_t^{\otimes j}$, $\hat{\rho}_t$
solving the nonlinear Heisenberg equation (\ref{eq: spohn3}) with
initial datum $\hat{\rho}_0=\vert \psi_0><\psi_0\vert$. Then,
$\hat{\rho}_t=\vert \psi_t><\psi_t\vert$, $\psi_t$ solving the Hartree
equation (\ref{eq: Hartree}) with initial datum $\psi_0$.
\end{dem}
\begin{rem} \label{Remark S.1}:
\textnormal{By looking at the the proof above it is clear that in
order to let Theorem \ref{teoSPOHN} to hold there is no need of
strict factorization of the initial datum as in (\ref{eq: spohn1}).
Instead asymptotic factorization in the sense of
\begin{eqnarray}\label{eq: REMS.1}
\lim_{N\to\infty}\left\|\hat{\rho}_{N,0}^{(j)}-\hat{\rho}_{0}^{\otimes
j}\right\|_{j}=0
\end{eqnarray}
would be sufficient. We remind that (\ref{eq: REMS.1}) is a reasonable
''physical'' condition because states close to the ground state of
$H_N^Q$,
which are the most accessible and thus the most interesting states,
can be approximately described
by factorized wave functions, and then, by factorized density matrices}.
\end{rem}
\begin{rem} \label{Remark S.2}:
\textnormal{In proving the
convergence of the series (\ref{eq: spohn7}) to (\ref{eq: spohn11})
the crucial tools have been the boundedness of the operator
$T_{N,j}^Q:\mathscr{L}^1\(L^2\(\R^{3j}\)\)\to\mathscr{L}^1\(L^2\(\R^{3j}\)\)$
(see (\ref{eq: spohn8opTbound})) and property $\mathbf{P1)}$ for the
RDM. In particular, the bound (\ref{eq: spohn8opTboundII}) on the
operator norm of $T_{N,j}^Q$ provides the rate of convergence to the
Hartree dynamics by means of (\ref{eq: nuova}).}
\textnormal{Then, by observing that the estimate obtained in (\ref{eq:
nuova}) is not uniform with respect to $\var$ and it fails when
$\var\to 0$, it follows that the error in approximating the
$N$-particle dynamics with the limiting one is diverging when $\var\to
0$ (for short times it is of the form $\frac{C_j}{N}e^{Ct/\var}$)}.
\end{rem}
In the next sections we will discuss some other results concerning the
mean-field limit starting from factorized initial datum as in
(\ref{eq: teoSPOHNwave}), both for bounded interactions and for the
Coulomb potential and we will see that considerations done in Remark
\ref{Remark S.2} still hold.
This means that
all results concerning the mean-field limit exhibit an error in
approximating the $N$-particle dynamics by the limiting one which is
not uniform with respect to $\var$ and diverging when $\var\to 0$.
This is a quite surprising feature because it seems that, roughly
speaking, the accuracy of the approximation depends on ''how much''
the system can be considered quantum or not and, except for fermionic
systems, there are no reasonable motivations for that. In fact, we
will see in Chapter 3 that in the fermionic case it is quite natural
to look at a joint limit: $N\to\infty$ and $\var\to 0$ as in \cite{NS}
and \cite{fermioni}. On the contrary, in looking at systems of
undistiguishable particles or even bosonic systems
the fact that the mean-field limit and the semiclassical approximation
seems to be so strictly connected is an open problem (except for
specific scalings of the potential as in \cite{Graffi}). Furthermore,
in the classical case (see Chapter 1) everything works, so it is quite
natural to ask if, at least for quantum systems having a reasonable
classical analogue, it is possible to realize the limit $N\to\infty$
uniformly with respect to $\var$. This is the main motivation of our
research and in the next section we will focus on that topic,
discussing some known results and presenting what we did in this
perspective.
\subsubsection{An alternative approach}
From the proof of Theorem \ref{teoSPOHN} presented above, we notice
that the expansion of the BBGKY hierarchy
in (\ref{eq: spohn7}) is much more involved than the corresponding
expansion (\ref{eq: spohn11}) of the infinite hierarchy
(\ref{eq: spohn12}). It turns out that it is possible to avoid the
expansion of the BBGKY hierarchy making use
of a simple compactness argument; this will be especially important
when dealing with singular potentials.
In the following we explain the main steps of this alternative proof
to Theorem \ref{teoSPOHN}. Then,
in the next section, we will illustrate how to extend it to potentials
with a Coulomb singularity.
The idea, which was first presented in \cite{BGM}, \cite{EY}, \cite{BEGMY},
consists in characterizing the limit of the RDM
$\hat{\rho}_{N,t}^{(j)}$
as the unique solution to the infinite hierarchy of equations
(\ref{eq: spohn12}); combined with the compactness,
this information provides a proof of Theorem \ref{teoSPOHN}. \\More
precisely, the proof is divided into three
main steps:
$i)$\ First of all, one shows the compactness of the sequence
$\{\hat{\rho}_{N,t}^{(j)}\}_{j= 1}^N$ with respect to an
appropriate weak topology.
$ii)$\ Then, one proves that an arbitrary limit point
$\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}$ of the sequence
$\{\hat{\rho}_{N,t}^{(j)}\}_{j= 1}^N$
is a solution to the infinite hierarchy (\ref{eq: spohn12}) (one
proves, in other words, the convergence
to the infinite hierarchy).
$iii)$\ Finally, one shows the uniqueness of the solution to the
infinite hierarchy
(\ref{eq: spohn12}).\\
We have already observed that the factorized family
$\{\hat{\rho}_{t}^{\otimes j}\}_{j\geq 1}$
is a solution of the infinite hierarchy with factorized initial datum
$\hat{\rho}_{0}^{\otimes j}$. In particular, if $\hat{\rho}_{0}=\vert
\psi_0><\psi_0\vert$, as in the present case, we find that
$\hat{\rho}_{t}=\vert \psi_t><\psi_t\vert$, $\psi_t$ solving the
Hartree equation (\ref{eq: Hartree}). Then, by proving that the
solution of the infinite hierarchy is unique, we are guaranteed that
it is factorized according to the solution of the Hartree equation.
Therefore, by $ii)$, it follows immediately that
$\hat{\rho}_{N,t}^{(j)}\to \hat{\rho}_{t}^{\otimes j}=\(\vert
\psi_t><\psi_t\vert\)^{\otimes j}$ as $N\to\infty$ (at first only in
the weak topology with respect to which we have compactness; since the
limit is an
orthogonal rank one projection, it is however simple to check that
weak convergence implies strong
convergence, in the sense (\ref{eq: spohn2})). Next, we discuss these
three main steps (compactness, convergence,
and uniqueness) in some more details in order to show that, even
following this approach, the estimates that ensure the convergence are
not uniform with respect to $\var$ and they fail if $\var\to 0$. \\
\underline{\textbf{Compactness}}: By knowing that, for any $j$ and
$N$, $\left\|\hat{\rho}_{N,t}^{(j)}\right\|_{\mathscr{L}^1}=1$ for
fixed $t$, thanks to standard abstract and compactness results of
functional analysis we prove that the sequence
$\Gamma_{N,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$ is compact with
respect to a suitable topology. More precisely, for an arbitrary fixed
$T>0$, we denote by
$\mathcal{C}\([0, T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ the space of
functions of $t \in [0, T]$ with values in
$\mathscr{L}^1\(L^2(\R^{3j})\)$ which are continuous in time with
respect to a suitable metric $\eta_j$ on
$\mathscr{L}^1\(L^2(\R^{3j})\)$ (it can be constructed explicitly in
such a way that the topology generated by $\eta_j$ is equivalent to
the weak*-topology of $\mathscr{L}^1\(L^2(\R^{3j})\)$). By $\eta_j$ we
can easily define a metric $\hat{\eta}_j$ on $\mathcal{C}\([0,
T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ and we consider the topology
$\tau_{prod}$ on
$\bigoplus_{j\geq 1}\mathcal{C}\([0,
T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ given by the product of the
topologies generated by the metrics $\hat{\eta}_j$ on
$\mathcal{C}\([0, T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$. The topology
$\tau_{prod}$ is precisely the topology with respect to which we prove
compactness of the sequence $\{\Gamma_{N,t}\}_{N\in \N}$ and this is
equivalent to the following
\begin{prop}\label{PROPcompactness}
Fix an arbitrary time $T > 0$.
For every sequence $\{M_m\}_{m\in\N}$ there exists a subsequence
$\{N_m\}_{m\in\N }\subset\{M_m\}_{m\in\N}$
and a limit point
$\Gamma_{\infty,t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}$ for
$\Gamma_{N_m,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^{N_m}$ such that
\begin{eqnarray}\label{eq: BenjiALT0}
\hat{\rho}_{\infty,t}^{(j)}\geq 0,\ \ \
\textnormal{\texttt{Tr}}\hat{\rho}_{\infty,t}^{(j)}\leq 1,\ \ \forall\
j\geq 1,
\end{eqnarray}
$\hat{\rho}_{\infty,t}^{(j)}$ (for any $j$) is symmetric with respect
to permutations of the labeling.
\end{prop}
Let $\mathcal{K}_j \equiv \mathcal{K}\(L^2(\R^{3j})\)$ be the space of
compact operators on $L^2(\R^{3j})$, equipped with the
operator norm. The claim of Proposition \ref{PROPcompactness} is
equivalent to the affirm that, passing to a subsequence,\\
\emph{
For every fixed $j\geq 1$ and for every fixed compact operator
$J^{(j)}\in \mathcal{K}_j$, }
\begin{eqnarray}\label{eq: BenjiALT1}
\textnormal{\texttt{Tr}}\
J^{(j)}\(\hat{\rho}_{N,t}^{(j)}-\hat{\rho}_{\infty,t}^{(j)}\)
\rightarrow 0\ \ \ as\ \ N\to\infty
\end{eqnarray}
\emph{uniformly in $t$ for $t \in [0, T]$.}
\vspace{0.5cm}
\underline{\textbf{Convergence}}: The second main step consists in
characterizing the limit points of the (compact) sequence
$\Gamma_{N,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$ as solutions to the
infinite hierarchy of equations (\ref{eq: spohn12}) with initial datum
$\hat{\rho}_0^{\otimes j}$, $\hat{\rho}_0=\vert \psi_0><\psi_0\vert$.\\
\begin{prop}\label{convALT}
Suppose that $\phi\in L^\infty(\R^3)$ such that $ \phi(x)\to 0$ as
$|x|\to \infty$.
Assume moreover
that $\Gamma_{\infty,t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}\in
\bigoplus_{j\geq 1}\mathcal{C}\([0,
T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ is a limit point of the sequence
$\Gamma_{N,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$ in the sense
(\ref{eq: BenjiALT1}). Then \begin{eqnarray}\label{eq: BenjiALT2}
\hat{\rho}_{\infty,t}^{(j)}=S_{j}(t)\hat{\rho}_{\infty,0}^{(j)}+
\(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\ S_j(t-t_1)C_{j,j+1}^Q\
\hat{\rho}_{\infty,t_1}^{(j+1)}.
\end{eqnarray}
for all $j\geq 1$, with
$\hat{\rho}_{\infty,0}^{(j)}=\hat{\rho}_{N,0}^{(j)}=\hat{\rho}_{0}^{\otimes
j}$. Here $S_{j}(t)$ is the flow associated with the $j$-particle free
dynamics and $C_{j,j+1}^Q$ is defined as in (\ref{eq: quant12}).
Therefore equation (\ref{eq: BenjiALT2}) evaluated for all $j\geq 1$
gives rise precisely to a solution of the Hartree hierarchy (\ref{eq:
spohn12}) with factorized initial datum.
\end{prop}
Note that in Proposition \ref{convALT} we assume the potential to
vanish at infinity. This condition, which
was not required in Theorem \ref{teoSPOHN}, is not essential but it
simplifies the proof and it is also satisfied by the Coulomb
interaction for which the derivation of the Hartree equation has been
proven (e.g. \cite{EY}) by following the present strategy (together
with crucial technical tools that are necessary to deal with the
singularity of the potential ).\\
\begin{dem}
Passing to a subsequence we can assume that
$\Gamma_{N,t}\to\Gamma_{\infty,t}$ as $N\to\infty$, in the sense
(\ref{eq: BenjiALT1}); this implies immediately that
$\hat{\rho}_{\infty,0}^{(j)}=\hat{\rho}_{N,0}^{(j)}=\hat{\rho}_{0}^{\otimes
j}$. To prove (\ref{eq: BenjiALT2}), on the other
hand, it is enough to show that for every fixed $j\geq 1$, and for
every fixed $J^{(j)}$ from a dense subset
of $\mathcal{K}_{j}$,
\begin{eqnarray}\label{eq: BenjiALT3}
\texttt{Tr}J^{(j)}\hat{\rho}_{\infty,t}^{(j)}=\texttt{Tr}J^{(j)}S_{j}(t)\hat{\rho}_{\infty,0}^{(j)}+
\(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\
\hat{\rho}_{\infty,t_1}^{(j+1)}.
\end{eqnarray}
To demonstrate (\ref{eq: BenjiALT3}), we start from the BBGKY
hierarchy (\ref{eq: spohn6}) which leads to
\begin{eqnarray}\label{eq: BenjiALT4}
\texttt{Tr}J^{(j)}\hat{\rho}_{N,t}^{(j)}&&=\texttt{Tr}J^{(j)}S_{j}(t)\hat{\rho}_{N,0}^{(j)}+
-\frac{i}{\var}\int_{0}^{t}\ud t_1\
\texttt{Tr}J^{(j)}S_j(t-t_1)T_{N,j}^Q
\hat{\rho}_{N,t_1}^{(j)}+\nonumber\\
&&\(-\frac{i}{\var}\)\frac{(N-j)}{N}\int_{0}^{t}\ud t_1\
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\ \hat{\rho}_{N,t_1}^{(j+1)}.
\end{eqnarray}
Since, by assumption, the l.h.s. and the first term on the r.h.s. of
the last equation converge, as
$N\to\infty$, to the l.h.s. and, respectively, to the first term on
the r.h.s. of (\ref{eq: BenjiALT3}) (for every compact
operator $J^{(j)}$), (\ref{eq: BenjiALT2}) follows if we can prove that
\begin{eqnarray}\label{eq: BenjiALT5}
-\frac{i}{\var}\int_{0}^{t}\ud t_1\
\texttt{Tr}J^{(j)}S_j(t-t_1)T_{N,j}^Q \hat{\rho}_{N,t_1}^{(j)}\to 0
\end{eqnarray}
and that
\begin{eqnarray}\label{eq: BenjiALT6}
&&\(-\frac{i}{\var}\)\frac{(N-j)}{N}\int_{0}^{t}\ud t_1\
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\
\hat{\rho}_{N,t_1}^{(j+1)}\to \(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\
\hat{\rho}_{\infty,t_1}^{(j+1)}\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. Eq. (\ref{eq: BenjiALT5}) follows because, by the
expression (\ref{eq: quant11}) of $T_{N,j}^Q$, we have
\begin{eqnarray}\label{eq: BenjiALT7}
&&\left\vert \frac{i}{\var} \texttt{Tr}J^{(j)}S_j(t-t_1)T_{N,j}^Q
\hat{\rho}_{N,t_1}^{(j)}\right\vert\leq \frac{1}{\var 2N}\sum_{k\neq
l}^{ j}\left\vert \texttt{Tr}J^{(j)}S_j(t-t_1)\[\phi(x_k -
x_l),\hat{\rho}_{N,t_1}^{(j)}\]\right\vert\leq\nonumber\\
&&\qquad\leq
\frac{j^2}{\var
N}\left\|J^{(j)}\right\|\left\|\phi\right\|\texttt{Tr}\left\vert\hat{\rho}_{N,t_1}^{(j)}\right\vert=
\frac{j^2}{\var N}\left\|J^{(j)}\right\|\left\|\phi\right\|\to 0
\end{eqnarray}
because the product $\left\|J^{(j)}\right\|\left\|\phi\right\|$ is
finite and uniformly bounded with respect to $N$
($\left\|J^{(j)}\right\|$ and $\left\|\phi\right\|$ being the operator
norms of $J^{(j)}$ and of the multiplication operator $\phi$). To
prove (\ref{eq: BenjiALT6}) one can use a similar argument, combined
with the
observation that, by the expression (\ref{eq: quant12}) of $C_{j,j+1}^Q $,
\begin{eqnarray}\label{eq: BenjiALT8}
&& \frac{i}{\var}
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\(\hat{\rho}_{N,t_1}^{(j+1)}-\hat{\rho}_{\infty,t_1}^{(j+1)}\)=\nonumber\\
&& = \frac{i}{\var}\sum_{1\leq k\leq j}
\texttt{Tr}\[\(J^{(j)}S_j(t-t_1)\),\phi(x_k -
x_{j+1})\]\(\hat{\rho}_{N,t_1}^{(j+1)}-\hat{\rho}_{\infty,t_1}^{(j+1)}\)\to
0,\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. This does not follow directly from the assumption
that $\Gamma_{N,t}\to \Gamma_{\infty,t}$ in the sense (\ref{eq:
BenjiALT1}) because the operator $\[\(J^{(j)}S_j(t-t_1)\),\phi(x_k -
x_{j+1})\]$ is not compact on $L^2(\R^{3(j+1}))$. Instead it is
necessary to apply an approximation argument which is made simpler by
the assumption that $\phi (x)\to 0$ as $|x|\to \infty$ (that is the
reason for which we did it).
The details of this approximation
argument can be found, for example, in
\cite{GP}.
\end{dem}
\underline{\textbf{Uniqueness}}: to conclude the proof of Theorem
\ref{teoSPOHN}, we still have to prove the uniqueness of the
solution to the infinite (Hartree) hierarchy (\ref{eq: BenjiALT2}).
\begin{prop}
Fix $\Gamma_{\infty, 0}=\{\hat{\rho}_{\infty,0}^{(j)}\}_{j\geq 1}\in
\bigoplus_{j\geq 1}\mathscr{L}^1 \(L^2(\R^3)\)$. Then there exists at
most one solution $\Gamma_{\infty,
t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}\in \bigoplus_{j\geq
1}\mathcal{C}\([0,T],\mathscr{L}^1 \(L^2(\R^3)\)\)$
to the infinite (Hartree) hierarchy (\ref{eq: BenjiALT2}) such that
$\hat{\rho}_{\infty,t}^{(j)}\vert_{t=0}=\hat{\rho}_{\infty,0}^{(j)}$
and
$\textnormal{\texttt{Tr}}\left\vert\hat{\rho}_{\infty,t}^{(j)}\right\vert\leq
1$
for all $j\geq 1$ and all $t \in [0, T]$.
\end{prop}
\begin{dem}
The proof is exactly the same we did in proving Theorem
\ref{teoSPOHN}. Indeed, we write the solution of the Hartree hierarchy
by iterating the Duhamel formula (\ref{eq: BenjiALT2}) and we observe
that the series we obtain is uniformly bounded in trace norm by a
geometric series converging for short times $t1$) and taking the
trace, one obtains estimates which are uniform with respect to $N$.
Such estimates are crucial in proving
the uniqueness of the solution of the Hartree hierarchy and even the
convergence of the BBGKY hierarchy to the limiting one.
In particular, they provide the rate of convergence to the Hartree
dynamics in terms of the number of particles $N$ and indipendently of
$\var$. Nevertheless,
by looking at the explicit computations, we find a factor $1/\var$ in
front of the interaction potential, thus we have again diverging
estimates when $\var\to 0$.\\\\
From now on, we will focus on the case of smooth pair interaction
potential, primarily, because in this case both the quantum and the
classical mean-field limit have been rigorously established, therefore
it is quite reasonable to look at that situation in investigating the
connection between mean-field limit and semiclassical approximation
(which we are going to discuss in Chapter 3 and 4). On the other side,
we will see that for our purposes we need to deal with a smooth
potential (see Chapter 4).
\section{Mean-field limit VS Semiclassical Approximation}
In this chapter we discuss the problem of ''connecting'' mean-field
limit and semiclassical approximation which, as we saw previously,
emerges quite naturally from the analysis of the quantum mean-field
limit results. If one wants to deal with the classical and quantum
case simultaneously, it is natural to
work in the classical phase space by using the so called ''Wigner formalism''.
\subsection{The Wigner formulation}
\setcounter{equation}{0}
\def\theequation{4.1.\arabic{equation}}
By the Heisenberg uncertainty principle, it follows that it is not
possible to determine simultaneously the position
and the momentum of a quantum particle, thus the concept of classical
phase space density
does not generalize directly to quantum mechanics. Nevertheless one
can define a substitute for
it, namely the Wigner transform. For any wave function $\psi\in
L^2\(\R^{d}\)$ we define the Wigner transform
of $\psi$ as
\begin{equation}\label{eq: WignerUNA}
f^\var_\psi\left(x, v\right)=(2\pi)^{-d}\int_{\R^{d}} \ud y\
e^{iy\cdot v}\overline{\psi}\left(x+\frac{\var
y}{2}\right)\psi\left(x-\frac{\var y}{2}\right),
\end{equation}
and we still interpret it as ''quantum phase space density'' (see
\cite{Wigner} ).
It is easy to check that $f^\var_\psi$ is always real but in general
is not positive (thus it cannot be
the density of a positive measure - in coincidence with the Heisenberg
principle). However, its
marginals reconstruct the position and momentum space densities, as
the following formulas
can be easily checked:
\begin{equation}\label{eq: WignerUNAerd}
\int f^\var_\psi(x, v)dv = |\psi(x)|^2,\ \ \ \ \ \int f^\var_\psi(x,
v)dx = |\hat{\psi}(v)|^2
\end{equation}
$\hat{\psi}(v)$ being the Fourier transform of $\psi$, namely, by
integrating versus the velocity variable we obtain the quantum spatial
probability density and by integrating with respect to the position
variable we find the velocity (or momentum) probability density.
In particular
\begin{equation}\label{eq: WignerUNAerd1}
\int f^\var_\psi(x, v)dvdx = 1
\end{equation}
for normalized wave functions.
More generally, if $J(x, v)$ is a classical phase space observable,
the scalar product
\begin{equation}\label{eq: WignerUNAerd2}
\left\langle J, f^\var_\psi\right\rangle=\int J(x,v)f^\var_\psi(x, v)dvdx
\end{equation}
can be interpreted as the expected value of $J$ in state described by
$\psi$. Recall that ''honest'' quantum
mechanical
observables are self-adjoint operators $O$ on $L^2(\R^d)$ and their
expected value is
given by
\begin{equation}\label{eq: WignerUNAerd3}
\left\langle O\right\rangle_\psi= \int \bar{\psi}(x)\(O\psi\)(x)\ud x
\end{equation}
For a large class of observables there is a natural relation between
observables $O$ and their
phase space representations (called symbols) that are functions on the
phase space like $J(x, v)$.
For example, if $J$ depends only on $x$ or only on $v$, then the
corresponding operator is just the
standard quantization, i.e.
\begin{equation}\label{eq: WignerUNAerd4}
\int J(x)f^\var_\psi(x, v)dxdv = \left\langle \psi,J\psi\right\rangle
\end{equation}
where $J$ is a multiplication operator on the right hand side,
\begin{equation}\label{eq: WignerUNAerd5}
\int J(v)f^\var_\psi (x, v)dxdv = \left\langle \psi,J(-i\var
\nabla)\psi\right\rangle
\end{equation}
and similar relations hold for the Weyl quantization of any symbol $J(x, v)$.
We also remark that the map $\psi\to f^\var_\psi$ is invertible, i.e.
one can fully reconstruct the
wave function from its Wigner transform. On the other hand, not every
real function of two
variables $(x, v)$ is the Wigner transform of some wave function.\\
The correspondence between wave functions and their Wigner transform
can be easily rephrased for density matrices. Indeed, if
$\hat{\rho}=\vert\psi><\psi\vert$ for some $\psi\in L^2\(\R^d\)$, then
formula (\ref{eq: WignerUNA}) can be rewritten as
\begin{equation}\label{eq: WignerUNAconRHO}
f^\var_\rho\left(x, v\right)=(2\pi)^{-d}\int_{\R^{3}} \ud y\
e^{iy\cdot v}\rho\left(x+\frac{\var y}{2}, x-\frac{\var y}{2}\right),
\end{equation}
where $\rho(x,y)=\bar{\psi}(x)\psi(y)$ is the integral kernel of $\hat{\rho}$.
Furthermore, formula (\ref{eq: WignerUNAconRHO}) holds for any density
matrix $\hat{\rho}\in \mathscr{L}^1\(L^2\(\R^d\)\)$, even for those
which are associated with mixed states and (\ref{eq: WignerUNAerd1})
holds because of positivity and trace norm normalization of the
density matrix. Vice versa, starting from a quantum system whose state
is described by a Wigner function $f^\var\left(x, v\right)$, it is
possible to compute the corresponding density matrix (actually, its
integral kernel) by the Weyl quantization rule
\begin{equation}\label{eq: Weyl}
\rho_{f^\var}\left(x, y\right)=(2\pi)^{-d}\int_{\R^{d}} \ud v\
e^{i\frac{v}{\var}\cdot (x-y)}f^\var\left(\frac{x+ y}{2}, v\right).
\end{equation}
Therefore the Wigner transform and the Weyl quantization rule provide
an invertible map $\hat{\rho}\leftrightarrow f^\var_\rho$ between
density matrices and Wigner functions and it is simple to check that
\begin{equation}\label{eq: propDENSisometry}
\left\|\rho\right\|_{L^2\(\R^d\times\R^d\)}=
\left\|f^\var_{\rho}\right\|_{L^2\(\R^d\times\R^d\)}.
\end{equation}
This is particularly meaningful because for any density matrix
$\hat{\rho}$ we have
\begin{equation}\label{eq: propDENS1}
\hat{\rho}\geq 0,\ \hat{\rho}\in \mathscr{L}^1\(L^2\(\R^d\)\),\
\text{with}\ \
\left\|\hat{\rho}\right\|_{\mathscr{L}^1\(L^2\(\R^d\)\)}=1\Rightarrow\left\|\hat{\rho}\right\|_{\mathscr{L}^2\(L^2\(\R^d\)\)}\leq
1,
\end{equation}
where $\mathscr{L}^2\(L^2\(\R^d\)\)$ is the Hilbert space of
Hilbert-Schmidt operators on $L^2\(\R^d\)$ and for any operator
$\Gamma\in\mathscr{L}^2\(L^2\(\R^d\)\)$ with kernel
$\gamma=\gamma(x,y)$ we find
\begin{equation}\label{eq: propDENS2}
\left\|\Gamma\right\|_{\mathscr{L}^2\(L^2\(\R^d\)\)}=\left\|\gamma\right\|_{L^2\(\R^d\times\R^d\)}.
\end{equation}
Therefore by (\ref{eq: propDENS1}), (\ref{eq: propDENS2}) and by
(\ref{eq: propDENSisometry}) it follows that
\begin{eqnarray}\label{eq: propDENS3}
&&\left\|\rho\right\|_{L^2\(\R^d\times\R^d\)}\leq 1\ \ \forall\ \
\text{density matrix}\ \ \hat{\rho}\ \Rightarrow\ \
\left\|f^\var\right\|_{L^2\(\R^d\times\R^d\)}\leq 1
\ \ \forall\ \ \text{Wigner function}\ \ f\nonumber\\
&&
\end{eqnarray}
\newpage
\begin{rem} \label{remW.1}
\textnormal{By (\ref{eq: Weyl}) it follows that one can fully
reconstruct a density matrix from its Wigner transform but, in
general, by knowing the Wigner function associated with the state of a
quantum system it is not possible to reconstruct such a state in the
wave function picture. More precisely, if we know that the system is
in a \emph{pure state} and we know that it is described by a certain
Wigner function $f^\var$, we can reconstruct the density matrix
$\hat{\rho}$ which will be given by $\hat{\rho}=\vert \psi><\psi\vert$
for some $L^2$-function $\psi$. On the contrary, if the system is in a
\emph{mixed state}, by knowing the Wigner function we can only
reconstruct the density matrix but there is no way to know which are
the wave functions ''composing'' it}.
\end{rem}
\begin{rem}\label{remW.2}
\textnormal{The correspondence between density matrices and Wigner
functions is quite useful but one has to be careful in using that. In
fact, by considering a density matrix $\hat{\rho}$ one can compute its
Wigner transform $f^\var_{\rho}$ and it will be for sure a real
function on the classical phase space with the properties specified
above. On the contrary, a real function on the classical phase space
does not correspond necessarily to an admissible quantum state,
namely, it is not necessarily the Wigner transform of a density
matrix}.\\
\end{rem}
Let us consider a density matrix $\hat{\rho}^0\in
\mathscr{L}^1\(L^2\(\R^d\)\)$ representing the initial state of a
system whose Hamiltonian $H$ is
\begin{eqnarray}\label{eq: propDENS4ante}
&& H=-\frac{\var^2}{2}\Delta_x + U(x)
\end{eqnarray}
and the potential $U$ is such that $H$ is a self-adjoint operator on
$L^2\(\R^d\)$. We know that the time evolution for the density matrix
$\hat{\rho}^0$ is determined by
\begin{eqnarray}\label{eq: propDENS4}
&&i\var \pa_t \hat{\rho}^t=[H,\hat{\rho}^t],
\end{eqnarray}
and it is easy to check that it preserves the Hilbert-Schmidt norm of
$\hat{\rho}^0$, namely the $L^2$-norm of the kernel $\rho^0$. Thus, by
looking at the initial Wigner function $f^\var_{\rho^0}(x,v)$ ($x,v\in
\R^d$) and at the time-evolved $f_t^\var(x,v)=f^\var_{\rho^t}(x,v)$,
the $L^2$-norm has to be also preserved in time (by (\ref{eq:
propDENSisometry}). We can verify this property by looking at the
equation solved by $f^\var_t$. By applying the Wigner transform
defined in (\ref{eq: WignerUNA}) to (\ref{eq: propDENS4}), we find the
equation
\begin{eqnarray}\label{eq: propDENS5}
&&\(\pa_t +v\cdot\nabla_x \)f^\var_t=T^\var f^\var_t,
\end{eqnarray}
where
\begin{eqnarray}\label{eq: propDENS6}
&&\(T^\var f^\var_t\)(x,v)=i\int_{-1/2}^{1/2}\ud \lambda\int \ud k
\hat{U}(k) e^{i\ k\cdot x}\(k\cdot \nabla_v\)f^\var_t(x,v+\var\lambda
k),
\end{eqnarray}
and we denoted by $\hat{U}$ the Fourier transforms of $U$, namely:
\begin{equation}\label{eq: fourierU}
\hat{U}(k)=\int_{\R^d}\ud x\ e^{-i\ k\cdot x} U(x).
\end{equation}
By noting that both $v\cdot\nabla_x$ and $T^\var$ are skewsymmetric
operators and reminding that $f_t^\var(x,v)\in \R$ for any $t$, we find
\begin{eqnarray}\label{eq: propDENS8}
&&\frac{1}{2}\frac{d}{d
t}\left\|f^\var_t\right\|_{L^2\(\R^d\times\R^d\)}^2=\(f_t^\var,\pa_t
f_t^\var\)=\(f_t^\var,-v\cdot\nabla_x f_t^\var\)+\(f_t^\var,T^\var
f_t^\var\)=0,\nonumber\\
&&
\end{eqnarray}
namely the $L^2$-norm is conserved. It can be also proved that
$H_s$-estimates hold for (\ref{eq: propDENS5})
($H_s(\R^{3N}\times\R^{3N})$ being the Sobolev space
$W^{s,2}(\R^{3N}\times\R^{3N})$) by assuming the potential $\phi$ to
be sufficiently smooth (see for example \cite{PULVIRENTI}) in the
sense that the $H_s$-norm of the time evolved Wigner function is
controlled by the $H_s$-norm of the initial datum, up to a constant
depending on time (but finite for any time interval) and of a suitable
norm of the potential.\\
Equation (\ref{eq: propDENS5}) looks like a classical kinetic equation
but the crucial facts are that $f^\var_t$ is not a probability density
in the phase space $\R^d\times\R^d$ and we have to deal with a
pseudodifferential operator instead of a differential one as it is
usual in kinetic theory. It is immediate to check that
\begin{eqnarray}\label{eq: propDENS7}
&&\int \ud x\int \ud v\ f^\var_t(x,v)=\int \ud x \rho^\var_t(x)=1 \ \
\forall\ t>0,
\end{eqnarray}
with
\begin{eqnarray}\label{eq: propDENS7bis}
\rho^\var_t(x)=\int \ud v\ f^\var_t(x,v), \ \ \rho^\var_t\geq 0\ \
\forall\ t,\ \ \ \rho^\var_t(x)\ud x :=\text{spatial probability
distribution, }
\end{eqnarray}
and (\ref{eq: propDENS7}) follows from conservation of ''mass'' and
from the fact that, because of the trace norm normalization of
$\hat{\rho}^0$, we have $\int \ud x\int \ud v\
f^\var_{\rho^0}(x,v)=\texttt{Tr}\hat{\rho}^0=1$.\\\\
\subsection{The Mean-Field system in the Wigner formalism}
\setcounter{equation}{0}
\def\theequation{4.2.\arabic{equation}}
The Wigner formalism introduced in the previous section is an
alternative way of describing the state and the dynamics of a quantum
system and it is precisely equivalent
to the density matrix (or Heisenberg) description, and, for pure
states, to the wave function (or Schr\"odinger) picture. As we have
observed, the advantage in using the Wigner formalism in looking at
semiclassical approximation of quantum systems is that Wigner
functions ''live'' on the classical phase space and for suitable
''semiclassical'' quantum states the Wigner functions can have a well
defined limit when $\var\to 0$ (see for example \cite{Paul}).\\
Thus, in the perspective of looking at the semiclassical limit, we
rephrase the quantum mean-field model discussed in Chapter 2 by using
the Wigner formulation.
By applying the Wigner transform (\ref{eq: WignerUNAconRHO}) to the
Heisenberg equation (\ref{eq: quant8}) we find
\begin{equation}\label{eq: WigLIOUhartreeNsec3}
\left(\pa_t+V_N\cdot \nabla_{X_N}\right)W_N^{\var}(t)=T_N^{\var}W_N^{\var}(t),
\end{equation}
where $W_N^{\var}(t):=W_N^{\var}(X_N,V_N;t)$ is the Wigner function
describing the state of the system (namely, the Wigner transform of
the density matrix $\hat{\rho}_{N,t}$),
$$
X_N=(x_1,\dots,x_N)\in \R^{3N}, \ \ V_N=(v_1,\dots,v_N)\in \R^{3N},
$$
and the pair $Z_N:=(X_N, V_N)$ denotes the generic point in the
classical $N$-particle phase space. Moreover,
\begin{eqnarray}\label{eq: opT_N}
&&\(T_N^{\var}W_N^{\var}\)(Z_N)=\frac{i}{(2\pi)^{3N}}\int_{-1/2}^{1/2}\ud
\lambda\int \ud K_N \hat{U}^Q(K_N) e^{iK_N\cdot V_N}\left(K_N\cdot
\nabla_{V_N}\right)W_N^{\var}(X_N,V_N+\lambda \var K_N),\nonumber\\
&&
\end{eqnarray}
where $K_N=(k_1,\dots,k_N)\in \R^{3N}$, $U^Q$ is the (mean-field)
interaction potential (\ref{eq: quant1}), and $\hat{U}^Q$ is the
Fourier transform of $U^Q$, namely:
\begin{equation}\label{eq: fourierUQ}
\hat{U}^Q(k)=\int_{\R^{3N}}\ud X_N\ e^{-i\ K_N\cdot X_N} \ U^Q(X_N).
\end{equation}
We note that (\ref{eq: WigLIOUhartreeNsec3}) is the analogue of the
classical Liouville equation (\ref{eq: class4}) and, roughly speaking,
by setting ''$\var=0$'' in (\ref{eq: opT_N}) we obtain precisely the
Liouville operator appearing in (\ref{eq: class4}). From now on, we
will refer to (\ref{eq: WigLIOUhartreeNsec3}) as ''$N$-particle
Wigner-Liouville equation''.\\
We remind that we are dealing with undistinguishable particles, then
we consider $N$-particle Wigner functions $W_N$ which are invariant in
the exchange of particle names, namely
\begin{equation}\label{eq: quant3NEWwig}
W_N(x_{\pi(1)},\dots,x_{\pi(N)},
v_{\pi(1)},\dots,v_{\pi(N)})=W_N(x_1,\dots,x_N, v_1,\dots,v_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$. It is easy to
verify that this property is preserved by the evolution (\ref{eq:
WigLIOUhartreeNsec3}).
\subsubsection{The Wigner BBGKY hierarchy}
For any fixed $j$ we introduce the $j$-particle ''marginals'':
\begin{equation}\label{eq: marginaliJ}
W_{N,j}^{\var}(t):=W_{N,j}^{\var}(X_j,
V_j;t)=\int_{\R^{3(N-j)}\times\R^{3(N-j)}} \ud X_{N-j}\ud
V_{N-j}W_{N}^{\var}(X_j, X_{N-j}, V_j, V_{N-j};t).
\end{equation}
It is easy to check that $\{W_{N,j}^{\var}(t)\}_{j=1}^{N}$ are
precisely the Wigner transforms of the RDM
$\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$.
Furthermore, by integrating the Wigner-Liouville equation (\ref{eq:
WigLIOUhartreeNsec3}) with respect to the last $N-j$ variables we find
the following sequence of equations:
\begin{eqnarray}\label{eq: hierarchyN}
&&\left(\pa_t +V_j\cdot
\nabla_{X_j}\right)W_{N,j}^{\var}(t)=T_{N,j}^{\var}W_{N,j}^{\var}(t)+\left(\frac{N-j}{N}\right)C_{j,j+1}^{\var}W_{N,j+1}^{\var}(t),\ \ j=1,2,\dots,
N,\nonumber\\
&&\\
&&\text{with}\ \ \ W_{N,N}^{\var}(t)=W_N^{\var}(t)\ \ \text{and}\ \
C_{N,N+1}^{\var}\equiv 0,\nonumber
\end{eqnarray}
which is precisely the BBGKY hierarchy (\ref{eq: quant10}) rephrased
in the Wigner formalism and it can be seen as the quantum analogue of
the classical BBGKY hierarchy (\ref{eq: class14}).\\
The operator $T_j^{\var}$ (for a fixed $j$), describing the
interaction of the first $j$ particles,
is given by
\begin{eqnarray}\label{eq: opT_j}
&&\left(T_{N,j}^{\var}W_{N,j}^{\var}\right)(X_j, V_j)=\nonumber\\
&&=\frac{i(2\pi)^{-3N}}{N}\sum_{l\neq r}^{j}\int_{-1/2}^{1/2}\ud
\lambda \int_{\R^3}\ud k\ \hat{\phi}(k)e^{ik\cdot (x_l-x_r)}(k\cdot
\nabla_{v_l})W_{N,j}^{\var}(X_j,V_{l-1},v_l+\lambda \var
k,V_{j-l}),\nonumber\\
&&
\end{eqnarray}
while the collision operator $C_{j,j+1}^{\var}$ is
\begin{eqnarray}\label{eq: opC_j+1}
&&\left(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\right)(X_j, V_j)=\nonumber\\
&&=i(2\pi)^{-3N}\sum_{l=1}^{j}\int_{-1/2}^{1/2}\ud \lambda
\int_{\R^3}\ud k \ \hat{\phi}(k)\int_{\R^3\times\R^3}\ud x_{j+1}\ud
v_{j+1}\ e^{ik\cdot( x_l-x_{j+1})}\nonumber\\
&&\qquad\qquad\qquad\qquad(k\cdot
\nabla_{v_l})W_{N,j+1}^{\var}(X_j,x_{j+1},V_{l-1},v_l+\lambda \var
k,V_{j-l},v_{j+1}),\nonumber\\
&&
\end{eqnarray}
and in (\ref{eq: opT_j}) and (\ref{eq: opC_j+1}) we denoted by
$\hat{\phi}$ the Fourier transform of the pair interaction potential
$\phi$, namely:
\begin{equation}\label{eq: fourierPERphi}
\hat{\phi}(k)=\int_{\R^3}\ud x\ e^{-i\ k\cdot x} \phi(x).
\end{equation}
By using (iteratively) the Duhamel formula, the solution
$W_{N,j}^{\var}(t)$ of the equations (\ref{eq: hierarchyN})
with initial datum $W_{N,j}^{\var}(0)$
can be written as
\begin{eqnarray}\label{eq: expFORwig}
W_{N,j}^{\var}(t)&&=\Phi_{j}^{(N)}(t)W_{N,j}^{\var}(0)+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud
t_n\dots\ud t_1\
\Phi_j^{(N)}(t-t_1)\(\frac{N-j}{N}\)C_{j,j+1}^\var\dots\nonumber\\
&&\qquad \quad\dots \(\frac{N-j-n+1}{N}\)C_{j+n-1,j+n}^\var\
\Phi_{j+n}^{(N)}(t_n)W_{N,j+n}^{\var}(0).
\end{eqnarray}
where $\Phi_{j}^{(N)}$ is the flow associated with the $j$-particle
operator $-V_j\cdot \nabla_{X_j}+T_{N,j}^\var$.
\subsection{The Hartree dynamics in the Wigner formalism}
\setcounter{equation}{0}
\def\theequation{4.3.\arabic{equation}}
This section is devoted to the description of the Hartree dynamics
discussed in Chapter 2 in terms of the Wigner formalism. \\
By applying the Wigner transform (\ref{eq: WignerUNAconRHO}) to the
Hartree equation (in the Heisenberg form) (\ref{eq: quant15}) we find
\begin{equation}\label{eq: WigLIOUhartree}
\left(\pa_t+v\cdot \nabla_x\right)f^{\var}(t)=T_{f^{\var}}^{\var}f^{\var}(t),
\end{equation}
where $f^{\var}(t):=f^{\var}(x, v;t)$ is the Wigner function
describing the state of the system (namely, the Wigner transform of
the density matrix $\hat{\rho}_{t}$ solving the Hartree equation
(\ref{eq: quant15})).
For any fixed $g$, the operator $T^\var_g$ acts as follows:
\begin{equation}\label{eq: operatorT}
T_g^{\var} f^{\var}(x, v)=(2\pi)^{-3}i\int_{-1/2}^{1/2}\ud
\lambda\int_{\R^3}\ud k \hat{\phi}(k)\hat{\rho}_g(k) e^{i \ k\cdot
x}(k\cdot \nabla_v)f^{\var}(x, v+\var\lambda k),
\end{equation}
where
\begin{equation}\label{eq: rho}
\rho_g(x)=\int_{\R^3}\ud v \ g(x, v),
\end{equation}
and $\hat{\rho}_g$ is the Fourier transform of $\rho_g$, namely:
\begin{equation}\label{eq: fourier}
\hat{\rho}_g(k)=\int_{\R^3}\ud x\ e^{-i\ k\cdot x} \rho_g(x).
\end{equation}
We observe that equation (\ref{eq: WigLIOUhartree}) is nonlinear \ (as
we can see by (\ref{eq: operatorT}) replacing $g$ with $f^{\var}$ )
because it arises from a nonlinear Heisenberg equation. Thus in the
following we will refer to (\ref{eq: WigLIOUhartree}) as ''(Hartree)
nonlinear Wigner-Liouville equation''.
Furthermore, we note that (\ref{eq: WigLIOUhartree}) is the analogue
of the classical Vlasov equation (\ref{eq: Vlasov}) and, roughly
speaking, by setting ''$\var=0$'' in (\ref{eq: operatorT}) we obtain
precisely the Vlasov operator appearing in (\ref{eq: Vlasov}).
By the analysis we did in the previous section, we know that the
linear equation (\ref{eq: propDENS5}) preserves the $L^2$-norm (see
(\ref{eq: propDENS8})). The same holds for the nonlinear equation
(\ref{eq: WigLIOUhartree}) and, by assuming the potential to be
sufficiently smooth, it can be proved that the $H_s$-norm is
controlled for any $s>0$. Indeed we have the following
\begin{prop}\label{PROPH_s}
Let $f^\var(t)$ be the solution of the nonlinear Wigner-Liouville
equation (\ref{eq: WigLIOUhartree})
whit initial datum $f_0^\var\in H_s\(\R^3\times\R^3\)$ with $s\in \mathbb{N}$.
Assuming the potential $\phi$ to satisfy
\begin{eqnarray}\label{eq: semicEXP2}
&&\int \ud k \ \hat{\phi}(k)\vert k\vert^n<+\infty \ \ \ \forall\
n=1,2,\dots,s
\end{eqnarray}
we find that
\begin{eqnarray}\label{eq: semicEXP3}
&&
\left\|f^\var(t)\right\|_{H_s\(\R^3\times\R^3\)}\leq
e^{Ct}\left\|f_0^\var\right\|_{H_s\(\R^3\times\R^3\)},
\end{eqnarray}
where $C$ is a positive constant depending on $s$ and on $\phi$ but
not on $\var$. For $s=0$ we have $C=0$ and $(3.3)$
becomes an equality (conservation of the $L^2$-norm).
\end{prop}
\begin{dem}
For any multi index $\alpha=\{\alpha_1,\alpha_2,\alpha_3\}$, we use
the standard notation
\begin{eqnarray}\label{eq: semicEXP4}
&& D^\alpha_x=\frac{\pa^{\vert
\alpha\vert}}{\pa^{\alpha_1}x_1\pa^{\alpha_2}x_2\pa^{\alpha_3}x_3},
\end{eqnarray}
where $\vert \alpha\vert=\alpha_1+\alpha_2+\alpha_3$. Analogously we set
\begin{eqnarray}\label{eq: semicEXP5}
&& D^\alpha_v=\frac{\pa^{\vert
\alpha\vert}}{\pa^{\alpha_1}v_1\pa^{\alpha_2}v_2\pa^{\alpha_3}v_3}.
\end{eqnarray}
It is well known that $H_s\(\R^3\times\R^3\)$ equipped with the scalar product
\begin{eqnarray}\label{eq: semicEXP6}
&& \(f,g\)_{s}=\sum_{\substack{\alpha,\beta\in \mathbb{N}:\\ \vert
\alpha\vert+\vert \beta\vert\leq s}}\(D^\alpha_v D^\beta_x
f,D^\alpha_v D^\beta_x g\)_{L^2\(\R^3\times\R^3\)}
\end{eqnarray}
is an Hilbert space and the corresponding norm is
$\left\|g\right\|_s:=\left\|g\right\|_{H_s\(\R^3\times\R^3\)}=\sqrt{\(g,g\)_{s}}$.
In order to estimate $\left\|f^\var(t)\right\|_s$, we compute the time
derivative $\pa_t D^\alpha_v D^\beta_x f^\var(t)$ with $\vert
\alpha\vert+\vert \beta\vert\leq s$. By (\ref{eq: WigLIOUhartree}) we
find:
\begin{eqnarray}\label{eq: semicEXP7}
&& \pa_t D^\alpha_v D^\beta_x f^\var(t)=D^\alpha_v D^\beta_x\(-v\cdot
\nabla_x +T^\var_{f^\var}\)f^\var(t)=\(-v\cdot \nabla_x
+T^\var_{f^\var}\)D^\alpha_v D^\beta_x f^\var(t)+\nonumber\\
&&\nonumber\\
&&+ \sum_{\substack{\alpha'<\alpha:\\ \vert
\alpha'\vert=1}}C_{\alpha,\alpha'}D^{\alpha'}_v v\cdot \nabla_x
D^{\alpha-\alpha'}_v D^\beta_x f^\var(t)+\nonumber\\
&&+\sum_{\substack{\beta'<\beta:\\ \vert \beta'\vert\geq 1}}\frac{i\
C_{\beta,\beta'}}{(2\pi)^{3}}\int_{-1/2}^{1/2}\ud \lambda\int \ud k \
\hat{\phi}(k)\hat{\rho}^\var(k;t)D^{\beta'}_x e^{i\ k\cdot
x}\(k\cdot\nabla_v\) D^\alpha_v D^{\beta-\beta'}_x
f^\var(x,v+\var\lambda k;t)\nonumber\\
&&
\end{eqnarray}
where $C_{\alpha,\alpha'}$, $C_{\beta,\beta'}$ are suitable
combinatorial coefficients, $\alpha'<\alpha$, $\beta'<\beta$ mean
$\alpha'_j<\alpha_j$, $\beta'_j<\beta_j$ (for $j=1,2,3$) respectively
and finally $\alpha-\alpha'=\{\alpha_j - \alpha'_j\}_{j=1}^{3}$,
$\beta-\beta'=\{\beta_j - \beta'_j\}_{j=1}^{3}$.
We observe now that, by virtue of the antisymmetry of the operators
$v\cdot\nabla_x$ and $T^\var_g$ (for any function $g$), we have
\begin{eqnarray}\label{eq: semicEXP8}
&& \(h,v\cdot \nabla_x h\)_{L^2\(\R^3\times\R^3\)}=\(h, T^\var_g
h\)_{L^2\(\R^3\times\R^3\)}=0,
\end{eqnarray}
for any $g$ and for each $h$ smooth enough. Moreover, reminding that
$f^\var(t)\in \R$ for all $t$, if $s>0$, for any $\alpha,\beta:\
0<\vert \alpha\vert+\vert \beta\vert\leq s$, we find:
\begin{eqnarray}\label{eq: semicEXP9}
&& \frac{1}{2}\frac{d}{d t}\(D^\alpha_v D^{\beta}_x
f^\var(t),D^\alpha_v D^{\beta}_x
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}=\(D^\alpha_v D^{\beta}_x
f^\var(t),\pa_t D^\alpha_v D^{\beta}_x
f^\var(t)\)_{L^2\(\R^3\times\R^3\)},\nonumber\\
&&
\end{eqnarray}
which for $s=0$ (namely $\vert \alpha\vert=\vert \beta\vert=0$) becomes:
\begin{eqnarray}\label{eq: semicEXP9s=0}
&& \frac{1}{2}\frac{d}{d t}\( f^\var(t),
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}=\( f^\var(t),\pa_t
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}.
\end{eqnarray}
Inserting (\ref{eq: semicEXP7}) in the right hand side of (\ref{eq:
semicEXP9s=0}), by virtue of (\ref{eq: semicEXP8}) we find:
\begin{eqnarray}\label{eq: semicCONSl^2}
&& \frac{1}{2}\frac{d}{d t}\( f^\var(t),
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}=\frac{d}{d t}\left\|
f^\var(t)\right\|_{L^2\(\R^3\times\R^3\)}=0,
\end{eqnarray}
namely, the $L^2$-norm is conserved.
On the contrary, for $s>0$, we insert (\ref{eq: semicEXP7}) in the
right hand side of (\ref{eq: semicEXP9}). We find the term involving
$D^\alpha_v D^{\beta}_x f^\var(t)$ does not give any contribution by
virtue of (\ref{eq: semicEXP8}). Thus, by using the shorthand notation
$\(\cdot,\cdot\)_{L^2\(\R^3\times\R^3\)}=\(\cdot,\cdot\)_{L^2}$, we
obtain
\begin{eqnarray}\label{eq: semicEXP10}
&& \frac{1}{2}\frac{d}{d t}\(D^\alpha_v D^{\beta}_x
f^\var(t),D^\alpha_v D^{\beta}_x
f^\var(t)\)_{L^2}=\sum_{\substack{\alpha'<\alpha:\\ \vert
\alpha'\vert=1}}C_{\alpha,\alpha'}\(D^\alpha_v D^{\beta}_x
f^\var(t),D^{\alpha'}_v v\cdot \nabla_x D^{\alpha-\alpha'}_v D^\beta_x
f^\var(t)\)_{L^2}+\nonumber\\
&&+\sum_{\substack{\beta'<\beta:\\ \vert \beta'\vert\geq 1}}\frac{i\
C_{\beta,\beta'}}{(2\pi)^{3}}\int_{-1/2}^{1/2}\ud \lambda\int \ud k \
\hat{\phi}(k)\hat{\rho}^\var(k;t)\(D^\alpha_v D^{\beta}_x
f^\var(t),D^{\beta'}_x e^{i\ k\cdot x}\(k\cdot\nabla_v\) D^\alpha_v
D^{\beta-\beta'}_x f^\var(x,v+\var\lambda k;t)\)_{L^2}.\nonumber\\
&&
\end{eqnarray}
We note that the first term on the right hand side of (\ref{eq:
semicEXP10}) is absent when $\vert \alpha\vert=0$. On the contrary, if
$\vert \alpha\vert\geq 1$, by using the Schwartz inequality we obtain:
\begin{eqnarray}\label{eq: semicEXP11}
&& \(D^\alpha_v D^{\beta}_x f^\var(t),D^{\alpha'}_v v\cdot \nabla_x
D^{\alpha-\alpha'}_v D^\beta_x f^\var(t)\)_{L^2}\leq C
\left\|f^\var(t)\right\|_{s}^2
\end{eqnarray}
because $\vert \alpha\vert-\vert \alpha'\vert+\vert \beta\vert+1=\vert
\alpha\vert+\vert \beta\vert\leq s$. Analogously, we find that the
second term in the right hand side of (\ref{eq: semicEXP10}) is
estimated by
\begin{eqnarray}\label{eq: semicEXP12}
&&\int \ud k \ \hat{\phi}(k)\hat{\rho}^\var(k;t)\vert k\vert^{\beta'
+1}\left\| D^{\alpha}_v D^{\beta}_x
f^\var(t)\right\|_{L^2}\left\|\nabla_v D^{\alpha}_v D^{\beta-\beta'}_x
f^\var(t)\right\|_{L^2}\leq\nonumber\\
&&\leq \int \ud k \ \hat{\phi}(k)\hat{\rho}^\var(k;t)\vert
k\vert^{\beta' +1}\left\|f^\var(t)\right\|_{s}^2,
\end{eqnarray}
where we used that $\vert \alpha\vert+1+\vert \beta\vert-\vert
\beta'\vert\leq s$. Now we remind that $\rho^\var(x;t)$ is the spatial
density associated with the Wigner function $f^\var(t)$, namely
$\rho^\var(x;t)\geq 0$ for all $x$ and $t$,
\begin{eqnarray}\label{eq: semicEXP13}
&&\rho^\var(x;t)=\int \ud v f^\var(x,v;t),
\end{eqnarray}
then the $L^1$-norm of $\rho^\var(t)\vert_{t=0}$ is preserved by the
evolution and it is equal to one. Thus, the Fourier transform
$\hat{\rho}^\var(t)$ is in $L^\infty(\R^3\times\R^3)$ for all $t$ and
we find:
\begin{eqnarray}\label{eq: semicEXP14}
&& \int \ud k \ \hat{\phi}(k)\hat{\rho}^\var(k;t)\vert k\vert^{\beta'
+1}\leq
\left\|\hat{\rho}^\var(t)\right\|_{L^\infty(\R^3\times\R^3)}\int \ud k
\ \hat{\phi}(k)\vert k\vert^{\beta' +1}<+\infty,
\end{eqnarray}
by virtue of the assumption (\ref{eq: semicEXP2}) on the pair
interaction potential $\phi$ (we remind that $1\leq \vert \beta'\vert<
\vert \beta\vert\leq s$). Finally, by (\ref{eq: semicEXP10}),
(\ref{eq: semicEXP11}), (\ref{eq: semicEXP12}) and (\ref{eq:
semicEXP14})
, it follows that:
\begin{eqnarray}\label{eq: semicEXP15}
&& \frac{1}{2}\frac{d}{d t}\sum_{\substack{\alpha,\beta:\\\vert
\alpha\vert+\vert \beta\vert\leq s}}\(D^\alpha_v D^{\beta}_x
f^\var(t),D^\alpha_v D^{\beta}_x
f^\var(t)\)_{L^2}=\frac{1}{2}\frac{d}{d
t}\left\|f^\var(t)\right\|_{s}^2\leq C
\left\|f^\var(t)\right\|_{s}^2,\ \ \ \forall\ t
\end{eqnarray}
$C$ depending on $\phi$ and $s$ but not on $\var$. We conclude
straightforward by observing that inequality (\ref{eq: semicEXP15}) is
equivalent to (\ref{eq: semicEXP3}).
\end{dem}
\subsubsection{The Wigner infinite hierarchy}
Let us consider the sequence $\{f_j^{\var}(t)\}_{j\geq 1}$, where
$f_j^{\var}(t)=f_j^{\var}(X_j, V_j;t)$ is given by:
\begin{eqnarray}\label{eq: factJ}
&&f_j^{\var}(X_j, V_j;t)=\prod_{k=1}^{j}f^{\var}(x_k,
v_k;t)=\(f^{\var}\)^{\otimes j}(X_j, V_j;t)
\end{eqnarray}
and $f^{\var}(t)$ is the solution of the nonlinear Wigner-Liouville
equation (\ref{eq: WigLIOUhartree}). By differentiating in time
(\ref{eq: factJ}) we easily deduce the following (infinite) hierarchy
of equations:
\begin{eqnarray}\label{eq: hierarchyINF}
&&\left(\pa_t +V_j\cdot
\nabla_{X_j}\right)f_{j}^{\var}(t)=C_{j,j+1}^{\var}f_{j+1}^{\var}(t),
\end{eqnarray}
where the operator $C_{j,j+1}^{\var}$ is the same of (\ref{eq:
opC_j+1}). This is precisely the Hartree hierarchy (\ref{eq: quant13})
rephrased in the Wigner formalism and it can be seen as the quantum
analogue of the Vlasov hierarchy (\ref{eq: class17}). Here we derived
the Hartree hierarchy by considering the $j$-particle Wigner function
(\ref{eq: factJ}) which is a product of solution of the nonlinear
Wigner-Liouville equation (\ref{eq: WigLIOUhartree}). Conversely, as
we observed in Chapter 2 for the Heisenberg formalism, by starting
from the hierarchy (\ref{eq: hierarchyINF}) and assuming the solution
to be factorized according to a one-particle time dependent Wigner
function $f^\var(t)$, it turns out that $f^\var(t)$ has to solve
equation (\ref{eq: WigLIOUhartree}). \\
By using (iteratively) the Duhamel formula, the solution
$f_{j}^{\var}(t)$ of the equations (\ref{eq: hierarchyINF})
with initial datum $f_{j}^{\var}(0)$
can be written as
\begin{eqnarray}\label{eq: expFORwigINF}
f_{j}^{\var}(t)&&=\Phi_{j}(t)f_{j}^{\var}(0)+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud
t_n\dots\ud t_1\ \Phi_j(t-t_1)C_{j,j+1}^\var\dots C_{j+n-1,j+n}^\var\
\Phi_{j+n}(t_n)f_{j+n}^{\var}(0).\nonumber\\
&&
\end{eqnarray}
where $\Phi_{j}$ is the flow associated with the $j$-particle operator
$-V_j\cdot \nabla_{X_j}$, namely it is the free $j$-particle flow
\begin{eqnarray}\label{eq: freeFLOW}
\Phi_{j}(t)f_j^\var(X_j,V_j)=f_j^\var(X_j-V_j t,V_j).
\end{eqnarray}
\subsection{The Limit $N\to\infty$}
\setcounter{equation}{0}
\def\theequation{4.4.\arabic{equation}}
By the analysis done in the previous section, it is quite natural to
rephrase the mean-field result discussed in Chapter 2 in the Wigner
formalism. This will be the subject of this section and, here and in
the sequel, we will always assume the interaction potential to be
sufficiently smooth.
Thanks to Theorem \ref{teoSPOHN}, we know that, for bounded
potentials, the sequence $\hat{\rho}_{N,t}^{(j)}$ of the RDM
associated with the $N$-particle mean-field dynamics is converging in
trace norm, as $N\to\infty$, to the $j$-fold product
$\hat{\rho}_t^{\otimes j}$ of solutions of the Hartree equation. By
reminding that the space $\mathscr{L}^1\(L^2\(\R^{3j}\)\)$ of trace
class operators on $L^2\(\R^{3j}\)$ is a subspace of the space
$\mathscr{L}^2\(L^2\(\R^{3j}\)\)$ of Hilbert-Schmidt operators on
$L^2\(\R^{3j}\)$, it follows that:
\begin{eqnarray}\label{eq: convergenceHS}
&&\left\|\hat{\rho}_{N,t}^{(j)} - \hat{\rho}_t^{\otimes j}
\right\|_{\mathscr{L}^2\(L^2\(\R^{3j}\)\)}
\leq \left\|\hat{\rho}_{N,t}^{(j)} - \hat{\rho}_t^{\otimes j}
\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\rightarrow 0,\ \ \ \text{as}\ \ N\to\infty.\nonumber\\
&&
\end{eqnarray}
Therefore, by virtue of the equality (\ref{eq: propDENS2}) concerning
the Hilbert-Schmidt norm and thanks to the property (\ref{eq:
propDENSisometry}) of the Wigner function, we can conclude that
\begin{eqnarray}\label{eq: convergenceL2STRONG}
&&\left\| W_{N,j}^{\varepsilon}(t) - \(f^\varepsilon(t)\)^{\otimes j}
\right\|_{L^2\(\R^{3j}\times \R^{3j}\)}
\leq \left\|\hat{\rho}_{N,t}^{(j)} - \hat{\rho}_t^{\otimes j}
\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\rightarrow 0,\ \ \ \text{as}\ \ N\to\infty,\nonumber\\
&&
\end{eqnarray}
where $W_{N,j}^{\varepsilon}(t)$ are the time evolved Wigner marginals
defined in (\ref{eq: marginaliJ}) and $f^\varepsilon(t)$ is the
solution of the (Hartree) nonlinear Wigner equation (\ref{eq:
WigLIOUhartree}). \\
Therefore, the mean-field theorem ensuring the convergence in trace
norm of the RDM, guarantees also the $L^2$-strong convergence of the
corresponding Wigner marginals. Nevertheless, by looking at (\ref{eq:
convergenceL2STRONG}), it is clear that the error in the approximation
for large $N$ is precisely the same we saw previously, and then, it is
depending on $\var$ and diverging as $\var\to 0$.
It turns out that, in the perspective of obtaining estimates on the
error in the mean-field approximation which are uniform with respect
to $\var$ or, at least, which exhibit a less singular dependence on
$\var$, a quite natural approach is to rephrase the whole mean-field
result discussed in Chapter 2 (by assuming the interaction to be
sufficiently smooth) in the Wigner formalism.\\
By looking at the Wigner BBGKY hierarchy (\ref{eq: hierarchyN}) we
observe that the operator $T_{N,j}^\var$ is of size
$O\(\frac{j^2}{N}\)$ while the operator $C_{j,j+1}^\var$ is $O(1)$
with respect to $N$ and it is properly the same appearing in the
infinite hierarchy (\ref{eq: hierarchyINF}). Therefore, in analogy to
what we did in proving Theorem \ref{teoSPOHN} one expects that the
flow $\Phi_j^{(N)}(t)$ appearing in (\ref{eq: expFORwig}) converges in
a suitable sense to the free flow $\Phi_j(t)$ as $N\to\infty$ so that,
this time by using the BBGKY hierarchy, one can prove that
\begin{eqnarray}\label{eq: convergenceWIG}
W_{N,j}^\var(t)\rightarrow f_j^\var(t),\ \ \ \text{as}\ \ N\to\infty,
\end{eqnarray}
in a sense to be made precise.
\\
In Chapters 1 and 2, to show the validity of propagation of chaos, we
considered as initial datum for the $N$-particle dynamics the
(bosonic) factorized state (\ref{eq: quant17bis}), or equivalently,
(\ref{eq: quant17}). We observe that the Wigner transform
$f^\var_\rho$ defined in (\ref{eq: WignerUNAconRHO}) is linear with
respect to the density matrix (kernel) $\rho$, thus we find that the
Wigner transform of the factorized state
$\hat{\rho}_{N,0}=\hat{\rho}_0^{\otimes N}$ considered in Theorem
\ref{teoSPOHN} is also factorized, namely
\begin{eqnarray}\label{eq: WigNinit}
&&W_{N}^{\var}(X_N,V_N)=\prod_{i=1}^{N}f_0^{\var}(x_i,v_i),
\end{eqnarray}
where $f_0^{\var}$ is the Wigner transform of $\hat{\rho}_0$. Moreover, being
$\hat{\rho}_0=\vert\psi_{0}><\psi_{0}\vert$, we find
\begin{eqnarray}\label{eq: ONEpartINDAT}
f_0^{\var}=f_{\rho_0}^{\var}\leftrightarrow\hat{\rho}_0 =\vert \psi_0
><\psi_0 \vert
\end{eqnarray}
and
\begin{eqnarray}\label{eq: ONEpartINDATII}
\left\|f_0^{\var}\right\|_{L^2(\R^3\times\R^3)}=\left\|\rho_0\right\|_{L^2(\R^3\times\R^3)}=\left\|\psi_0\right\|_{L^2(\R^3)}^2=1.
\end{eqnarray}
By taking the $j$-particle marginal associated with
$W_{N}^{\var}(X_N,V_N)$ we straightforward obtain
\begin{eqnarray}\label{eq: margINDAT}
&&W_{N,j}^{\var}(X_j,V_j)=\prod_{i=1}^{j}f_0^{\var}(x_i,v_i)=\(f_0^{\var}\)^{\otimes j}(X_j,
V_j),
\end{eqnarray}
then, by (\ref{eq: expFORwig}), the solution of the equations
(\ref{eq: hierarchyN}) with initial datum (\ref{eq: margINDAT}) is
given by
\begin{eqnarray}\label{eq: expFORwigFACT}
W_{N,j}^{\var}(t)&&=\Phi_{j}^{(N)}(t)\(f_0^{\var}\)^{\otimes j}+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud
t_n\dots\ud t_1\
\Phi_j^{(N)}(t-t_1)\(\frac{N-j}{N}\)C_{j,j+1}^\var\dots\nonumber\\
&&\qquad \quad\dots \(\frac{N-j-n+1}{N}\)C_{j+n-1,j+n}^\var\
\Phi_{j+n}^{(N)}(t_n)\(f_0^{\var}\)^{\otimes j+n},
\end{eqnarray}
while the hierarchy (\ref{eq: expFORwigINF}) with initial datum
$\(f_0^{\var}\)^{\otimes j}$ is
\begin{eqnarray}\label{eq: expFORwigINFfact}
f_{j}^{\var}(t)&&=\Phi_{j}(t)\(f_0^{\var}\)^{\otimes j}+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud
t_n\dots\ud t_1\ \Phi_j(t-t_1)C_{j,j+1}^\var\dots C_{j+n-1,j+n}^\var\
\Phi_{j+n}(t_n)\(f_0^{\var}\)^{\otimes j+n}.\nonumber\\
&&
\end{eqnarray}
\vspace{0.2cm}
Following the line of the proof of Theorem \ref{teoSPOHN}, to prove
the convergence of the series (\ref{eq: expFORwigFACT}) to (\ref{eq:
expFORwigINFfact})
we must find a norm $\left|\cdot\right|_j$ for the marginals
$W_{N,j}^{\var}(t)$ which plays the role of the trace norm on
$L^2\(\R^{3j}\)$ in Theorem \ref{teoSPOHN}. First of all, it has to be
controlled by the flows $\Phi_j^{(N)}$ and $\Phi_j$ in the sense that
for any $T>0$ and for fixed $j$
\begin{eqnarray}\label{eq: contrTHEflow}
\left|\Phi_j^{(N)}(t)W_{N,j}^{\var}\right|_j\leq
C_{t,j}\left|W_{N,j}^{\var}\right|_j,\ \ \ \ C_{t,j}>0:
\sup_{t\in[0,T]}C_{t,j}<+\infty,
\end{eqnarray}
and
\begin{eqnarray}\label{eq: contrTHEflowLIB}
\left|\Phi_j(t)W_{N,j}^{\var}\right|_j\leq
C'_{t,j}\left|W_{N,j}^{\var}\right|_j,\ \ \ \ C'_{t,j}>0:
\sup_{t\in[0,T]}C'_{t,j}<+\infty,
\end{eqnarray}
(note that for the flows $S_j^{(N)}$ and $S_j$ involved Theorem
\ref{teoSPOHN} we had properly conservation of the trace norm,
actually estimates of the form (\ref{eq: contrTHEflow}) and (\ref{eq:
contrTHEflowLIB}) would have been sufficient).
Thus, by (\ref{eq: contrTHEflow}) we could have the following bound
for the $n$-th term of the (formal) series (\ref{eq: expFORwigFACT})
\begin{eqnarray}\label{boundFORser}
\frac{t^n}{n!}j(j+1)\dots(j+n-1)\ (C_t)^n
\left|\(f_0^{\var}\)^{\otimes j+n}\right|_{j+n},\ \ \ \
C_t=C_t(\phi,j)>0:\ \forall\ T>0 \sup_{t\in[0,T]}C_t<+\infty\nonumber\\
&&
\end{eqnarray}
provided that the operator $C_{j,j+1}^\var$ satisfies
\begin{eqnarray}\label{eq: contrOPC}
\left|C_{j,j+1}^\var W_{N,j+1}^{\var}\right|_j\leq j \ C
\left|W_{N,j+1}^{\var}\right|_{j+1},\ \ \ \ C=C(\phi)>0.
\end{eqnarray}
Clearly (\ref{boundFORser}) and (\ref{eq: contrOPC}) would hold even
for the $n$-th term of the series (\ref{eq: expFORwigINFfact}) by
virtue of (\ref{eq: contrTHEflowLIB}).\\
By (\ref{boundFORser}), it would follow that $\left|\cdot\right|_j$
has to be such that
\begin{eqnarray}\label{eq: proprP1}
\left|\(f_0^{\var}\)^{\otimes
j}\right|_j=\(\left|f_0^{\var}\right|_1\)^j\leq a^j \ \ \text{for any
$j$},
\end{eqnarray}
where $a$ is some positive constant. Then
we could conclude that the $n$-th term of the (formal) series
(\ref{eq: expFORwigFACT}) and the $n$-th term of (\ref{eq:
expFORwigINFfact}) are bounded by:
\begin{eqnarray}\label{boundFORserII}
\frac{t^n}{n!}j(j+1)\dots(j+n-1)\ (C_t)^n a^{j+n}< t^n (C_j\ a^{j})\
(2a C_t)^n ,\ \ \ \ C=C(\phi)>0
\end{eqnarray}
and then we would have convergence for short times $\vert t\vert <\psi_t\vert,
\end{eqnarray}
$\psi_t$ solving the Hartree equation (\ref{eq: Hartree}) with initial
datum $\psi_0$.
To prove convergence of (\ref{eq: expFORwigFACT}) to (\ref{eq:
expFORwigINFfact}), the norm $\left|\cdot\right|_j$ has to be such that
\begin{eqnarray}\label{eq: contrOPT}
\left\|T_{N,j}^\var
\right\|\rightarrow 0\ \ \ \text{as}\ \ N\to\infty,
\end{eqnarray}
where $\left\|\cdot\right\|$ is the operator norm on the space of
$j$-particle functions with finite norm $\left|\cdot\right|_j$. In
fact, we observe that
\begin{eqnarray}\label{eq: convFLOWSNUOVA}
\left|\Phi_j^{(N)}(t)\ f_j-\Phi_{j}(t)\ f_j\right|_j\leq \int_0^t \ud
\tau\left| \Phi_j(t-\tau)T_{N,j}^\var\ f_j(\tau)\right|_j,
\end{eqnarray}
for any $j$-particle Wigner function $f_j$. Thus,
by virtue of (\ref{eq: contrTHEflowLIB}) and (\ref{eq: contrOPT}) we
would obtain
\begin{eqnarray}\label{eq: convFLOWS}
\lim_{N\to\infty}\left\|\Phi_j^{(N)}(t)-\Phi_{j}(t)\right\|=0,
\end{eqnarray}
implying convergence of (\ref{eq: expFORwigFACT}) to (\ref{eq:
expFORwigINF}) with respect to the norm $\left|\cdot\right|_j$,
namely, propagation of chaos, for short times $\vert t\vert< t_0$ in
the sense that, for any fixed $j$,
\begin{eqnarray}\label{eq: propOFchaosSHORT}
\left|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes j}\right|_j\rightarrow
0,\ \ \text{as}\ \ N\to\infty \ \ \forall\ t0$) if we could prove that
\begin{eqnarray}\label{eq: NONLINwigCONTROLLAnorma}
\left|f_{j}^\var(t_0-\delta)\right|_j=\(\left|f^\var(t_0-\delta)\right|_1\)^j\leq C a^j
.
\ \ \ \text{for any fixed}\ j,\nonumber\\
&&
\end{eqnarray}
In fact, the bound (\ref{eq: NONLINwigCONTROLLAnorma}), together with
the convergence proved previously up to time $t_0$, would
imply estimate (\ref{eq: proprP1}) to hold for
$W_{N,j}^\varepsilon(t)$ where $t=t_0-\delta$. Then, by iteration we
could conclude that propagation of chaos in the sense of (\ref{eq:
propOFchaosSHORT}) holds for all $t$.\\
By looking at the scheme we have just presented it turns out that the
accuracy of the mean-field approximation
would be provided by the speed of convergence of the operator norm of
$T_{N,j}^\var$ to zero as $N\to\infty$ (see (\ref{eq: contrOPT})).
Then, if one was able to provide an estimate uniform in $\var$ for
$\left\|T_{N,j}^\var\right\|$, the convergence (\ref{eq:
propOFchaosSHORT}) would be also uniform with respect to $\var$ and
then, by iteration, we would have uniformity in $\var$ for all times. \\
\subsubsection{Choice of the norm $\left|\cdot\right|$}
Since by (\ref{eq: convergenceL2STRONG}) we already know that the
$j$-particle Wigner marginals are converging strongly in $L^2$ to the
$j$-fold product of solutions of the (Hartree) nonlinear
Wigner-Liouville equation, it would be reasonable to choose the
$L^2$-norm as $\left|\cdot\right|$ to check if it is possible to
improve the ''bad'' dependence of the error (in the limit
$N\to\infty$) with respect to $\var$.
Moreover, on the basis of (\ref{eq: ONEpartINDATII}) one could think
to choose the $L^2$-norm because (\ref{eq: proprP1}) would be
satisfied (with $a=1$) and this would hold for each $t$ because the
Hartree dynamics preserves the $L^2$-norm (see (\ref{eq:
semicCONSl^2})). Furthermore, the flows $\Phi_j^{(N)}(t)$ and
$\Phi_j(t)$ not only control the $L^2$-norm in the sense of (\ref{eq:
contrTHEflow}) and (\ref{eq: contrTHEflowLIB}) but even preserve it. \\
Nevertheless, it turns out that the operator $C_{j,j+1}^\var$ is
unbounded from $L^2\(\R^{3(j+1)}\times\R^{3(j+1)}\)$ to
$L^2\(\R^{3j}\times\R^{3j}\)$ (it can be verified easily by looking at
(\ref{eq: opC_j+1})), thus property (\ref{eq: contrOPC}) fails.
Actually, one can verify that, by assuming $H_s$ regularity at time
$t=0$, it is propagated by the flow $\Phi_j^{(N)}(t)$ (see
\cite{PULVIRENTI}), by the free flow $\Phi_j(t)$ and also by the
nonlinear Wigner-Liouville equation (\ref{eq: WigLIOUhartree})
(according to Proposition \ref{PROPH_s}). So, this choice could be
appropriate for the preservation in time of property (\ref{eq:
proprP1}) but, as for the $L^2$-norm, the boundness of
$C_{j,j+1}^\var$ fails.
By taking into account the (formal) analogy between the $N$-particle
Wigner-Liouville equation (\ref{eq: WigLIOUhartreeNsec3}) and the
Liouville equation (\ref{eq: class4}) one could think to use the
$L^1$-norm. Indeed it is easy to check that the operators
$T_{N,j}^\var$ and $C_{j,j+1}^\var$ are bounded in $L^1$ and it can be
also verified that the flow $\Phi_j^{(N)}(t)$ controls the $L^1$-norm
in the sense of (\ref{eq: contrTHEflow}). The free flow $\Phi_j(t)$
clearly preserves the $L^1$-norm. Furthermore, by assuming property
(\ref{eq: proprP1}) to hold, namely $f_0^\var\in
L^1\(\R^3\times\R^3\)$, it is easy to check that it is verified for
all $t$ because the $L^1$-norm is controlled by the Hartree dynamics.
In other words, property (\ref{eq: NONLINwigCONTROLLAnorma}) would be
satisfied and we could iterate the procedure presented above to prove
convergence for all times. Therefore the $L^1$-norm could seem a good
choice but the point is that Wigner functions are, in general, not in
$L^1$. More precisely, by only knowing that $f_0^\var$ is the Wigner
transform of a wave function $\psi_0\in L^2\(\R^3\)$
(as in the present situation), we are not guaranteed that $f_0^\var\in
L^1\(\R^3\times\R^3\)$. Indeed, in general
the $L^1$-norm of Wigner functions is not related to any norm of the
wave functions from which they arise.
We will come back on this topic in Remark \ref{RemarkSUl^1}.
It turns out that a fruitful approach is to use a norm which, from on
side, is ''good'' for estimating $C_{j,j+1}^\var$, it is controlled by
$\Phi_j^{(N)}(t)$, $\Phi_j(t)$ and even by the Hartree dynamics, and,
on the other side, it somehow ''relates'' Wigner functions to the wave
functions from which they arise. \\
Let us to consider the Fourier transform $\mathscr{F}_x$ of the
$N$-particle Wigner function with respect to position variables, namely
\begin{eqnarray}\label{eq: normaNUOVA}
&&\(\mathscr{F}_x W_N^\var\)(P_N,V_N):=\tilde{W}_N^\var(P_N,V_N)=\int
\ud X_N\ e^{-i\ P_N\cdot X_N} W_N^\var(X_N,V_N),
\end{eqnarray}
with $P_N=(p_1,\dots, p_N)\in \R^{3N}$
and let us define the $\tilde{L}^1$-norm as
\begin{eqnarray}\label{eq: normaNUOVA1}
&&\left\|W_N^\var\right\|_{\tilde{L}^1(\R^{3N}\times\R^{3N})}:=\left\|\tilde{W}_N^\var\right\|_{L^1(\R^{3N}\times\R^{3N})}=\int \ud P_N\int \ud V_N\ \vert
\tilde{W}_N^\var(P_N,V_N)\vert.
\end{eqnarray}
We can verify that the operator
$T_{N,j}^\var:\tilde{L}^1(\R^{3j}\times\R^{3j})\to\tilde{L}^1(\R^{3j}\times\R^{3j})$ is bounded under the assumption $\left\|\hat{\phi}\right\|_{L^1\(\R^{3}\)}<+\infty$. Indeed, by
computing
\begin{eqnarray}\label{eq: opT_jTRASF}
&&\mathscr{F}_x\(T_{N,j}^{\var}W_{N,j}^{\var}\)(P_j,V_j)
=\int \ud P_j\ e^{-i\ P_j\cdot X_j}\
T_{N,j}^{\var}W_{N,j}^{\var}(X_j,V_j),\nonumber\\
&&
\end{eqnarray}
by manipulating (\ref{eq: opT_j}) we find
\begin{eqnarray}\label{eq: opT_jTRASF1}
&&\mathscr{F}_x\(T_{N,j}^{\var}W_{N,j}^{\var}\)(P_j,V_j):=
\(\tilde{T}_{N,j}^{\var}\tilde{W}_{N,j}^{\var}\)(P_j,V_j)=\frac{i(2\pi)^{-3N}}{\var N}\sum_{ l\neq r}^{j}\ \sum_{\sigma=\pm 1} \sigma \int \ud k\
\hat{\phi}(k)\nonumber\\
&&\quad \tilde{W}_{N,j}^{\var}\(p_1,\dots,p_l - k
,\dots,p_j,\ v_1,\dots,v_l +\frac{\sigma\var k}{2},\dots
, v_j\),\nonumber\\
&&
\end{eqnarray}
then
\begin{eqnarray}\label{eq: opT_jBOUND}
&&\left\|T_{N,j}^{\var}W_{N,j}^{\var}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}=\left\|\tilde{T}_{N,j}^{\var}\tilde{W}_{N,j}^{\var}\right\|_{L^1\(\R^{3j}\times\R^{3j}\)}\leq
\frac{1}{(2\pi)^{3N}}\frac{2 j^2}{\var N}
\left\|\hat{\phi}\right\|_{L^1\(\R^{3}\)}\left\|W_{N,j}^{\var}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}.\nonumber\\
&&
\end{eqnarray}
In a similar way, we verify that the operator
$C_{j,j+1}^\var:\tilde{L}^1(\R^{3(j+1)}\times\R^{3(j+1)})\to\tilde{L}^1(\R^{3j}\times\R^{3j})$ is
bounded
under the assumption
$\left\|\hat{\phi}\right\|_{L^\infty\(\R^{3}\)}<+\infty$. In fact we
compute
\begin{eqnarray}\label{eq: opC_j+1TRASF}
&&\mathscr{F}_x\(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\)(P_{j},V_{j})
=\int \ud P_j\ e^{-i\ P_j\cdot X_j}\
\(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\)(X_j,V_j),\nonumber\\
&&
\end{eqnarray}
obtaining by (\ref{eq: opC_j+1}) that
\begin{eqnarray}\label{eq: opC_j+1TRASF1}
\mathscr{F}_x\(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\)(P_j,V_j)&&:=
\(\tilde{C}_{j,j+1}^{\var}\tilde{W}_{N,j+1}^{\var}\)(P_j,V_j)=\nonumber\\
&&=\frac{i(2\pi)^{-3N}}{\var}\(\frac{N-j}{N}\)\sum_{l=1}^j\
\sum_{\sigma=\pm 1} \sigma \int \ud v_{j+1}\int \ud k\
\hat{\phi}(k)\nonumber\\
&&\quad \tilde{W}_{N,j+1}^{\var}\(p_1,\dots,p_l - k,\dots,p_j,k,\
v_1,\dots,v_l +\frac{\sigma\var k}{2},\dots, v_{j+1}\),\nonumber\\
&&
\end{eqnarray}
then
\begin{eqnarray}\label{eq: opC_j+1BOUND}
\left\|C_{j,j+1}^{\var}W_{N,j+1}^{\var}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}&&=\left\|\tilde{C}_{j,j+1}^{\var}\tilde{W}_{N,j+1}^{\var}\right\|_{L^1\(\R^{3j}\times\R^{3j}\)}\leq\nonumber\\
&&\leq
(2\pi)^{-3N}\frac{(2j)}{\var}\(\frac{N-j}{N}\)
\left\|\hat{\phi}\right\|_{L^\infty\(\R^{3}\)}\left\|W_{N,j+1}^{\var}\right\|_{\tilde{L}^1\(\R^{3(j+1)}\times\R^{3(j+1)}\)}.\nonumber\\
&&
\end{eqnarray}
Furthermore, concerning the initial datum $\(f_0^\var\)^{\otimes j}$,
by (\ref{eq: ONEpartINDAT}) we have
\begin{eqnarray}\label{eq: propP1perL^1tilde}
\left\|\(f_0^{\var}\)^{\otimes
j}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}=\(\left\|f_0^{\var}
\right\|_{\tilde{L}^1\(\R^{3}\times\R^{3}\)}\)^j\leq C
\(\left\|\hat{\psi}_0\right\|_{L^1\(\R^{3}\)}\)^{2j}\leq C
\(\left\|\psi_0\right\|_{H_s\(\R^{3}\)}\)^{2j},\ \ s>3/2\nonumber\\
&&
\end{eqnarray}
where the last inequalities are simply obtained by explicit
computations. For any $t>0$, by (\ref{eq: controparteWAVE}) we have
\begin{eqnarray}\label{eq: propP1perL^1tildeITER}
\left\|f^{\var}(t) \right\|_{\tilde{L}^1\(\R^{3}\times\R^{3}\)}\leq C
\left\|\hat{\psi}_t\right\|_{L^1\(\R^{3}\)}^2\leq C
\left\|\psi_t\right\|_{H_s\(\R^{3}\)}^2,\ \ s>3/2,
\end{eqnarray}
and by using standard energy methods it is easy to check that, under
suitable smoothness assumption on the potential $\phi$, the $H_s$-norm
of $\psi_t$ is controlled by the $H_s$-norm of $\psi_0$ for any $s$.
Furthermore, even by looking at the (Hartree) nonlinear
Wigner-Liouville equation (\ref{eq: WigLIOUhartree}), it is easy to
check that the $\tilde{L}^1$-norm of $f^{\var}(t)$ is controlled by
the $\tilde{L}^1$-norm of $f_0^{\var}$.
Finally, by virtue of (\ref{eq: opT_jBOUND}), (\ref{eq:
opC_j+1BOUND}), (\ref{eq: propP1perL^1tilde}) and (\ref{eq:
propP1perL^1tildeITER}), it follows that by setting
\begin{eqnarray}\label{eq: defNORM}
\left|\cdot\right|_{j}:=\left\|\cdot\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)},
\end{eqnarray}
and by assuming $\psi_0\in H_s\(\R^3\)$ with $s>3/2$ and the potential
$\phi$ to be sufficiently smooth (in order to make all constants
appearing in the estimates finite), we have
\begin{eqnarray}\label{eq: propOFchaos}
\left|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes j}\right|_j\rightarrow
0,\ \ \text{as}\ \ N\to\infty \ \ \forall\ t.
\end{eqnarray}
\vspace{0.3cm}
Therefore, for smooth potentials, we can show propagation of chaos in
the Wigner formulation by following the same strategy of Theorem
\ref{teoSPOHN}. Nonetheless, we note that the error in the
approximation (\ref{eq: propOFchaos}) is still not uniform with
respect to $\var$ and diverging when $\var\to 0$ because from
(\ref{eq: opT_jBOUND}) we see that the operator norm of $T_{N,j}^\var$
is of order $1/\var$ (as in Theorem \ref{teoSPOHN}).\\
We conclude the present analysis by observing that (\ref{eq:
propOFchaos}) implies straightforward that
\begin{eqnarray}\label{eq: propOFchaosII}
\int_{\R^{3j}}\ud V_j\
\sup_{X_j}\left|W_{N,j}^\var(X_j,V_j;t)-\(f^\var(t)\)^{\otimes
j}(X_j,V_j)\right| \rightarrow 0,\ \ \text{as}\ \ N\to\infty \ \
\forall\ t,
\end{eqnarray}
namely
\begin{eqnarray}\label{eq: propOFchaosIII}
\left\|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes
j}\right\|_{L^\infty\(\R^{3j}_{X_j}\)\cap L^1\(\R^{3j}_{V_j}\)}
\rightarrow 0,\ \ \text{as}\ \ N\to\infty \ \ \forall\ t.
\end{eqnarray}
Despite the fact that (\ref{eq: propOFchaosII}) is a quite ''strong''
convergence, it is not related to any convergence for the reduced
density matrices and it does not imply any convergence for the
expected value of $j$-particle observables (namely, it does not
provide informations about macroscopic values of physically
interesting quantities).
Nevertheless, one can verifies that the convergence (\ref{eq:
propOFchaos}) and the uniform bounds
\begin{eqnarray}\label{eq: unifBOUNDL^2}
\left\|W_{N,j}^\var(t)\right\|_{L^2\(\R^{3j}\times \R^{3j}\)}\leq 1,\
\ \ \left\|f^\var(t)\right\|_{L^2\(\R^{3}\times \R^{3}\)}\leq 1,
\end{eqnarray}
imply
\begin{eqnarray}\label{eq: propOFchaosINL^2weak}
W_{N,j}^\var(t)\rightarrow \(f^\var(t)\)^{\otimes j},\ \ \text{as}\ \
N\to\infty \ \ \forall\ t,\ \ \ L^2-\text{weakly}.
\end{eqnarray}
By virtue of property (\ref{eq: WignerUNAerd2}), (\ref{eq:
propOFchaosINL^2weak}) ensures the convergence of expected values of
suitable observables. More precisely, (\ref{eq: propOFchaosINL^2weak})
allows to compute ''macroscopic'' (or ''effective'') expected value of
$j$-particle observables $O_j$ whose phase space representations
(symbols) are in $L^2\(\R^{3j}\times\R^{3j}\)$ (see also \cite{FO}).
Indeed, for any $j$-particle observable $O_j$ with symbol
$O_j(X_j,V_j)$, we have the following estimate
\begin{eqnarray}\label{eq: distrINEQinL^2weak}
\(O_j,W_{N,j}^\var(t)\)_{L^2}\approx\(O_j,\(f^\var(t)\)^{\otimes
j}\)_{L^2}+\frac{C_j(\var)}{N},\ \ \ \forall\ \ t,\ \ \text{as}\ \
N\to\infty,
\end{eqnarray}
where $C_j(\var)\to \infty\ \ \text{as}\ \var\to 0$.\\
%\vspace{0.3cm}
\begin{rem}\label{RemarkSUl^1}
\textnormal{We observe that all estimates we did by using the
$\tilde{L}^1$-norm would be also valid for the $L^1$-norm. Thus, by
assuming $f_0^\var\in L^1\(\R^3\times\R^3\)$ and following exactly the
same strategy leading to the $\tilde{L}^1$-convergence (\ref{eq:
propOFchaos}), we can prove that
\begin{eqnarray}\label{eq: propOFchaosINL^1}
\left\|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes
j}\right\|_{L^1\(\R^{3j}\times\R^{3j}\)} \rightarrow 0,\ \ \text{as}\
\ N\to\infty \ \ \forall\ t,
\end{eqnarray}
and, as for the $\tilde{L}^1$-convergence, it can be verified that
(\ref{eq: propOFchaosINL^1}) together with the uniform bounds
(\ref{eq: unifBOUNDL^2}) leads to the $L^2$-weak convergence (\ref{eq:
propOFchaosINL^2weak}) and, in particular, to the estimate (\ref{eq:
distrINEQinL^2weak}). Then, apparently, there is no reason for
considering the $\tilde{L}^1$-norm instead of the $L^1$-norm. In fact,
in both cases we can realize the limit $N\to\infty$ in the $L^2$-weak
sense and in both cases we find that the error in the mean-field
approximation is not uniform with respect to $\var$, indeed diverging
as $\var\to 0$ (by looking at the constant $C_j(\var)$ in (\ref{eq:
distrINEQinL^2weak})). Nevertheless,
as we have already noticed, the crucial point is: which assumptions
one has to do on the wave function $\psi_0$ to ensure that its Wigner
transform $f_0^\var$ is in $L^1\(\R^{3}\times\R^{3}\)$?
We remind that
\begin{eqnarray}\label{eq: noNORMAL^1}
\int\ud x\int \ud v\ f_0^\var(x,v)=\int \ud x\vert
\psi_0(x)\vert^2=\texttt{Tr}\hat{\rho}_0=1\ \ \ \ \ \text{with}\ \
\hat{\rho}_0=\vert. \psi_0><\psi_0\vert
\end{eqnarray}
We know that the integral on the phase space of $f_0^\var(x,v)$ does
not correspond to its $L^1$-norm being $f_0^\var$ not positive in
general. But, by considering a wave function $\psi_0$ such that
$f_0^\var(x,v)\geq 0$ for any $x,v$ we could identify the $L^2$-norm
of $\psi_0$ (which is taken equal to one) with the $L^1$-norm of
$f_0^\var$ and we are guaranteed that property (\ref{eq: proprP1}) is
verified (with $a=1$). The only way for having a positive Wigner
function is to choose $\psi_0\approx e^{-x^2}$ (see for example
\cite{FO}), in particular we can consider coherent states of the form
$\psi(x)=N_\var e^{-\frac{(x-x_0)^2}{\var}}e^{i\frac{v_0 x}{\var}}$,\
for some\ $x_0,v_0\in \R^3$. }\\
\textnormal{In the end, we found that propagation of chaos in the
mean-field limit by using the Wigner formalism can be proven, for
smooth potentials, in the $L^2$-norm, directly by the mean-field
result for the RDM (the use of the BBGKY hierarchy is prevented by the
unboundedness of the operators involved).
On the other side, by treating the Wigner BBGKY hierarchy, it can be
proven in the $L^1$-norm by choosing initial gaussian states, and, in
the $\tilde{L}^1$-norm, by choosing initial wave functions in $H_s$,
$s>3/2$ (if the dimension of the system is assumed to be equal to 3;
in general, in any dimension $d$,
we have $s>d/2$). In each of the three cases we obtain convergence of
expected values of $j$-particle observables $O_j$ with symbol in
$L^2\(\R^{3j}\times\R^{3j}\)$.
Furthermore, in each of these cases the error in the mean-field
approximation is not uniform with respect to $\var$ and it is
diverging as $\var\to 0$.}
\end{rem}
\vspace{0.2cm}
\subsection{Alternative approaches}
\setcounter{equation}{0}
\def\theequation{4.5.\arabic{equation}}
The validity of propagation of chaos in the mean-field limit has been
established also in \cite{FGS} by using the ''second-quantization
formalism''. For fixed $\var$, the authors provide an alternative
proof of the emergence of the Hartree dynamics for bounded potential
$\phi$ and, even if obtained by using a different formalism, the
general strategy of the proof is analogous to that of \cite{SPOHN} and
the result can be formulated in terms of convergence of reduced
density matrices to products of solutions of the Hartree equation.
Then, by passing to the Wigner formalism, for a restricted class of
two-body interactions the following (distributional) estimate in
$\mathcal{S}'\(\R^{3j}\times\R^{3j}\)$ is proven
\begin{eqnarray}\label{eq: distrINEQfro}
W_{N,j}^\var(t)\approx\(f^\var(t)\)^{\otimes
j}+\frac{C_j}{N}+O\(e^{-1/\sqrt{\left\|\phi\right\|_\infty t}}\),\ \ \
\forall\ t,\ \ \text{as}\ \ \ N\to\infty,
\end{eqnarray}
where
$\left\|\phi\right\|_\infty:=\left\|\phi\right\|_{L^\infty\(\R^3\)}$,
$C_j$ is a positive constant only depending on $j$ and
$W_{N,j}^\var(t)$ and $f^\var(t)$ are defined as in (\ref{eq:
distrINEQinL^2weak}). It turns out that the error in approximating the
$N$-particle evolution with the Hartree dynamics is indeed uniform
with respect to $\var$ but the exponential remainder appearing in
(\ref{eq: distrINEQfro}) is small only if $\left\|\phi\right\|_\infty
t <<1$, namely, by looking at very short times or by considering an
interaction potential having very small $L^\infty$-norm.\\
\subsubsection{Joint limit $N\to\infty$ and $\var\to 0$}
In looking at the connection between mean-field limit and
semiclassical approximation,
a joint limit $N\to\infty$ and $\var\to 0$ can be considered. Indeed,
there are systems in which this kind of limit arises quite naturally
by the scaling properties of the Hamiltonian.
%
A remarkable example is provided by the model considered in
\cite{fermioni} and, previously, in \cite{NS} (with a somewhat
different interpretation). The model considered in \cite{fermioni} is
a system of $N$ fermions interacting by the mean-field potential
(\ref{eq: quant1}) with initial data localized in a cube of size of
order one and at energy comparable with the ground state
energy of the system.
The Hamiltonian of the system is given by (\ref{eq: quant2}), thus all
the potential energy arises from the interaction term (\ref{eq:
quant1}) and it follows straightforward that the potential energy per
particle is of order one. As regard to the kinetic energy, it
%is well-known
can be verified that the
kinetic energy per particle of $N$ fermions, i.e.,
$-\frac{1}{2}\var^2\Delta_{x_k}\ (k=1,\dots,N)$, in a cube of size one
scales like $\var^2 N^{2/3}$ in the ground state. Therefore, in order
to look at the limit $N\to\infty$ keeping the kinetic energy per
particle of order one, one has to multiply the kinetic energy in
(\ref{eq: quant2}) by $N^{-2/3}$. Then, by defining the ''effective
Planck constant'' $\var_{eff}$ such that
$\var_{eff}=\var N^{-1/3}$,
the Hamiltonian of the systems becomes
\begin{equation}\label{eq: quant2fermi}
H_{N,eff}^{Q}=-\sum_{k=1}^{N}\frac{\var_{eff}^2\Delta_k}{2}+U^{Q}(X_N),\ \ \
\text{$\var_{eff}\approx N^{-1/3}\to 0$ as $N\to\infty$}.
\end{equation}
Therefore the kinetic and the
potential energy per particle in the Hamiltonian $H_{N,eff}^Q$ are
comparable and, as we already observed in introducing the mean-field
model, this is the basic physical criterion to obtain a non trivial
limiting dynamics (as $N\to\infty$) that captures the nonlinear effect
of the interaction. Clearly, the limit $N\to\infty$ for the system
whose Hamiltonian is $H_{N,eff}^{Q}$ entails the limit $\var_{eff}\to
0$ which is a semiclassical limit for (\ref{eq: quant2}). Thus, one
expects to find a limiting dynamics which is ruled by a classical
equation. On the other side, it is known (and it is validated by
numerous applications) that the equation governing the macroscopic
(physically observable) dynamics of a Fermi gas in states close to the
ground state is the Hartree-Fock equation:
\begin{eqnarray}\label{eq: hartreeFock}
&&i\var\pa_t\hat{\rho}_{t}=\[-\frac{\var^2}{2}
\Delta,\hat{\rho}_{t}\]+\texttt{Tr}_{2}\left\{\[\phi(x-x_{2})
%+\phi(x_{2}-x)
,\hat{\rho}_{t}\otimes\hat{\rho}_{t}\]\right\} - \int \ud z
\[\phi(x-z)-\phi(y-z)\]\rho_t(x,z)\rho_t(z,y).\nonumber\\
&&
\end{eqnarray}
Equation (\ref{eq: hartreeFock}) differs from the Hartree equation
(\ref{eq: quant15}) because of the presence of the so called
''exchange term'' which is the main effect of the correlations induced
by the Fermi-Dirac statistics (see (\ref{eq: quant4})).
In \cite{fermioni} it has been proven that there exists a fixed time
$T>0$ such that the difference, in a suitable weak sense,
between the $j$-particle marginal associated with the $N$-particle
Wigner function of this system and the solution of the (Hartree)
nonlinear Wigner-Liouville equation (\ref{eq: WigLIOUhartree})
is of order $N^{-1}\approx \var^3$ for any time $t \leq T$, provided that
the potential $\phi$ is real analytic.
In other words, all $\var^2$ corrections come from the difference between
the Vlasov equation (\ref{eq: Vlasov}) and the Hartree equation
(\ref{eq: quant15}); hence they are related to the accuracy of
the semiclassical approximation in the one-body theory. In particular
it is proven that all correlation
effects (the main of them is precisely the exchange term) are of order
at most $O(\var^3)$.\\
We observe that
the case of undistinguishable particles (in the sense specified by
(\ref{eq: quant3NEW}) and (\ref{eq: quant3NEWwig})) and even the
bosonic case are
crucially different from the fermionic case discussed above. Indeed in
these situations the
kinetic energy per particle
, i.e., $-\frac{1}{2}\var^2\Delta_{x_k}\ (k=1,\dots,N)$, in a cube of
size one scales like $\var^2 $ in the ground state. Thus the
Hamiltonian of an $N$-particle system interacting by the potential
(\ref{eq: quant1}) is precisely (\ref{eq: quant2}) because no further
scaling is needed. Therefore, there is no reason for considering a
joint limit and the problem of realizing the (mean-field) limit $N\to
\infty$ uniformly in $\var$ arises quite naturally.
On the other side, even
for undistinguishable particles there are models in which the scaling
of the potential
somewhat leads to define a rescaled Hamiltonian which exhibits an
effective Planck constant going to $0$ as $N\to\infty$ (as in the
fermionic case).
These kind of systems are taken into account in \cite{Graffi} (and
previously in \cite{NS} with the specific scaling $\var\approx
N^{-1/3}$) and an example is provided by systems interacting by Kac
potentials defined below.\\
\emph{Example: the Kac potential}. Consider a system of $N$ identical
bosons of mass
$m = 1$ interacting through the (Kac) potential
\begin{equation}\label{eq: Kac}
\phi_\lambda(x)=\frac{1}{\lambda}
\phi\(\frac{x}{\lambda}\),
\end{equation}
where $\lambda$ is a large parameter of the same order of $N$ and
$\phi$ is a given smooth
potential. The Hamiltonian is:
\begin{equation}\label{eq: KacHAM}
H_{N}=-\frac{\var^2}{2}\sum_{k=1}^{N}\Delta_{x_k}+\sum_{1\leq k0
\end{equation}
in a suitable sense.
This is what we are going to show in the present chapter and it is
contained in our recent paper \cite{MIO}. \\
Note that the term by term convergence (\ref{eq: convergenceCAP4})
does not provide the uniformity in $\var$ of the limit $N\to\infty$
because this would require a control of the remainder of the expansion
(\ref{eq: WignerJNexpcap4}), and for the moment we are not able to do
it. On the other side, in proving (\ref{eq: convergenceCAP4}),
we provide quantum corrections to the classical mean-field limit
result and, by characterizing explicitly both coefficients
$W_{N,j}^{(k)}\left(t\right)$ and $f_j^{(k)}\left(t\right)$, we prove
that those corrections are given in terms of the classical Liouville
flow and, in particular, of suitable derivatives of the classical
trajectories.
We note that to prove (\ref{eq: convergenceCAP4}) we make use of
coherent states (see Section 5.5) and in that framework it is somewhat
expected to find that quantum corrections to the classical dynamics
can be expressed in terms of derivatives of the classical trajectories
(see for example \cite{HEPP}, \cite{HAG}).\\
\subsection{Semiclassical expansion for the Hartree dynamics}
\setcounter{equation}{0}
\def\theequation{5.1.\arabic{equation}}
We want to determine an expansion in power series of $\var$ of the
solution $f^{\var}(x,v;t)$ of the (Hartree) nonlinear Wigner-Liouville
equation (\ref{eq: WigLIOUhartree}) for a given initial datum
$f_0^\var(x,v)$, namely:
\begin{equation}\label{eq: fEXP}
f^{\var}(t)=f^{(0)}(t)+\var f^{(1)}(t)+\var^2 f^{(2)}(t)+\dots
\end{equation}
by knowing that the initial datum $f_0^\var$ is expanded as follows
\begin{equation}\label{eq: fEXPzeroTEMPO}
f_0^{\var}(x,v)=f_0^{(0)}(x,v)+\var f_0^{(1)}(x,v)+\var^2 f_0^{(2)}(x,v)+\dots
\end{equation}
Indeed an expansion like (\ref{eq: fEXPzeroTEMPO}) holds for general
quantum states. For example, in \cite{PULVIRENTI} the semiclassical
expansion for various kinds of states is presented, both gently
varying with respect to $\var$ (such as pure states whose wave
function is not depending on $\var$) and singularly behaving as
$\var\to 0$ (such as states of semiclassical type: WKB and coherent
states). In the first situation we find an expansion of the form
(\ref{eq: fEXPzeroTEMPO}) where the coefficients $f_0^{(k)}$ are
smooth, on the contrary, for WKB and coherent states we find
distributional coefficients (precisely Dirac $\delta$-functions and
suitable derivatives of it) which apparently are more difficult to
treat (with respect to the smooth case). Nevertheless, by manipulating
them in a suitable way, such kinds of ''singular'' expansions can be
very useful to deal with problems of semiclassical approximation (see
Section 5.5 and Chapter 6).
Following \cite{PULVIRENTI}, for a fixed $g$, the operator
$T_g^{\var}$ appearing in equation (\ref{eq: WigLIOUhartree}) can be
expanded as
\begin{equation}\label{eq: operatorTexp}
T_g^{\var}=T^{(0)}_g+\var T^{(1)}_g+\var^2 T^{(2)}_g+\dots
\end{equation}
where
\begin{equation}\label{eq: operatorT^nEVEN}
T^{(n)}_g =c_n(2\pi)^{-3}i\int_{\R^3}\ud k
\hat{\phi}(k)\hat{\rho}_g(k) e^{i \ k\cdot x}(k\cdot \nabla_v)^{n+1},
\end{equation}
\begin{equation}\label{eq: operatorT^nCOSTANTE}
c_n=\frac{1}{2^n (n+1)!},
\end{equation}
for $n$ even and
\begin{equation}\label{eq: operatorT^nODD}
T^{(n)}_g =0,
\end{equation}
for $n$ odd. The operator $T^{(n)}_g$, for $n$ even, can be also written as
\begin{equation}\label{eq: operatorT^nEVEN1}
T^{(n)}_g =(-1)^{n/2}c_n\left(D_x^{n+1}\phi\ast g\right)\cdot D_v^{n+1},
\end{equation}
where, as in (\ref{eq: Vlasov}), $\ast$ denotes the convolution with
respect to both $x$ and $v$ and we used the notation:
\begin{eqnarray}\label{eq: derivate PRODOTTO}
&&D_x^{n}\nu\cdot D_v^{n}\zeta=\sum_{\substack{n_1,n_2,n_3\in
\mathbb{N}:\\\sum_j n_j=n}}\frac{\pa^n
\nu}{\pa^{n_1}{x^1}\pa^{n_2}{x^2}\pa^{n_3}{x^3}}\frac{\pa^n
\zeta}{\pa^{n_1}{v^1}\pa^{n_2}{v^2}\pa^{n_3}{v^3}},\\
&&\text{with}\ \ x=(x^1,x^2,x^3)\in\R^3\ \text{and}\
v=(v^1,v^2,v^3)\in\R^3\nonumber
\end{eqnarray}
for the one-particle functions $\nu$ and $\zeta$.
Inserting (\ref{eq: fEXP}) in (\ref{eq: operatorTexp}) and setting:
\begin{equation}\label{eq: opTnot}
T_k^{(n)}=T_{f^{(k)}(t)}^{(n)},
\end{equation}
we readily arrive to the following sequence of problems for the
coefficients $ f^{(k)}(t)$ of the expansion (\ref{eq: fEXP}):
\begin{equation}\label{eq: WigLIOUseqk=0}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)f^{(0)}(t)=T_{0}^{(0)}f^{(0)}(t),\\
&\left.f^{(0)}(x,v;t)\right\vert_{t=0}=f_0^{(0)}(x,v),
\end{aligned}
\right.
\end{equation}
and
\begin{equation}\label{eq: WigLIOUseq}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot
\nabla_x\right)f^{(k)}(t)=L(f^{(0)}(t))f^{(k)}(t)+\Theta^{(k)}(t),\\
&\left.f^{(k)}(x,v;t)\right\vert_{t=0}=f_0^{(k)}(x,v),
\end{aligned}
\right.
\end{equation}
for $k\geq1$, where
\begin{equation}\label{eq: opL}
L(h)f=T_{h}^{(0)}f+T_{f}^{(0)}h=\left(\nabla_x\phi\ast h\right)\cdot
\nabla_v f+\left(\nabla_x\phi\ast f\right)\cdot \nabla_v h,
\end{equation}
and
\begin{equation}\label{eq: Teta}
\Theta^{(k)}(t)=\sum_{\substack{l,p,r:\\l+p+r=k\\l0$:
\begin{eqnarray}\label{eq: claim}
\sup_{t\in[0,T]}\left\|f^{(k)}(t)\right\|_{L^1(\R^3\times\R^3)}<+\infty\ \ \
\forall\ \ k\geq 2.
\end{eqnarray}
Thus, we note that a recursive argument is not ''well-posed'' because
it is not possible to provide a uniform bound for the
$\mathcal{C}^0\(L^1\)$-norm of $f^{(k)}$, by assuming the same to hold
for $f^{(n)}$ with $n0$. Note that
$f_{j}^{(k)}(t)$ is factorized only for $k=0$.
Thus by (\ref{eq: INFindatcoeff0})
we find that, as expected, the zero order term of the expansion of
$\(f^{\var}(t)\)^{\otimes j}$ is given by the $j$-fold product
$\(f^{(0)}(t)\)^{\otimes j}$ of solutions of the Vlasov initial value
problem (\ref{eq: WigLIOUseqk=0}).\\
By the analysis done in Chapter 3, we know that
$\{f_j^{\var}(t)\}_{j\geq 1}$ solves the infinite (Hartree) hierarchy
(\ref{eq: hierarchyINF}) with factorized initial datum
$\{\(f_0^{\var}\)^{\otimes j}\}_{j\geq 1}$. On the other hand, the
sequence of $j$-particle marginals $\{W_{N,j}^\var(t)\}_{j=1}^N$
associated with the solution of the $N$-particle Wigner-Liouville
equation (\ref{eq: WigLIOUhartreeNsec3}) with factorized initial datum
$\(f_0^{\var}\)^{\otimes N}$, solves the Wigner BBGKY hierarchy
(\ref{eq: hierarchyN}) with initial datum $\{\(f_0^{\var}\)^{\otimes
j}\}_{j=1}^N$. Moreover, in Chapter 3 we proved also that, for any
$j$, $W_{N,j}^\var(t)\to \(f^{\var}(t)\)^{\otimes j}$ $L^2$-weakly
as $N\to\infty$ and the error in approximating the $N$-particle
dynamics with the limiting one is not uniform with respect to $\var$
and diverging as $\var\to 0$.
We recall that the reason for that arises from the fact that the
operator $T_{N,j}^\var$ involved in the BBGKY hierarchy (\ref{eq:
hierarchyN}) is bounded in the norm appropriate to study the
convergence (namely $\tilde{L}^1\(\R^{3j}\times\R^{3j}\)$ or
$L^1\(\R^{3j}\times\R^{3j}\)$), but its norm is diverging when $\var$
goes to zero (in particular it is $O(1/\var)$).
This suggests to consider the semiclassical equations described in
Sections 4.1 and 4.2. In this way, considering equations at each order
in $\var$ and analyzing the hierarchies associated with each of those
equation, we have to deal with operators which are clearly independent
of $\var$ (e.g. $T_N^{(n)}$), %for the $N$-particle case, and
$T_k^{(n)}$ for the expansion associated with the Hartree dynamics)
, and we have to investigate only the limit $N\to\infty$ without any
dependence on $\var$. The price we have to pay is that now those
operators are unbounded, as it comes out for the classical mean-field
limit we faced in Chapter 1.
Thus, if we want to prove that the coefficient of order $\var^k$ of
the expansion of the $j$-particle marginals $W_{N,j}^{\var}(t)$, namely:
\begin{eqnarray}\label{eq: coeffN}
&&W_{N,j}^{(k)}(X_j, V_j;t)=\int_{\R^{3(N-j)}\times \R^{3(N-j)}}\ud
X_{N-j}\ud V_{N-j}W_{N}^{(k)}(X_j,X_{N-j}, V_j,V_{N-j};t),\nonumber\\
&&
\end{eqnarray}
converges to the corresponding object $f_j^{(k)}(t)$ arising from the
Hartree dynamics (i.e (\ref{eq: coeffHAR})), the use of the hierarchy
solved by $W_{N,j}^{(k)}(t)$ does not seem a good idea. In fact, even
at level zero, when we have to deal with the classical mean-field
limit, the hierarchy is very difficult to handle with (see Section
1.4) because it involves derivation operators which are clearly
unbounded, unless to make them act on analytic functions (see
(\ref{eq: class14})-(\ref{eq: class16})). The obstacle which occurs in
facing the higher order terms is precisely the same. \\
However, as we saw in Chapter 1,
%remarked
in the classical case we can treat the convergence in a more natural
way, avoiding to use the hierarchy. Indeed we can control the
$j$-particle marginals associated with the Liouville equation
(\ref{eq: class4}) in terms of the expectation of the $j$-fold product
of empirical measures with respect to the initial $N$-particle
probability distribution (see Section 1.4).
Then, to use this strategy to establish the convergence of
$W_{N,j}^{(0)}(t)$ to $\(f^{(0)}(t)\)^{\otimes j}$, we have to choose
the one-particle initial Wigner function $f_0^\var$ in such a way that
the zeroth order coefficient $f_0^{(0)}$ is a one-particle probability
distribution. As a consequence, the (factorized) zeroth order
coefficient $W_{N,0}^{(0)}$ (\ref{eq: W_Nindatcoeff0}) of the
$N$-particle expansion is also a probability distribution (we will
discuss this choice in Section 4.5). Then, we will follow a similar
strategy in dealing with the convergence of the higher order terms of
the expansion. More precisely, we will express $W_{N,j}^{(k)}(t)$ in
terms of the expectation, with respect to $W_{N,0}^{(0)}$, of suitable
(derivation) operators acting on empirical measures. The control of
these objects will be obtained thanks to some estimates of the
derivatives of the classical flow with respect to the initial data
(see Proposition \ref{DERIVATE1}).
\subsection{Idea of the proof}
\setcounter{equation}{0}
\def\theequation{5.4.\arabic{equation}}
As we already noticed in the previous section, the convergence of the
$j$-particle marginal at zeroth order in $\var$ is ensured by our
assumption on the initial datum (to be specified in Section 4.5) and
by the classical mean-field theory.
Thus, the first non-trivial term is that of order one in $\var$. By
looking at (\ref{eq: WigLIOUseq}) for $k=1$, we realize that the first
correction to the Vlasov equation in the Hartree dynamics satisfies
\begin{equation}\label{eq: WigLIOUseqONE}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)f^{(1)}=L(f^{(0)})f^{(1)},\\
&\left.f^{(1)}(x,v;t)\right\vert_{t=0}=f_0^{(1)}(x,v),
\end{aligned}
\right.
\end{equation}
(looking at the expression (\ref{eq: Teta}) for the source terms
$\Theta^{(k)}$, we straightforward verify that $\Theta^{(1)}\equiv
0$). As we shall see in detail in the following section, our choice
for the initial one-particle datum is a mixture of coherent states
such that each coefficient of the expansion is given by suitable
derivatives of the zeroth order term which, as we already observed, is
a probability distribution. In particular, the explicit form for
$f_0^{(1)}$ is:
\begin{equation}\label{eq: PRIMOcoef}
f_0^{(1)}(x, v)=D^2_G f_0^{(0)}(x, v),
\end{equation}
where $D_G^2$ is a suitable second order derivation operator (see
formula (\ref{eq: operatorD_G}) below in the case $k=2$)
%involving derivatives
with respect to the variable $z\in \R^6$ (we recall the notation
$z=(x,v)\in \R^3\times\R^3$ introduced in Chapter 1).
%initial variables $z_1,\dots, z_N$ (where $z_i=(x_i,v_i)$ for $i=1,\dots,N$.
As regard to the $N$-particle dynamics, looking at (\ref{eq:
W_Nindatcoeffk}) in the case $k=1$, we know that the initial datum for
the coefficient of order one in $\var$
%, namely $W_{N,0}^{(1)}$,
is:
\begin{eqnarray}\label{eq: PRIMOcoefN}
W_{N,0}^{(1)}(Z_N)&&=\sum_{j=1}^{N}f_0^{(1)}(z_j)\prod_{l\neq
j}^{N}f_0^{(0)}(z_l)=\mathcal{D}^2W_{N,0}^{(0)}(Z_N),
\end{eqnarray}
where
\begin{eqnarray}\label{eq: opDstorto}
\mathcal{D}^2=\sum_{j=1}^{N}D_{G,j}^2,
\end{eqnarray}
and $D_{G,j}^2$ is the operator $D_G^2$ acting on the variable $z_j\in
\R^6$ . Let us consider the time evolved empirical measure $\mu_N(t)$
(see Section 1.3) associated with the flow generated by the Newton
equations (\ref{eq: class3}) and let us define $\mathcal{D}^2\mu_N(t)$
as the distribution acting on a test function $u$
%$\in C_b^{\infty}(\R^6)$
in the following way:
\begin{eqnarray}\label{eq: opDstortoMu}
\(u,\mathcal{D}^2\mu_N(t)\)&&=\mathcal{D}^2\(\frac{1}{N}\sum_{l=1}^{N}u(z_l(t))\)=\frac{1}{N}\sum_{l,j=1}^{N}D_{G,j}^2
u(z_l(t)).
\end{eqnarray}
We know that the operators $D_{G,j}^2$ involve derivatives with
respect to the initial variables $z_j$, $j=1,\dots,N$,
% $z_1,\dots,z_N$,
thus, if at time $t=0$ we have $\mu_N\to f_0^{(0)}$ when $N\to\infty$
%a.e with respect to the (factorized) measure
$W_{N,0}^{(0)}=\(f_0^{(0)}\)^{\otimes \infty}$
in the weak sense of probability measures,
%(see (\ref{eq: class11})),
it follows that:
\begin{eqnarray}\label{eq: opDstortoMut=0}
\(u,\mathcal{D}^2\mu_N\)&&=\mathcal{D}^2\frac{1}{N}\sum_{l=1}^{N}u(z_l)=
\frac{1}{N}\sum_{l,j=1}^{N}D_{G,j}^2
u(z_l)=\frac{1}{N}\sum_{j=1}^{N}D_{G,j}^2 u(z_j)=\nonumber\\
&& =\(D_{G}^2 u,\mu_N \)\ \rightarrow\ \(D_{G}^2 u,f_0^{(0)}\)=
\(u,D_{G}^2f_0^{(0)}\)=\(u,f_0^{(1)}\)\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. By the Strong Law of Large Numbers (\ref{eq:
class11}) we know that the convergence (\ref{eq: opDstortoMut=0}) holds
a.e with respect to the product measure $\(f_0^{(0)}\)^{\otimes
\infty}$, then, by (\ref{eq: PRIMOcoefN}) and (\ref{eq:
opDstortoMut=0}), we can conclude that:
\begin{eqnarray}\label{eq: marg1ord1t=0}
\(u,W_{N,1}^{(1)}(t)\vert_{t=0}\)=\(u,\mathbb{E}_N\left[\mathcal{D}^2\mu_N\right]\) \ \rightarrow\ \(u,f_0^{(1)}\)\ \text{as}\
N\to\infty,\nonumber\\
&&
\end{eqnarray}
where $\mathbb{E}_N\left[\cdot\]$ denotes the expectation with respect
to the $N$-particle probability distribution
$W_{N,0}^{(0)}=\(f_0^{(0)}\)^{\otimes N}$ (see (\ref{eq:
W_Nindatcoeff0})).\\
In the sequel, as in Chapter 1, we will say that a configuration $Z_N$
is ''typical'' with respect to the probability measure $f_0^{(0)}$, if
the corresponding empirical measure $\mu_N(z|Z_N)$ converges to
$f_0^{(0)}$ in the weak topology of probability measures.\\
By equation (\ref{eq: WigLiouN}) for $k=1$, we have:
\begin{eqnarray}\label{eq: W_N^1eq}
&&\left(\pa_t +V_N\cdot
\nabla_{X_N}\right)W_{N}^{(1)}=\nabla_{X_N}U\cdot
\nabla_{V_N}W_{N}^{(1)},\nonumber\\
&&W_{N}^{(1)}(Z_N;t)\vert_{t=0}=W_{N,0}^{(1)}(Z_N),
\end{eqnarray}
namely, the classical Liouville equation (\ref{eq: class4}), where, to
simplify the notation, we omitted the superscript ''$cl$''. Therefore:
\begin{eqnarray}\label{eq: W_N^1espress}
&&W_{N}^{(1)}(Z_N;t)=S_N(t)W_{N,0}^{(1)}(Z_N).
\end{eqnarray}
Finally,
%putting (\ref{eq: W_N^1espress}) in (\ref{eq: W_N,1sufunztest}) and
by virtue of (\ref{eq: W_N^1espress}) and (\ref{eq: PRIMOcoefN}), we obtain
\begin{eqnarray}\label{eq: W_N,1sufunztestA}
\(u,W_{N,1}^{(1)}(t)\)&&=\int_{\R^{3N}\times \R^{3N}}\ud
Z_{N}S_N(t)W_{N,0}^{(1)}(Z_N)\(u,\mu_N\)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud
Z_{N}W_{N,0}^{(1)}(Z_N)\(u,\mu_N(t)\)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud
Z_{N}\mathcal{D}^2W_{N,0}^{(0)}(Z_N)\(u,\mu_N(t)\)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud
Z_{N}W_{N,0}^{(0)}(Z_N)\(u,\mathcal{D}^2\mu_N(t)\)=\nonumber\\
&&=\(u,\mathbb{E}_N\left[\mathcal{D}^2\mu_N(t)\right]\).
%\ \ \text{for any}\ t\geq 0.\nonumber\\
%&&
\end{eqnarray}
Therefore, the behavior of $W_{N,1}^{(1)}(t)$ is determined by that of
$\mathcal{D}^2\mu_N(t)$ for any initial configuration $Z_N$ which is
typical with respect to $f_0^{(0)}$.
%the $N$-particle probability measure $W_{N,0}^{(0)}$.
Finally, since $\mu_N(t)$ solves the Vlasov equation in the weak form
(see Chapter 1):
\begin{equation}\label{eq: VlasovWEAK}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\mu_N(t)=\left(\nabla \phi \ast
\mu_N(t)\right)\cdot \nabla_v \mu_N(t)\\
&\left. \mu_N(t)\right\vert_{t=0}=\mu_N,
\end{aligned}
\right.
\end{equation}
applying $\mathcal{D}^2$, we get:
\begin{equation}\label{eq: VlasovWEAKconD^2}
\left\{
\begin{aligned}
&\left(\pa_t + v\cdot \nabla_x
\right)\mathcal{D}^2\mu_N(t)=L\left(\mu_N(t)\right)
\mathcal{D}^2\mu_N(t)+R_N,\\
&\left. \mathcal{D}^2\mu_N(t)\right\vert_{t=0}=\mathcal{D}^2\mu_N,
\end{aligned}
\right.
\end{equation}
where $R_N$ is a term involving objects of the form $\sum_j
\(D_{G,j}\mu_N(t)\)\( D_{G,j}\mu_N(t)\)$ which, as we shall see later,
are of order $1/N$ when tested versus smooth functions.
The equation (\ref{eq: VlasovWEAKconD^2}) is similar to (\ref{eq:
WigLIOUseqONE}), except for the presence of the term $R_N$ and for the
fact that we have $L\left(\mu_N(t)\right) $ instead of
$L\left(f^{(0)}\right)$. Therefore, the proof of the convergence of
$W_{N,1}^{(1)}(t)$ to $f^{(1)}(t)$ reduces to that of a stability
property for the solution of (\ref{eq: WigLIOUseqONE}) with respect to
suitable weak topologies. Proposition \ref{PROP6.2} in the forthcoming
Section 4.5 will provide us such property.
The general case $k>1$ is only technically more complicated because of
the presence of source terms, but the main ideas are those presented
here.
\begin{rem}\label{REMARKsuTIPOdiCONV}
\textnormal{By looking at the strategy of the proof for $k=1$ we
realize that the basic idea is to apply the classical mean-field
theory at suitable derivatives of the empirical measure. Therefore, if
in the classical framework we have to deal with convergence with
respect to the weak topology of probability measures, namely, with
continuous and uniformly bounded test functions, now we need to deal
with test functions whose derivatives are also continuous and
uniformly bounded (e.g (\ref{eq: opDstortoMut=0})). Therefore we can
argue that we will establish the term by term convergence in a
suitable distributional sense.}
\end{rem}
We conclude by establishing a Proposition controlling the size of the
derivatives of the Hamilton flow associated with (\ref{eq: class3})
with respect to the initial data.
From now on we shall denote by $C$ a positive constant, independent
of $N$, possibly changing from line to line.
\begin{prop}\label{DERIVATE1}
Let $z_i(t)=\left(x_i(t),v_i(t)\right),\ i=1,\dots,N$ be the solution
of equations (\ref{eq: class3}) with initial datum
$z_i=\left(x_i,v_i\right),\ i=1,\dots,N$. %given at a certain time
$t_00$. Actually $\frac{\pa z_i^\beta(t)}{\pa
z_j^\alpha}=O\(\frac{1}{N}\)$ while $\frac{\pa z_i^\beta(t)}{\pa
z_i^\alpha}=O\(1\)$ and these two estimates give rise to (\ref{eq:
derivateK}) in the case $k=1$. Estimate (\ref{eq: derivateK}) says
that for each derivative of any order with respect to some $z_j$ of
$z_i(t)$ , we gain a factor $1/N$.
We have also the following corollary whose straightforward proof will
be omitted.
\begin{cor}\label{PROP5.2}
Let $\mathcal{U}=\mathcal{U}(Z_N(t))$ be a function of the time
evolved configuration $Z_N(t)$ of the form:
$$
\mathcal{U}(Z_N(t))=\frac{1}{N}\sum_{i=1}^{N}u(z_i(t)),
$$
where $u\in C_b^{\infty}(\R^3\times\R^3)$. Then, if the pair
interaction potential $\phi$ satisfies Hypotheses H, the following
estimate holds:
\begin{equation}\label{eq: corollario}
\left\vert\frac{\pa^k \mathcal{U}(Z_N(t))}{\pa
z_{j_1}^{\alpha_{1}}\dots \pa z_{j_k}^{\alpha_{k}}} \right\vert\leq
\frac{C}{N^{d_k}},
\end{equation}
where $d_k$ is the number of different indices in the sequence
$I=(j_1,\dots, j_k)$.
\end{cor}
The proof of Proposition \ref{DERIVATE1} will be given in Appendix A.
\subsection{Results and technical preliminaries}
\setcounter{equation}{0}
\def\theequation{5.5.\arabic{equation}}
We choose, as initial condition for the one-particle Wigner function,
a mixture of coherent states. The Wigner function associated with a
pure coherent state centered at the point $(x_0, v_0)$ is given by:
\begin{equation}\label{eq: coeherent}
w(x,v\vert
x_0,v_0)=\frac{1}{(\pi\var)^{3}}e^{-\frac{(x-x_0)^{2}}{\var}}e^{-\frac{(v-v_0)^{2}}{\var}}.
\end{equation}
Let now $g=g(x,v)$ be a smooth probability density on the one-particle
phase space independent of $\var$
(see Hypotheses $\text{H}^1$ below) . Then we define:
\begin{equation} \label{eq: mix}
f_{0}^\var(x, v)=\int_{\R^3\times\R^3}\ud x_0 \ud v_0 \ w(x, v\vert
x_0,v_0)g(x_0, v_0).
\end{equation}
Using the standard notation $z=(x,v)$ and $z_0=(x_0,v_0)$, (\ref{eq:
mix}) is equivalent to:
\begin{eqnarray} \label{eq: mix2}
f_{0}^\var(z)&&=\frac{1}{(\pi\var)^{3}}\int_{\R^6}\ud z_0\
e^{-\frac{(z-z_0)^{2}}{\var}} g(z_0)=\nonumber\\
&&=\frac{1}{(\pi)^{3}}\int_{\R^6}\ud \zeta \ e^{- \zeta^{2}}
g(z-\sqrt{\var}\zeta).
\end{eqnarray}
Expanding
\begin{eqnarray} \label{eq: expG}
g(z-\sqrt{\var}\zeta)&&=g(z)-\left(\zeta \cdot \nabla_z
\right)g(z)\sqrt{\var}+\left(\zeta \cdot \nabla_z
\right)^2g(z)\frac{(\sqrt{\var})^2}{2}+\dots\nonumber\\
&& \dots -\left(\zeta \cdot \nabla_z
\right)^{2n-1}g(z)\frac{(\sqrt{\var})^{2n-1}}{(2n-1)!}+\left(\zeta
\cdot \nabla_z
\right)^{2n}g(z)\frac{(\sqrt{\var})^{2n}}{(2n)!}+\dots,\nonumber\\
&&
\end{eqnarray}
and performing the gaussian integrations (which cancels the terms with
the odd powers of $\sqrt{\var}$), we readily arrive to the following
expansion for the Wigner function $f_0^\var$:
\begin{eqnarray} \label{eq: expf_0}
&& f_0^\var=f_0^{(0)}+\var f_0^{(1)}+\dots +\var^n f_0^{(n)}+\dots,
\end{eqnarray}
where
\begin{eqnarray}
&& f_0^{(0)}=g,\label{eq: expf_0COEFF}\\
&& f_0^{(n)}=D_G^{2n}f_0^{(0)}\ \ \text{for}\ n\geq 1,\label{eq: expf_0COEFFk}
\end{eqnarray}
and $D_G^{k}$ ($G$ stands for ''Gaussian''), for each $k>0$, is the
following derivation operator with respect to the variable $z=(x,v)$:
\begin{eqnarray} \label{eq: operatorD_G}
&&D_G^{k}=\sum_{\substack{\alpha_1\dots\alpha_k:\\ \alpha_j=1,\dots,6}}\
\sum_{\substack{s_1\dots s_k:\\ 0\leq s_j\leq k\\s_1+\dots +s_k=k}}
C_G(\alpha_1\dots \alpha_k)\frac{\pa^k}{\pa z^{\alpha_1}\dots \pa
z^{\alpha_k}},
\end{eqnarray}
where
\begin{eqnarray} \label{eq: operatorD_Gcost}
&&C_G(\alpha_1\dots \alpha_k)=\frac{1}{k!}\int_{\R^6}\ud \zeta\
e^{-\zeta^2} \prod_{j=1}^{k} \zeta^{\alpha_j}.
\end{eqnarray}
Therefore, $C_G(\alpha_1\dots \alpha_k)$ is equal to zero for each
sequence $\alpha_1\dots \alpha_k$ in which at least one index appears
an odd number of times.\\
\\
{\bf Hypotheses $\text{H}^1$}:
We assume that
$g=f_0^{(0)}\in \mathcal{S}(\R^3\times\R^3)$, thus (\ref{eq:
expf_0COEFFk}) make sense for any $n\geq 1$ and, in particular,
$f_0^{(n)}\in \mathcal{S}(\R^3\times\R^3)$ for any $n$. By the
analysis done in Section 4.1, this allows to identify the time-evolved
coefficients $f^{(n)}(t)$, $n\geq 1$, as the unique solutions of the
initial value problems (\ref{eq: WigLIOUseq}).
\begin{rem} \label{Remark6.1}
\textnormal{Here we consider a completely factorized $N$-particle
initial state (see (\ref{eq: meanFIELDinCOND})), then property
(\ref{eq: quant3NEWwig}) is satisfied. Furthermore the one-particle
state is a mixture and this automatically excludes the Bose statistics.}
\end{rem}
\begin{rem} \label{Remark6.2}
\textnormal{We made the choice to expand fully the initial state
$f_0^\var$ according to equation (\ref{eq: expf_0}). Another
possibility is to assume the ($\var$ dependent) state $f_0^\var$
(which is a probability measure in the present case) as initial
condition for the Vlasov problem and, consequently, $f_0^{(k)}=0$ for
the problems (\ref{eq: WigLIOUseq}). Now the coefficients $f^{(k)}(t)$
are $\var$ dependent but this does not change deeply our analysis
because $f_0^\var$ is smooth, uniformly in $\var$}.
\end{rem}
As we explained at the beginning of the present Chapter, our goal is
to compare the $j$-particle semiclassical expansion associated with
the $N$-particle flow, namely $W_{N,j}^{(k)}(t)$, $k=0,1,2,\dots$,
with the corresponding coefficients $f_j^{(k)}(t)$ of the expansion:
\begin{equation}\label{eq: expHartreeJ}
f_j^{\var}(t)=f_j^{(0)}(t)+\var f_j^{(1)}(t)+\dots +\var^k f_j^{(k)}(t)+\dots,
\end{equation}
where $f_j^{(k)}(t)$ is given by (\ref{eq: coeffHAR}). The main result
is the following.
\begin{teo}\label{THEO6.1}
Let us consider the (Hartree) nonlinear Wigner-Liouville equation
(\ref{eq: WigLIOUhartree}) as in (\ref{eq: mix}) where the probability
distribution $g$ satisfies Hypotheses $\text{H}^1$. Moreover, let us
consider the $N$-particle Wigner-Liouville equation (\ref{eq:
WigLIOUhartreeNsec3}) with factorized initial datum as in (\ref{eq:
meanFIELDinCOND}). If the pair interaction potential $\phi$ is assumed
to verify Hypotheses H, for all $t> 0$, for any integers $k$ and $j$,
the following limit holds in $\mathcal{S}'(\R^{3j}\times\R^{3j})$:
\begin{equation}\label{eq: Th6.1}
W_{N,j}^{(k)}(t)\rightarrow f_j^{(k)}(t).
\end{equation}
as $N\to \infty$.
\end{teo}
\begin{rem} \label{Remark6.3}
\textnormal{As we shall see in the sequel, the convergence (\ref{eq:
Th6.1}) is slightly stronger than the convergence in
$\mathcal{S}'(\R^{3j}\times\R^{3j})$. Indeed, the sequence
$W_{N,j}^{(k)}(t)$ converges also when it is tested on functions in
$\mathcal{C}_b^{\infty}(\R^{3j}\times\R^{3j})$.
Such kind of convergence, which is natural in the present context,
will be called $\mathcal{C}_b^{\infty}$-weak convergence}.
\end{rem}
A crucial tool in proving Theorem \ref{THEO6.1} is provided by the following
\begin{prop}\label{PROP6.2}
Let $\gamma_N(x,v;t)$ be a sequence in
$\mathcal{S}'(\R^{3}\times\R^{3})$ (for each $t$) satisfying:
\begin{equation}\label{eq: Prop6.2}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\gamma_N=L(h_N)\gamma_N+\Theta_N,\\
&\left. \gamma_N(x,v;t)\right\vert_{t=0}=\gamma_{N,0}(x,v),
\end{aligned}
\right.
\end{equation}
where $\gamma_{N,0}$, $\Theta_N$ are sequences in
$\mathcal{S}'(\R^{3}\times\R^{3})$. We assume that:\\
{\bf i)}\ $h_N(x,v;t)$ is a sequence of probability measures
converging, as $N\to\infty$, to a measure $h(t)\ud x\ud v$ with a
density $h(t)\in \mathcal{C}^{\infty}_b\(\R^3\times\R^3\)$ and such that
$\left\vert \nabla_v h\right\vert \in \mathcal{C}^0\(
L^1(\R^{3}\times\R^{3}),\R^+\)$.\\
{\bf ii)}\ for all $u_1,u_2$ in
$\mathcal{C}_b^{\infty}(\R^{3}\times\R^{3})$ , there exists a constant
$C=C(u_1, u_2)>0$, not depending on $N$, such that:
\begin{equation}\label{eq: Prop6.2hyp}
\left\| u_1\ast \(u_2\gamma_N\)\right\|_{L^{\infty}(\R^3\times\R^3)}<
C<+\infty\ \ \ for \ any \ t.
\end{equation}
{\bf iii)}\ $\gamma_{N,0}\rightarrow \gamma_0\text{,\ as}\ \
N\to\infty$,\ \ $\mathcal{C}_b^{\infty}$-weakly ,
$\gamma_0=\gamma_0(x,v)$ is a function belonging to
$L^1(\R^{3}\times\R^{3})$.
\\
{\bf iv)}\ $\Theta_N \rightarrow \Theta\text{,\ as}\ \ N\to\infty$,\
\ $\mathcal{C}_b^{\infty}$-weakly , $\Theta=\Theta(x,v;t)$ is a
function belonging to \ $\mathcal{C}^0\(L^1(\R^{3}\times\R^{3}),
\R^+\)$.
\\
\emph{Then:}
\begin{eqnarray}\label{eq: Prop6.2thesis}
&&\text{$\gamma_N \rightarrow\gamma\text{,\ as}\ \ N\to\infty$ \ \
$\mathcal{C}_b^{\infty}$-weakly},\nonumber\\
&&
\end{eqnarray}
where $\gamma$ is the unique solution of the problem (\ref{eq:
Prop6.1}) in $\mathcal{C}^0\(L^1\(\R^{3}\times\R^{3}\),\R^+\)$.
\end{prop}
For the proof, see Appendix B.
\subsection{Convergence}
\setcounter{equation}{0}
\def\theequation{5.6.\arabic{equation}}
This section is devoted to the proof of Theorem \ref{THEO6.1}.\\
By (\ref{eq: DuhamelN}) and (\ref{eq: Teta_N}), for $k\geq 0$ we have:
\begin{eqnarray}\label{eq: expW_N^k}
W_N^{(k)}(Z_N;t)&&=\sum_{n\geq
0}\sum_{r=0}^{k}\sum_{\substack{r_1\dots r_n:\\r_j>0\\\sum
r_j=k-r}}\int_{0}^{t}\ud t_1\int_{0}^{t_1}\ud t_2\dots
\int_{0}^{t_{n-1}}\ud t_n \nonumber\\
&&\quad \quad \quad S_N(t-t_1)T_N^{(r_1)}S_N(t_1-t_2)\dots
T_N^{(r_n)}S_N(t_n)W_{N,0}^{(r)}(Z_N).
\end{eqnarray}
It is useful to remind that, the only non-vanishing terms in (\ref{eq:
expW_N^k}) are those for which all $r_1,\dots,r_n$ are even (because
the odd terms in the expansion for the operator $T_N^\var$ appearing
in (\ref{eq: WigLIOUhartreeNsec3}) are vanishing (see (\ref{eq:
T_NcoeffODD})).
According to (\ref{eq: W_Nindatcoeffk}) and (\ref{eq: expf_0COEFFk}),
\begin{eqnarray}\label{eq: W_NindatcoeffkSEC7}
&&W_{N,0}^{(r)}(Z_N)=\sum_{\substack{s_1\dots s_N\\0\leq s_j\leq r\\
\sum_j s_j=r}}\prod_{j=1}^{N}\(D_{G,j}^{2 s_j}f_0^{(0)}(z_j)\),
\end{eqnarray}
where $D_{G,j}^{k}$ is defined in (\ref{eq: operatorD_G}) and the
extra symbol $j$ means that this operator acts on the variable $z_j$.
Defining the operator $\mathcal{D}^{2r}$ as:
\begin{eqnarray}\label{eq: opDstortoGEN}
&&\mathcal{D}^{0}=\texttt{1},\nonumber\\
&&\mathcal{D}^{2r}=\sum_{\substack{s_1\dots s_N:\\0\leq s_j\leq
r\\\sum_j s_j=r}}\prod_{j=1}^{N}D_{G,j}^{2s_j},\ \ \ r\geq 1,
\end{eqnarray}
we have:
\begin{eqnarray}\label{eq: W_N,0^r}
W_{N,0}^{(r)}(Z_N)=\mathcal{D}^{2r}W_{N,0}^{(0)}(Z_N)\ \ \forall\ r\geq 0.
\end{eqnarray}
In order to investigate the behavior of the $j$-particle functions
$W_{N,j}^{(k)}(Z_j;t)$ when $N\to\infty$, we consider the following
object, for a given configuration $Z'_j=(z'_1\dots z'_j)$:
\begin{eqnarray}\label{eq: W_N,j^k}
&&\omega_{N,j}^{(k)}(Z'_j;t)=\int_{\R^{6N}}\ud Z_N\
W_{N}^{(k)}(Z_N;t)\mu_N(z'_1\vert Z_N)\dots \mu_N(z'_j\vert Z_N).
\end{eqnarray}
In the end of the section, we will show that (\ref{eq: W_N,j^k}) is
asymptotically equivalent to $W_{N,j}^{(k)}(Z'_j;t)$.
From (\ref{eq: expW_N^k}), (\ref{eq: W_N,0^r}) and (\ref{eq:
W_N,j^k}), it follows that:
\begin{eqnarray}\label{eq: expW_N^kBIS}
&&\omega_{N,j}^{(k)}(Z'_j;t)=\sum_{n\geq
0}\sum_{r=0}^{k}\sum_{\substack{r_1\dots r_n:\\r_j>0\\\sum
r_j=k-r}}\int_{0}^{t}\ud t_1\int_{0}^{t_1}\ud t_2\dots
\int_{0}^{t_{n-1}}\ud t_n \int_{\R^{6N}}\ud Z_N \mu_{N,j}(Z'_j\vert
Z_N)\nonumber\\
&&\qquad \quad
S_N(t-t_1)T_N^{(r_1)}S_N(t_1-t_2)\dots
T_N^{(r_n)}S_N(t_n)\mathcal{D}^{2r}W_{N,0}^{(0)}(Z_N),
\end{eqnarray}
where
\begin{eqnarray}\label{eq: muperJ}
&&\mu_{N,j}(Z'_j\vert Z_N)=\mu_{N}(z'_1\vert Z_N)\dots \mu_{N}(z'_j\vert Z_N).
\end{eqnarray}
Integrating by parts,
reminding that each $r_j$ is even and that each $T_N^{(r_j)}$ involves
derivatives of order $r_j +1$,
we have:
\begin{eqnarray}\label{eq: expW_N^kTRIS}
\omega_{N,j}^{(k)}(Z'_j;t)&&=\sum_{n\geq 0}(-1)^n\sum_{r=0}^{k}\
\sum_{\substack{ {\bf \underline{r}}_n:\ r_j>0\\\vert {\bf
\underline{r}}_n\vert =k-r}}\ \int_{ord}^{t}\ud {\bf
\underline{t}}_n\nonumber\\
&&\qquad
\mathbb{E}_N\left[\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N,j}(Z'_j\vert
Z_N(t))\right],
\end{eqnarray}
where ${\bf \underline{r}}_n$ is the sequence of positive integers
$r_1,\dots ,r_n$, $\vert {\bf \underline{r}}_n\vert=\sum_{j=1}^{n}r_j$
and $Z_N(t)$ is the Hamiltonian flow associated with (\ref{eq:
class3}). Moreover ${\bf \underline{t}}_n=t_1\dots t_n$ and
$\int_{ord}^{t}\ud {\bf \underline{t}}_n$ denotes the integral over
he simplex $00\\\vert {\bf
\underline{r}}_n\vert =k-r}}\ \int_{ord}^{t}\ud {\bf
\underline{t}}_n\eta_j(Z'_j;t,r,{\bf \underline{r}}_n,{\bf
\underline{t}}_n,Z_N),\nonumber\\
&&
\end{eqnarray}
(for any configuration $Z_N$, typical with respect to $f_0^{(0)}$),
where $\eta_j$ is given by:
\begin{eqnarray}\label{eq: eta_j}
&&\eta_j(Z'_j;t,r,{\bf \underline{r}}_n,{\bf
\underline{t}}_n,Z_N)=\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N,j}(Z'_j\vert
Z_N(t)).\nonumber\\
&&
\end{eqnarray}
Note that:
\begin{eqnarray}\label{eq: senza MEDIA k=0}
\nu_j^{(0)}(Z'_j;t)=
\mu_{N,j}\(Z'_j\vert Z_N(t)\).
\nonumber\\
&&
\end{eqnarray}
We start by analyzing the behavior of $\nu_{j}^{(k)}$ in the cases
$j=1,2$, thus we are lead to consider:
\begin{eqnarray}
&&\eta_1(z'_1;t,r,{\bf \underline{r}}_n,{\bf
\underline{t}}_n,Z_N)=\mathcal{D}^{2r} T_N^{(r_n)}(t_n)
T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N}(z'_1\vert
Z_N(t)),\label{eq: aUnCorpo}\nonumber\\
&&
\end{eqnarray}
and
\begin{eqnarray}
&&\eta_2(z'_1,z'_2;t,r,{\bf \underline{r}}_n,{\bf
\underline{t}}_n,Z_N)=\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N,2}\(Z'_2\vert
Z_N(t)\).\nonumber\\
&&\label{eq: aDueCorpi}
\end{eqnarray}
It is useful to stress that the operators $T_N^{(r_j)}(t_j)$
($j=1,\dots,n$) and $\mathcal{D}^{2r}$ act as suitable distributional
derivatives with respect to the variables $Z_N$. To evaluate $\eta_1$,
let us first analyze the action of $T_N^{(r)}(\tau)$. By (\ref{eq:
opTtempo}) and (\ref{eq: restoOPt}), for any function $G=G(Z_N)$, we
have:
\begin{eqnarray}\label{eq: opTtemposuG}
&&\left(T_N^{(r)}\(\tau\)G\right)\(Z_N\)=
%S_N\(-\tau\)T_N^{(r)}S_N\(\tau\)G\(Z_N\)=\nonumber\\
S_N\(-\tau\)\(\hat{T}_N^{(r)}
%\left(S_N\(-\tau\)G\right)\(Z_N\(\tau\)\)
+R_N^{(r)}\)\(S_N\(\tau\)G\)\(Z_N\)=\nonumber\\
&&=(-1)^{r/2}\frac{c_r}{N}\sum_{j,l}S_N\(-\tau\)D_x^{r+1}\phi(x_j-x_l)\cdot
D_{v_j}^{r+1}\(S_N\(\tau\)G\)\(Z_N\)+\nonumber\\
&&+\frac{1}{N}\sum_{l,j=1}^{N}\sum_{\substack{k_1,k_2\in
\mathbb{N}^{3}\\ \vert k_1\vert+\vert k_2\vert=r+1}}C_{k_1,
k_2}S_N\(-\tau\)\frac{\pa^{r+1}}{\pa_{x_l}^{\vert
k_1\vert}\pa_{x_j}^{\vert k_2\vert}}\phi(x_l -
x_j)\cdot\frac{\pa^{r+1}}{\pa_{v_l}^{\vert k_1\vert}\pa_{v_j}^{\vert
k_2\vert}}\(S_N\(\tau\)G\)\(Z_N\).\nonumber\\
&&
\end{eqnarray}
Note that the derivatives involved here are done with respect to the
variables at time $t=0$.
Denoting by $D_{z_j}^{r}$ any derivative of order $r$ with respect to
a variable $z_j$ at time $t=0$, we observe that:
\begin{eqnarray}\label{eq: derTEMPsec7}
&&
S_N(-t)D_{z_j}^{r}G(Z_N)=\(D_{z_j}^{r}G\)(Z_N(t))=D_{z_j}^{r}(t)\(S_N(-t)G\)(Z_N),
\end{eqnarray}
where, by $D_{z_j}^{r}(t)$, we denote the same derivative of order $r$
with respect to the variable $z_j(t)$. Then, by (\ref{eq:
derTEMPsec7}) and (\ref{eq: opTtemposuG}):
\begin{eqnarray}\label{eq: opTtemposuGbis}
&&\left(T_N^{(r)}\(\tau\)G\right)\(Z_N\)=S_N\(-\tau\)\(\hat{T}_N^{(r)}
%\left(S_N\(-\tau\)G\right)\(Z_N\(\tau\)\)
+R_N^{(r)}\)S_N\(\tau\)G\(Z_N\)=\nonumber\\
&&=(-1)^{r/2}\frac{c_r}{N}\sum_{j,l}\(D_x^{r+1}\phi\)(x_j(\tau)-x_l(\tau))\cdot
D_{v_j}^{r+1}(\tau)G\(Z_N\)+\nonumber\\
&&+\frac{1}{N}\sum_{l,j=1}^{N}\sum_{\substack{k_1,k_2\in
\mathbb{N}^{3}\\ \vert k_1\vert+\vert k_2\vert=r+1}}C_{k_1,
k_2}\(\frac{\pa^{r+1}}{\pa_{x_l}^{\vert k_1\vert}\pa_{x_j}^{\vert
k_2\vert}}\phi\)(x_l(\tau) -
x_j(\tau))\cdot\frac{\pa^{r+1}}{\pa_{v_l}^{\vert
k_1\vert}\pa_{v_j}^{\vert k_2\vert}}(\tau)G\(Z_N\).\nonumber\\
&&
\end{eqnarray}
Therefore, in computing the action of $T_N^{(r)}(\tau)$, we have to
consider derivatives with respect to the variables at time $\tau$.
As a consequence, we have to deal with a complicated function of the
configuration $Z_N$ which, however, we do not need to make explicit,
as we shall see in a moment.
On the basis of the previous considerations,
we compute the time derivative of $\eta_1$ by applying the operators
$\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots
T_N^{(r_{1})}(t_{1}) $ to the Vlasov equation:
\begin{eqnarray}\label{eq: VlasovWEAKcap7}
&&\left(\pa_t+v'_1\cdot
\nabla_{x'_1}\right)\mu_N(t)=\(\nabla_{x'_1}\phi\ast\mu_N(t)\)\cdot\nabla_{v'_1}\mu_N(t).
\end{eqnarray}
In doing this we have to compute
\begin{eqnarray}\label{eq: VlasovWEAKcap7BIS}
&&\mathcal{D}^{2r}T_N^{(r_{n})}(t_{n})T_N^{(r_{n-1})}(t_{n-1}) \dots
T_N^{(r_{1})}(t_{1}) \mu_N(z'_1\vert Z_N(t))\mu_N(z'_2\vert Z_N(t)).
\end{eqnarray}
Now we select the contribution in which each
$T_N^{(r_{\ell})}(t_{\ell})$ and $\mathcal{D}^{2r}$ apply either on $
\mu_N(z'_1\vert Z_N(t))$ or to $\mu_N(z'_2\vert Z_N(t))$. The other
contribution involves terms in which are present products of
derivatives with respect to the same variable. By Proposition
\ref{DERIVATE1} and Corollary \ref{PROP5.2} we expect those terms to
be negligible (in the $\mathcal{C}^{\infty}_{b}$-weak sense) in the
limit $N\to\infty$.
Therefore we obtain the following equation:
\begin{eqnarray}\label{eq: eqPEReta}
&&\left(\pa_t+v'_1\cdot \nabla_{x'_1}\right)\eta_1(z'_1,t,r,{\bf
\underline{r}}_n,{\bf
\underline{t}}_n,Z_N)=L(\mu_N(t))\eta_1(z'_1,t,r,{\bf
\underline{r}}_n,{\bf \underline{t}}_n,Z_N)+\nonumber\\
&&\quad +\sum_{0\leq \ell\leq r}\ \sum_{0\leq m\leq
n}\sum_{\substack{I\subset I_n:\\ \vert I\vert=m,\\0<\vert{\bf
\underline{r}}_I\vert+\ell0\\\vert {\bf \underline{r}}_n\vert
=k-r}}\left.\int_{0}^t \ud t_2\int_0^{t_2}\ud t_3\dots
\int_0^{t_{n-1}}\ud t_n \eta_1\(z'_1;t,r,{\bf \underline{r}}_n,{\bf
\underline{t}}_n,Z_N\)\right \vert_{t_1=t}+\nonumber\\
&&+\sum_{n\geq 0}(-1)^n\sum_{r=0}^{k}\sum_{\substack{\vert {\bf
\underline{r}}_n\vert:\\r_j>0\\ \vert {\bf \underline{r}}_n\vert
=k-r}}\int_{ord}^{t}\ud {\bf \underline{t}}_n \left(\pa_t+v'_1\cdot
\nabla_{x'_1}\right)\eta_1(z'_1;t,r,{\bf \underline{r}}_n,{\bf
\underline{t}}_n,Z_N).
\end{eqnarray}
In evaluating the first term on the right hand side of (\ref{eq:
eqPERnu}), we are lead to consider $\eta_1$ evaluated in $t=t_1$.
Thus, according to the expression of $\eta_1$ (see (\ref{eq:
aUnCorpo})), we have to deal with:
\begin{eqnarray}\label{eq: sorgPERnu}
T_N^{(r_1)}(t)\mu_N(z'_1\vert Z_N(t))=S_N(-t)T_N^{(r_1)}\mu_N(z'_1\vert Z_N).
\end{eqnarray}
Therefore:
\begin{eqnarray}\label{eq: sorgPERnuCONTO}
&&T_N^{(r_1)}(t)\mu_N(z'_1\vert Z_N(t))
= (-1)^{r_1/2}c_{r_1}\left(D_{x'_1}^{r_1
+1}\phi\ast\mu_N(t)\right)(x'_1)\cdot D_{v'_1}^{r_1 +1}\mu_N(z'_1\vert
Z_N(t))=\nonumber\\
&& = (-1)^{r_1/2}c_{r_1}\int
\ud x'_2\ \ud v'_2\ D_{x'_1}^{r_1 +1}\phi(x'_1-x'_2)\cdot
D_{v'_1}^{r_1 +1}\mu_N(x'_1,v'_1\vert Z_N(t))\mu_N(x'_2,v'_2\vert
Z_N(t)),\nonumber\\
&&
\end{eqnarray}
where the term involving off-diagonal derivatives, namely
$R_N^{(r_1)}$ (see (\ref{eq: OFFdiagOPt})),
disappears because both the derivatives and the empirical distribution
are evaluated at time $t$.
Hence we compute $\eta_1$ in $t=t_1$ and, inserting it in the first
term of the right hand side of (\ref{eq: eqPERnu}), we obtain:
\begin{eqnarray}\label{eq: sorgPERnuRESULT}
&& \sum_{n\geq 0}(-1)^n\sum_{r=0}^{k}\ \sum_{\substack{ {\bf
\underline{r}}_n:\ r_j>0\\\vert {\bf \underline{r}}_n\vert =k-r}}\
\left.\int_{0}^{t}\ud t_2\int_{0}^{t_2}\ud t_3 \ud t_3\dots
\int_{0}^{t_{n-1}}\ud t_n \eta_1\(z'_1;t,r,{\bf \underline{r}}_n,{\bf
\underline{t}}_n,Z_N\)\right \vert_{t_1=t}=\nonumber\\
&&=\sum_{ \substack{00$, independent of $N$,
such that:}\nonumber\\
&&\left\| u_1 \ast \ \(u_2\ \nu_{1}^{(k)}(t)\)
\right\|_{L^{\infty}\(\R^3\times\R^3\)}0\\\vert {\bf \underline{r}}_n\vert =k-r}}\
\int_{ord}^{t}\ud {\bf \underline{t}}_n\sup_{x'_1,v'_1}\left\vert\int
\ud y\ud w\ u_1(x'_1-y, v'_1-w)u_2(y,w)\eta_{1}(y, w; t;r,{\bf
\underline{r}}_n ,{\bf \underline{t}}_n ,Z_N)\right\vert=\nonumber\\
&& =\sum_{n\geq 0}\sum_{r=0}^{k}\ \sum_{\substack{{\bf
\underline{r}}_n:\ r_j>0\\\vert {\bf \underline{r}}_n\vert =k-r}}\
\int_{ord}^{t}\ud {\bf \underline{t}}_n\nonumber\\
&&\ \ \ \sup_{x'_1,v'_1}\left\vert\int \ud y\ud w\ \left(u_1(x'_1-y,
v'_1-w)u_2(y,w)\right)\mathcal{D}^{2r}T_N^{(r_n)}(t_n)\dots
T_N^{(r_1)}(t_1)\mu_N(y, w\vert Z_N(t))\right\vert=\nonumber\\
&& =\sum_{n\geq 0}\sum_{r=0}^{k}\ \sum_{\substack{{\bf
\underline{r}}_n:\ r_j>0\\\vert {\bf \underline{r}}_n\vert =k-r}}\
\int_{ord}^{t}\ud {\bf \underline{t}}_n\nonumber\\
&&\ \ \ \sup_{x'_1,v'_1}\left\vert\int \ud y\ud w\
g(x'_1,v'_1,y,w)\mathcal{D}^{2r}T_N^{(r_n)}(t_n)\dots
T_N^{(r_1)}(t_1)\mu_N(y, w\vert Z_N(t))\right\vert,\nonumber\\
&&
\end{eqnarray}
where we used the notation $g(x'_1,v'_1,y,w):=u_1(x'_1-y,
v'_1-w)u_2(y,w)$ and,
%in order to emphasize that, in this context, $x$ and $v$ play the
role of parameters and the variables are indeed $y$ and $w$.
clearly, we have $g(x'_1,v'_1,\cdot,\cdot)\in
\mathcal{C}^{\infty}_b(\R^3\times\R^3)$ for any $x'_1$ and $v'_1$ and
$g(\cdot,\cdot,y,w)\in \mathcal{C}^{\infty}_b(\R^3\times\R^3)$ for any
$y$ and $w$.
By some estimates which will be proven in Appendix C (see Lemma C.2),
we are guaranteed that, applying the operator
$\mathcal{D}^{2r}T_N^{(r_n)}(t_n)\dots T_N^{(r_1)}(t_1)$ on the
empirical measure $\mu_N(t)$ and integrating versus a function in
$\mathcal{C}^{\infty}_b(\R^3\times\R^3)$ we obtain a quantity
uniformly bounded in $N$. This feature, by virtue of the good
properties of the function $g$
ensures that
(\ref{eq:uniformBOUNDNESSproof}) is finite.
Let us now look at the initial datum for $\nu_1^{(k)}(t)$, in order to
verify assumption {\bf iii)}.
From (\ref{eq: eqPERnuINDAT}) we know that
$\nu_1^{(k)}(0)=\mathcal{D}^{2k}\mu_N\in
\mathcal{S}'(\R^{3}\times\R^{3})$. As regard to its limiting behavior,
we find that:
\begin{eqnarray}\label{eq: convINDATstruttura}
&&\left. \nu_{1}^{(k)}(t)\right\vert_{t=0}=\mathcal{D}^{2k}\mu_N=
\sum_{n=1}^{N}\ \sum_{\substack{I\subset I_N\\ \vert I\vert=n}}\
\sum_{\substack{s_j: j\in I\\1\leq s_j\leq k\\\sum_{j}
s_j=k}}\prod_{j\in I}D_{G,j}^{2 s_j}\ \mu_N,
\end{eqnarray}
where $I_N=\{1,\dots,N\}$.
For our convenience, we have written the action of the operator
$\mathcal{D}^{2k}$ in a equivalent and slightly different way from
that we used in (\ref{eq: opDstortoGEN}).
We realize that the only surviving term in the sum (\ref{eq:
convINDATstruttura}) is that with $n=1$.
Hence:
\begin{eqnarray}\label{eq: convINDATstruttura3}
&& \ \left. \nu_1^k(t)\right\vert_{t=0}=\sum_{j=1}^N D_{G,j}^{2
k}\mu_N=\frac{1}{N}\sum_{j=1}^{N}D_{G,j}^{2
k}\delta(z'_1-z_j)=D_{G}^{2 k}\mu_N.
\end{eqnarray}
Therefore we can conclude, by using the mean-field limit:
\begin{eqnarray}\label{eq: convINDAT}
\left(u, \nu_{1}^{(k)}(t)\vert_{t=0}\right)&&=\left(u,D_{G}^{2
k}\mu_N\right)=\nonumber\\
&&=\left( D_{G}^{2 k}u,\mu_N\right)\rightarrow \left(D_{G}^{2 k}
u,f_0^{(0)}\right)=\(u,D_{G}^{2
k}f_0^{(0)}\right)=\(u,f_0^{(k)}\right),\ \text{as}\ \ N\to\infty,\ \
\nonumber\\
&&\ \forall\ u\ \text{in}\ \
\mathcal{C}_{b}^{\infty}\left(\R^3\times\R^3\right).
\end{eqnarray}
Thus, $f_0^{(k)}$ plays the role of $\gamma_0$ in Proposition
\ref{PROP6.2} and it is in $L^1\(\R^3\times \R^3\)$ because
$f_0^{(0)}\in \mathcal{S}\(\R^3\times \R^3\)$.
We conclude the convergence proof (for the one and two-particle
functions) by induction.
For $k=0$ we know that, for any configuration $Z_N$ which is typical
with respect to $f_0^{(0)}$,
we have:
\begin{eqnarray}\label{eq:convergenzaQOclas}
&&\nu_{1}^{(0)}(t)=\mu_N(t)\rightarrow f^{(0)}(t),\ \ \text{as}\ \ N\to\infty
\end{eqnarray}
in the weak sense of probability measures (see (\ref{eq: class11}) and
(\ref{eq: class12})) and, as a consequence,
the convergence holds $\mathcal{C}_{b}^{\infty}-\text{weakly}$. Moreover
\begin{eqnarray}\label{eq:convergenzaQOclasAdue}
&&\nu_{2}^{(0)}(t)=\mu_N(t)\otimes\mu_N(t)\rightarrow
f_2^{(0)}(t)=f^{(0)}(t)\otimes f^{(0)}(t),\ \text{as}\ \ N\to\infty,
\end{eqnarray}
in the weak sense of probability measures
and, as previously,
the convergence holds $\mathcal{C}_{b}^{\infty}-\text{weakly}$.
We make the following inductive assumptions for all $h0$, is the distribution acting as
\begin{eqnarray} \label{eq: WKB3}\\
\(u,D_v^m\delta(v-v_0)\)&&=\int
%\ud x
\ud v\ u(v) D_v^m\delta(v-v_0)=\nonumber\\
&&=(-1)^m\int \ud v\ D_v^m u(v)\delta(v-v_0)=(-1)^m D_v^m u(v_0),\nonumber\\
&&
\end{eqnarray}
for any smooth test function $u$.
Then, we consider the Wigner function
\begin{eqnarray}\label{eq: WKB4}
f^\var_{W}(x,v)=\int \ud v_0\ g_W(v_0)f^\var_{WKB}(x,v\vert v_0), \ \
\ \ \ \ g_W\in\mathcal{S}\(\R^3\)
\end{eqnarray}
associated with the (continuum) mixed state (see paragraph ''Mixed
states'' in Section 2.1) described by the density matrix (kernel)
\begin{eqnarray}\label{eq: WKB5}
\rho_{W}(x,v)= \int \ud v_0 g_W(v_0)\overline{\psi}_{WKB}(x\vert v_0)
\psi_{WKB}(y\vert v_0)
\end{eqnarray}
and we assume $g_W$ to be a probability density with respect to the
velocity variable, $g_W$ not depending on $\var$ (states similar to
(\ref{eq: WKB4}) have been considered in \cite{Graffi}).
By (\ref{eq: WKB2}), (\ref{eq: coeff_0WKB}), (\ref{eq: coeffODDWKB})
and (\ref{eq: coeffEVENwkb}) we obtain that (\ref{eq: WKB4}) is
expanded as:
\begin{eqnarray}\label{eq: WKB6}
f^\var_{W}(x,v)=f^{(0)}_{W}(x,v)+\var f^{(1)}_{W}(x,v)+\var^2
f^{(2)}_{W}(x,v)+\dots
\end{eqnarray}
where
\begin{eqnarray}
&& f_{W}^{(0)}(x,v)=a^2(x)g_W(v),\label{eq: coeff_0WKBmix}\\
&& f_{W}^{(2n+1)}(x,v)=0\ \ \ \ \ \ \forall\ \
n=0,1,2,\dots,\label{eq: coeffODDWKBmix}\\
&& f_{W}^{(2n)}(x,v)=-\frac{1}{(2)^{2n}}\sum_{l=0}^{2n}
\frac{1}{l!}\frac{1}{(2n-l)!}(-1)^l D^l a(x) D^{2n-l }a(x)
D_v^{2n}g_W(v).\ \ \ \forall\ n=0,1,2,\dots \nonumber\\
&& \label{eq: coeffEVENwkbMIX}
\end{eqnarray}
By virtue of our assumptions on $a$ and $g_W$ we find that
$f_{W}^{(0)}$ defined by (\ref{eq: coeff_0WKBmix}) is a one-particle
probability density and $f_{W}^{(0)}\in\mathcal{S}\(\R^3\)$.
Furthermore, we choose $a$ in such a way that
\begin{eqnarray} \label{eq: A}
&& D^m a(x)= \alpha^{(m)}(x) a(x), \ \forall\ m\geq 1\ \ \ \
\text{with}\ \ \ \alpha^{(m)}(x)\in C^0\(\R^3\),
\end{eqnarray}
$C^0\(\R^3\)$ being the space of continuous functions. Therefore, by
(\ref{eq: coeffEVENwkbMIX}) we find
\begin{eqnarray}
&& f_{W}^{(2n)}(x,v)=-\frac{1}{(2)^{2n}}\sum_{l=0}^{2n}
\frac{1}{l!}\frac{1}{(2n-l)!}(-1)^l D^l a(x) D^{2n-l }a(x)
D_v^{2n}g_W(v)=\nonumber\\
&&\qquad\qquad\ \ = \beta^{(2n)}(x)
a^2(x)D_v^{2n}g_W(v)=\beta^{(2n)}(x) D_v^{2n}f_{W}^{(0)}(x,v)\ \ \
\forall\ n=0,1,2,\dots \nonumber\\
&& \label{eq: coeffEVENwkbMIXagain}
\end{eqnarray}
where $\beta^{(2n)}(x)=-\frac{1}{(2)^{2n}}\sum_{l=0}^{2n}
\frac{1}{l!}\frac{1}{(2n-l)!}(-1)^l \alpha^{(l)}(x)\alpha^{(2n-l)}(x)$
for any $n\geq 1$ and in the last equality of (\ref{eq:
coeffEVENwkbMIXagain}) we used (\ref{eq: coeff_0WKBmix}). By the
smoothness of $\alpha$ (see (\ref{eq: A})) it follows that
$\beta^{(2n)}(x)\in C^0\(\R^3\)$ for all $n$, thus (\ref{eq:
coeffEVENwkbMIXagain}) together with the smoothness of $f_{W}^{(0)}$
implies that $f_{W}^{(2n)}\in \mathcal{S}\(\R^3\times\R^3\)$ for all
$n\geq 1$.
Therefore, we have
\begin{eqnarray}
&& f_{W}^{(0)}\in \mathcal{S}\(\R^3\times\R^3\)
\label{eq: coeff_0WKBmixREG}\\
&& f_{W}^{(2n+1)}(x,v)=0\ \ \ \ \ \ \ \ \forall\
n=0,1,2,\dots,\label{eq: coeffODDWKBmixREG}\\
&& f_{W}^{(2n)}\in \mathcal{S}\(\R^3\times\R^3\)
\ \ \ \forall\ n=1,2,\dots \nonumber\\
&& \label{eq: coeffEVENwkbMIXREG}
\end{eqnarray}
and, by applying Proposition \ref{PROP6.1} as in Section 4.1, we can
identify the time-evolved coefficients $f_W^{(k)}(t)$, for each
$k\geq 0$, as the unique solutions of problems (\ref{eq:
WigLIOUseqk=0}) and (\ref{eq: WigLIOUseq}) in $L^1
\(\R^3\times\R^3\)$. Clearly, by (\ref{eq: coeffODDWKBmixREG}) we find
\begin{eqnarray}\label{eq: dispariNULLI}
&& f_{W}^{(2n+1)}(x,v;t)\equiv 0,\ \ \ \text{for each}\ n\geq 0 \
\text{and}\ \forall\ \ t,
\end{eqnarray}
while $f_W^{(2n)}(t)\in\mathcal{S}
\(\R^3\times\R^3\)$ for any $n\geq 0$. In particular, $f_W^{(0)}(t)$
is the unique solution of the Vlasov equation (\ref{eq: Vlasov}) with
initial datum $f_W^{(0)}$, thus we are guaranteed that $f_W^{(0)}(t)$
is a one-particle probability density for all times.
Let us consider the following factorized initial datum for the
$N$-particle Wigner-Liouville equation (\ref{eq: WigLIOUhartreeNsec3})
\begin{eqnarray}\label{eq: WKBindatN}
&& W_{N,W}^\var(X_N,V_N)=\(f_W^{\var}\)^{\otimes N}(X_N,V_N).
\end{eqnarray}
Then we find
\begin{eqnarray}\label{eq: WKBindatNexp}
&& W_{N,W}^\var(X_N,V_N)=W_{N,W}^{(0)}(X_N,V_N)+\var
W_{N,W}^{(1)}(X_N,V_N)+\var^2 W_{N,W}^{(2)}(X_N,V_N)+\dots,\nonumber\\
&&
\end{eqnarray}
where, by (\ref{eq: WKB6}),
\begin{eqnarray}
&& W_{N,W}^{(0)}(X_N,V_N)=\(f_W^{(0)}\)^{\otimes N}(X_N,V_N)\label{eq:
coeff_0WKBmixFACT}\\
&& W_{N,W}^{(n)}(X_N,V_N)=\sum_{\substack{s_1\dots s_N\\0\leq s_j\leq
n\\ \sum_j s_j=n}}\prod_{j=1}^{N}f_W^{(s_j)}(x_j, v_j)\ \ \
\text{for}\ n\geq 1,\label{eq: coeffNindat}
\end{eqnarray}
and we note that, as in the case we considered in Chapter 4,
factorization holds only for the zero order coefficient which,
furthermore, turns to be an $N$-particle probability density.
By (\ref{eq: coeffNindat}), thanks to (\ref{eq: coeffODDWKBmix}) and
(\ref{eq: coeffEVENwkbMIXagain}), we find
\begin{eqnarray}
&& W_{N,W}^{(0)}(X_N,V_N)=\(f_W^{(0)}\)^{\otimes N}(X_N,V_N)\label{eq:
coeff_0WKBmixFACTII}\\
&& W_{N,W}^{(n)}(X_N,V_N)=\sum_{\substack{s_1\dots s_N
\text{even}\\0\leq s_j\leq n\\ \sum_j
s_j=n}}\prod_{j=1}^{N}\beta^{(s_j)}(x_j)D_{v_j}^{s_j}W_{N,W}^{(0)}(X_j,V_j)\ \
\ \text{for}\ n\geq 1.\label{eq: coeffNindatII}
\end{eqnarray}
Defining the operator $\hat{\mathcal{D}}^{r}$, for any $r$ even, as:
\begin{eqnarray}\label{eq: opDstortoGENwkb}
&&\hat{\mathcal{D}}^{0}=\texttt{1},\nonumber\\
&&\hat{\mathcal{D}}^{r}=\sum_{\substack{s_1\dots
s_N\text{even}:\\0\leq s_j\leq k\\\sum_j
s_j=r}}\prod_{j=1}^{N}\beta^{(s_j)}(x_j)D_{v_j}^{s_j},\ \ \ r\geq 2,
\end{eqnarray}
we have:
\begin{eqnarray}
&&W_{N,W}^{(2n+1)}(X_N,V_N)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \text{for}\ n=0,1,2,\dots\label{eq: W_N,W^kODD}\\
&&W_{N,W}^{(2n)}(X_N,V_N)=\hat{\mathcal{D}}^{2n}W_{N,W}^{(0)}(X_N,V_N)\ \ \ \
\text{for}\ n=0,1,2,\dots.\label{eq: W_N,W^kEVEN}
\end{eqnarray}
Now, let us consider the factorized $j$-particle Wigner function
$\(f_W^{\var}(t)\)^{\otimes j}$, where $f_W^{\var}(t)$ is the solution
of the (Hartree) nonlinear Wigner-Liouville equation (\ref{eq:
WigLIOUhartree}) with initial datum $f_W^\var$ given by (\ref{eq:
WKB4}). The product $\(f_W^{\var}(t)\)^{\otimes j}$ can be expanded as
\begin{eqnarray}\label{eq: WKBindatINFexp}
&& \(f_W^{\var}(t)\)^{\otimes j}=f_{j,W}^{(0)}(t)+\var
f_{j,W}^{(1)}(t)+\var^2 f_{j,W}^{(2)}(t)+\dots,\nonumber\\
&&
\end{eqnarray}
where, by the analysis done in Section 4.3 and thanks to (\ref{eq:
coeffODDWKBmix}), we have
\begin{eqnarray}
&& f_{j,W}^{(0)}(t)=\(f_W^{(0)}(t)\)^{\otimes j}\label{eq: coeffF_jwkb}\\
&& f_{j,W}^{(2n+1)}(t)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \forall\ n\geq 0\label{eq: coeffF_jwkbODD}\\
&& f_{j,W}^{(2n)}(t)=\sum_{\substack{s_1\dots s_j \text{even}\\0\leq
s_m\leq 2n\\ \sum_m s_m=2n}}\prod_{m=1}^{j}f_{W}^{(s_m)}(t),\ \ \
\forall\ n\geq 1,
\end{eqnarray}
$f_W^{(0)}(t)$ solving the Vlasov equation (\ref{eq: Vlasov}) with
initial datum $f_W^{(0)}$ and $f_{W}^{(s_m)}(t)$, with $1\leq s_m\leq
2n$, obtained by (\ref{eq: WigLIOUseq}).
By the analysis done in Section 4.2, we find that the $N$-particle
zero order coefficient $W_{N,W}^{(0)}(t)$ solves
\begin{equation}\label{eq: WigLiouNk=0WKB}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot
\nabla_{X_N}\right)W_{N,W}^{(0)}(t)=T_{N}^{(0)}W_{N,W}^{(0)}(t),\\
&\left.W_{N,W}^{(0)}(X_N,V_N;t)\right\vert_{t=0}=\(f_W^{(0)}\)^{\otimes
N}(X_N,V_N)
\end{aligned}
\right.
\end{equation}
the odd coefficients $W_{N,W}^{(2n+1)}(t)$ are determined by
\begin{equation}\label{eq: WigLiouWKBODD}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot
\nabla_{X_N}\right)W_{N,W}^{(2n+1)}(t)=T_N^{(0)}W_{N,W}^{(2n+1)}(t),\\
&\left.W_{N,W}^{(2n+1)}(X_N,V_N;t)\right\vert_{t=0}=0,\ \ \ \ \ \ \
k=0,1,2,\dots
\end{aligned}
\right.
\end{equation}
and the even terms $W_{N,W}^{(2n)}(t)$ solve
\begin{equation}\label{eq: WigLiouNWKBeven}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot
\nabla_{X_N}\right)W_{N,W}^{(2n)}(t)=T_N^{(0)}W_{N,W}^{(2n)}(t)+\Theta_{N,W}^{(2n)}(t),\\
&\left.W_{N,W}^{(2n)}(X_N,V_N;t)\right\vert_{t=0}=\hat{\mathcal{D}}^{2n}W_{N,W}^{(0)}(X_N,V_N),\ \ \ \ \ \ \
n=1,2,\dots
\end{aligned}
\right.
\end{equation}
where
\begin{equation}\label{eq: Teta_Neven}
\Theta_{N,W}^{(2n)}(t)=\sum_{\substack{0\leq
l<2n}}T_N^{(2n-l)}W_{N,W}^{(l)}(t).
\end{equation}
We observe that the odd coefficients $W_{N,W}^{(2n+1)}(t)$ solve
homogeneous Liouville equations with zero initial data, then
$W_{N,W}^{(2n+1)}(t)\equiv 0$ for all $n\geq 0$. On the contrary, the
zero order term $W_{N,W}^{(0)}(t)$ solve the Liouville problem
(\ref{eq: WigLiouNk=0WKB}), thus, denoting by
$\{W_{j,W}^{(0)}(t)\}_{j=1}^N$ the corresponding $j$-particle
marginals, by the classical mean-field theory we obtain
\begin{equation}\label{eq: convWKBzero}
W_{j,W}^{(0)}(t)\rightarrow \(f_W^{(0)}(t)\)^{\otimes j}, \ \ \
\text{as}\ N\to\infty,\ \ \ \ \text{for any fixed}\ \ j
\end{equation}
in the weak topology of probability measures and, in particular, in
$\mathcal{S}'\(\R^{3j}\times\R^{3j}\)$, where
$\(f_W^{(0)}(t)\)^{\otimes j}$ is given by (\ref{eq: coeffF_jwkb}).
As regard to the higher order terms, we can apply exactly the same
strategy presented in Chapter 4, replacing the operator $D^k_G$
defined in (\ref{eq: operatorD_G}) with $\beta^{(k)}(x)D_v^k$ (see
(\ref{eq: coeffEVENwkbMIXagain}) ) and the operator $\mathcal{D}^r$
(see Section 4.6) with $\hat{\mathcal{D}}^r$ (see (\ref{eq:
opDstortoGENwkb})). In the end we prove that, for $k\geq 1$
\begin{equation}\label{eq: convWKB}
W_{j,W}^{(k)}(t)\rightarrow f_{W,j}^{(k)}(t), \ \ \ \text{as}\
N\to\infty,\ \ \ \ \text{for any fixed}\ \ j
\end{equation}
in $\mathcal{S}'\(\R^{3j}\times\R^{3j}\)$. Clearly, (\ref{eq:
convWKB}) is trivially verified for $k=2n+1,$
because we have already noticed that
\begin{equation}\label{eq: DISPARItrivial}
W_{j,W}^{(2n+1)}(t)\equiv f_{j,W}^{(2n+1)}(t)\equiv 0,\ \ \ \forall \
n\geq 0.
\end{equation}
\vspace{0.3cm}
As we observed in Remark \ref{Remark6.1}, the assumption on the
initial $N$-particle Wigner function to be a product of mixed states
prevents the possibility of considering bosonic states.
Indeed,
factorized states could be compatible with the bosonic statistics if
pure states were considered.\\
It is easy to check that the classical limit is equivalent to the
limit of heavy particles. In fact, if we set $\var= 1$ in the
$N$-particle Hamiltonian (\ref{eq: quant2}) but we let the particle
mass $m$ (previously chosen equal to 1) become large, by imposing the
condition that the kinetic
energy per particle is independent of $m$, we find exactly the
mean-field hamiltonian (\ref{eq: quant2}) where $\var$ is replaced by
the ''effective Planck constant'' $\var_{m}=1/\sqrt{m}$ going to zero
as $m\to\infty$. Then a possible application of our result would be
that of studying the approximations of dynamics of this type, in which
one looks at a particular scaling which corresponds to the
semiclassical one, even if interpreted in a different sense.
\newpage
\section*{Appendix A}
\setcounter{equation}{0}
\def\theequation{A.\arabic{equation}}
{\bf Proof of Proposition \ref{DERIVATE1}}\\
To avoid inessential notational complications, we deal with the
one-dimensional case. \\
By the Newton equations, we have:
\begin{eqnarray}\label{eq:est7v}
&&\frac{\pa x_{i}(t)}{\pa v_r}=\delta_{ir}t+\int_{0}^{t}\ud s
(t-s)\frac{1}{N}\sum_{ j\neq i}^N
\pa_{x}F\left(x_{i}(s)-x_{j}(s)\right)\left(\frac{\pa x_{i}(s)}{\pa
v_{r}}-\frac{\pa x_{j}(s)}{\pa v_{r}}\right),\\
&&\nonumber\\
&&\frac{\pa v_{i}(t)}{\pa v_{r}}=\delta_{ir} + \int_{0}^{t} \ud
s\frac{1}{N}\sum_{j\neq i}^{N}\pa_{x}
F\left(x_{i}(s)-x_{j}(s)\right)\left(\frac{\pa x_{i}(s)}{\pa
v_r}-\frac{\pa x_{j}(s)}{\pa v_{r}}\right),\label{eq:est7vBIS}
\end{eqnarray}
where:
\begin{eqnarray}\label{eq:est1}
F=-\nabla_{x}\phi,
\end{eqnarray}
is the force associated with the potential $\phi$.
Let us analyze in detail the derivative of $x_{i}(t)$. From
(\ref{eq:est7v}), we get:
\begin{eqnarray}\label{eq:estAppA}
&&
\max_{\substack{i,r\\t\leq T}}\left\vert \frac{\pa x_{i}(t)}{\pa
v_r}\right\vert\leq C.
\end{eqnarray}
Inserting this estimate again in (\ref{eq:est7v}), we realize that we
can obtain a better bound for $\frac{\pa v_{i}(t)}{\pa v_{r}}$ in the
case $r\neq i$ (see \cite{Graffi}), namely:
\begin{eqnarray}\label{eq:estAppA1}
\left\vert \frac{\pa x_{i}(t)}{\pa v_r}\right\vert &&\leq C
\int_{0}^{t}\ud s (t-s) \left\vert\frac{\pa x_{i}(s)}{\pa
v_{r}}\right\vert+\\
&&+C \int_{0}^{t}\ud s (t-s)\frac{1}{N}\left\vert \frac{\pa
x_{r}(s)}{\pa v_{r}}\right\vert +\nonumber\\
&&+C \int_{0}^{t}\ud s (t-s)\frac{1}{N}\sum_{ \substack{j\neq i\\j\neq
r}}^N \pa_{x}F\left(x_{i}(s)-x_{j}(s)\right)\left\vert \frac{\pa
x_{j}(s)}{\pa v_{r}}\right\vert.\nonumber\\
&&
\end{eqnarray}
Hence, by virtue of the Gronwall lemma, we find:
\begin{eqnarray}\label{eq:estAppA2}
&&
\max_{\substack{i\neq r\\ t\leq T}}\left\vert \frac{\pa x_{i}(t)}{\pa
v_r}\right\vert\leq \frac{C}{N}.
\end{eqnarray}
By (\ref{eq:est7vBIS}), we find that the same estimate holds for the
derivative of $v_i(t)$ with respect to $v_r$.
Analogous estimates hold for the derivatives with respect to the
initial positions (see also \cite{Graffi}).\\
Therefore the claim of Proposition \ref{DERIVATE1} is proven for
derivatives of order one.
Now, let us consider a sequence $I:=(j_1,\dots, j_k)$ of possibly
repeated indices. We show that:
\begin{eqnarray}\label{eq:estAppA3}
%\text{sup}_{t >0}
\frac{1}{N}\sum_{i=1}^N
\left\vert \frac{\pa^k x_{i}(t)}{\pa v_{j_1}\dots \pa
v_{j_k}}\right\vert \leq \frac{C}{N^{d_k}},
\end{eqnarray}
where $d_k$ is the number of different indices in the sequence
$j_1,\dots, j_k$.
We know that (\ref{eq:estAppA3}) is verified for $k=1$ (it follows
directly by (\ref{eq:estAppA}) and (\ref{eq:estAppA2})), thus we prove
(\ref{eq:estAppA3}) by induction on $k$.
Denoting by:
\begin{eqnarray}\label{eq:estAppA4}
D(I):= \frac{\pa^k }{\pa v_{j_1}\dots \pa v_{j_k}},
\end{eqnarray}
estimate (\ref{eq:estAppA3}) can be rewritten as:
\begin{eqnarray}\label{eq:estAppA3bis}
\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I)x_i(t)\right\vert \leq
\frac{C}{N^{d_k}}.
\end{eqnarray}
By (\ref{eq:est7v}) we derive the following estimate for $ D(I)x_i(t)$:
\begin{eqnarray}\label{eq:estAppA5prima}
&&
%\frac{1}{N}\sum_{i=1}^{N}
\left\vert D(I) x_{i}(t)\right\vert \leq \int_{0}^{t} \ud s (t-s)
%\frac{1}{N^2}
\frac{C}{N}
%\sum_{i=1}^{N}
\sum_{ j\neq i}^N
%\pa_{x}F\left(x_{i}(s)-x_{j}(s)\right)
\left\vert D(I)\left( x_{i}(s)-x_{j}(s)\right)\right\vert +M_i(t),\nonumber\\
&&
\end{eqnarray}
where the term $M_i(t)$ can be computed from (\ref{eq:est7v})
according to the Leibniz rule. Let $\mathcal{P}_n:=\{I_1,\dots, I_n\}$
be a partition of the set $I$ of cardinality $n$, with $2\leq n\leq
k$, then we have:
\begin{eqnarray}\label{eq:estAppA6prima}
&& M_i(t) \leq\int_{0}^{t} \ud s (t-s)
%\frac{1}{N^2}
\frac{1}{N}
\sum_{ j\neq i}^N
\sum_{n=2}^{k}\sum_{\mathcal{P}_n}C(\mathcal{P}_n)\left\vert\prod_{H\in
\mathcal{P}_n} \[D(H)
\(x_{i}(s)-x_{j}(s)\right)\]\right\vert\leq\nonumber\\
&&\leq\int_{0}^{t} \ud s
(t-s)\sum_{n=2}^{k}\sum_{\mathcal{P}_n}C(\mathcal{P}_n)\frac{1}{N}
\sum_{ j=1}^N \left\vert\prod_{H\in \mathcal{P}_n} \[D(H)
\(x_{i}(s)-x_{j}(s)\right)\]\right\vert,
\end{eqnarray}
where $D(H):= \prod_{h\in H}\frac{\pa}{\pa v_{h}}$ and
$C(\mathcal{P}_n)$ are coefficients depending on the partition
$\mathcal{P}_n$ and on suitable derivatives of $F$.
By
%(\ref{eq:estAppA6prima}) and
(\ref{eq:estAppA5prima}), it follows that:
\begin{eqnarray}\label{eq:estAppA5}
&&
\frac{1}{N}\sum_{i=1}^{N}
\left\vert D(I) x_{i}(t)\right\vert \leq \int_{0}^{t} \ud s (t-s)
\frac{C}{N}
\sum_{i=1}^{N} \left\vert D(I)x_{i}(s)\right\vert
+M(t),
\end{eqnarray}
where $M(t)=\frac{1}{N}\sum_{i=1}^{N}M_i(t)$ and, by
(\ref{eq:estAppA6prima}), we have:
\begin{eqnarray}\label{eq:estAppA6}
&& M(t) \leq\int_{0}^{t} \ud s
(t-s)\sum_{n=2}^{k}\sum_{\mathcal{P}_n}C(\mathcal{P}_n)\frac{1}{N^2}
%\frac{1}{N}
\sum_{i=1}^{N}\sum_{ j=1}^N \left\vert\prod_{H\in \mathcal{P}_n}
\[D(H) \(x_{i}(s)-x_{j}(s)\right)\]\right\vert,\nonumber\\
&&
\end{eqnarray}
We observe that:
\begin{eqnarray}\label{eq:estAppA6bis}
&&\frac{1}{N^2}
%\frac{1}{N}
\sum_{i,j=1}^{N}
%\sum_{ j=1}^N
\left\vert\prod_{H\in \mathcal{P}_n}\left[D(H)
\(x_{i}(s)-x_{j}(s)\right)\]\right\vert\leq\frac{1}{N}
\sum_{i=1}^{N}\prod_{H\in\mathcal{P}_n}\left\vert D(H)x_{i}(s)\right\vert+
%(-1)^n
\frac{1}{N}\sum_{j=1}^{N}\prod_{H\in\mathcal{P}_n}\left\vert
D(H)x_{j}(s)\right\vert+\nonumber\\
&&+\sum_{\mathcal{Q}\subset \mathcal{P}_n}C(\mathcal{Q})\(\frac{1}{N}
\sum_{i=1}^{N}\prod_{Q\in\mathcal{Q}}\left\vert
D(Q)x_{i}(s)\right\vert\)\(\frac{1}{N}
\sum_{j=1}^{N} \prod_{J\in\mathcal{P}_n\setminus\mathcal{Q}
}\left\vert D(J)x_{j}(s)\right\vert\),\nonumber\\
&&
%&&+\sum_{J,H:J\sqcup H=I }C(J,H)D(J)x_{i}(s)D(H)x_{i}(s).
\end{eqnarray}
where $\mathcal{Q}$ is any subpartition of $\mathcal{P}_n$ and
$C(\mathcal{Q})$ are coefficients depending on $\mathcal{Q}$.\\
We assume that the estimate (\ref{eq:estAppA3bis}) holds for any
$m\leq k-1$, namely:
\begin{eqnarray}\label{eq:estAppA7}
%%\text{sup}_{t >0}
%\max_{\substack{i,\ M\subset I:\\\vert M\vert=m}}
\frac{1}{N}\sum_{i=1}^{N}\left\vert D(M) x_{i}(t)\right\vert \leq
\frac{C}{N^{d_m}},\ \ \text{for any } M\subset I\ \text{s.t.\ } \vert
M\vert=m\leq k-1,
\end{eqnarray}
where $d_m$ is the number of different indices in the sequence $M$.\\
%which are different from $i$ (counted without their multiplicity).\\
Indeed, if we consider a partition $\mathcal{P}_n$ of cardinality
$n\geq 2$, we are guaranteed
%by (\ref{eq:estAppA7})
that $\vert M\vert\leq k-1$ for each $M\in\mathcal{P}_n$. Then, by
noting that:
\begin{eqnarray}\label{eq:estAppAnuova}
\frac{1}{N}\sum_{i=1}^{N}\prod_{H\in\mathcal{H}}\left\vert D(H)
x_{i}(t)\right\vert \leq\prod_{H\in\mathcal{H}}
\frac{1}{N}\sum_{i=1}^{N}\left\vert D(H) x_{i}(t)\right\vert,\
\forall\ \text{subpartition} \ \mathcal{H}\subseteq \mathcal{P}_n,
\end{eqnarray}
we can apply the inductive hypotheses (\ref{eq:estAppA7}) to estimate
the derivatives of $x_i(s)$ and $x_j(s)$ appearing in
(\ref{eq:estAppA6bis}). Thus, we obtain:
%in estimating (\ref{eq:estApp6bis}) and we get:
\begin{eqnarray}\label{eq:estAppA8}
\frac{1}{N}\sum_{i=1}^{N}\prod_{H\in \mathcal{P}_n}\left\vert
D(H)x_{i}(s)\right\vert&&\leq \prod_{H\in \mathcal{P}_n}
\frac{1}{N}\sum_{i=1}^{N}\left\vert
D(H)x_{i}(s)\right\vert\leq\nonumber\\
&& \leq\prod_{H\in \mathcal{P}_n}\frac{C}{N^{d_h}}=\frac{C}{N^{\sum
d_h}}\leq \frac{C}{N^{d_k}} ,
\end{eqnarray}
where $d_h$ is the number of different indices in the sequence $H$ and
we used that $\sum_{H\in \mathcal{P}_n}d_h\geq d_k$.\\
% because $H\subset I$.\\
In a similar way, we find
\begin{eqnarray}\label{eq:estAppA9}
\frac{1}{N}\sum_{i=1}^{N}\prod_{Q\in \mathcal{Q}}\left\vert
D(Q)x_{i}(s)\right\vert\leq \prod_{Q\in \mathcal{Q}}\frac{C}{N^{d_q}},
%\leq \frac{C}{N^{d}},
\end{eqnarray}
where $d_q$ is the number of different indices in the sequence
$Q$.\\Moreover, we have:
\begin{eqnarray}\label{eq:estAppA10}
&&\frac{1}{N}\sum_{j=1}^{N}\prod_{H\in \mathcal{P}_n}\left\vert
D(H)x_{j}(s)\right\vert\leq \prod_{H\in
\mathcal{P}_n}\frac{C}{N^{d_h}}=\frac{C}{N^{\sum d_h}}\leq
\frac{C}{N^{d_k}},
\end{eqnarray}
and
\begin{eqnarray}\label{eq:estAppA11}
&&\frac{1}{N}\sum_{j=1}^{N}\prod_{J\in
\mathcal{P}_n\setminus\mathcal{Q}}\left\vert D(J)x_{j}(s)\right\vert
\leq \prod_{J\in \mathcal{P}_n\setminus\mathcal{Q}} \frac{C}{N^{d_j}},
\end{eqnarray}
where $d_j$ is the number of different indices in the sequence $J$.
Then, putting together (\ref{eq:estAppA9}) and (\ref{eq:estAppA11}), we find:
\begin{eqnarray}\label{eq:estAppA12}
&&\sum_{\mathcal{Q}\subset
\mathcal{P}_n}C(\mathcal{Q})\(\frac{1}{N}\sum_{i=1}^{N}\prod_{Q\in\mathcal{Q}}\left\vert D(Q)x_{i}(s)\right\vert\)\(\frac{1}{N}\sum_{j=1}^{N}\prod_{J\in\mathcal{P}_n\setminus\mathcal{Q} }\left\vert
D(J)x_{j}(s)\right\vert\)\leq\nonumber\\
&&\sum_{\mathcal{Q}\subset
\mathcal{P}_n}C(\mathcal{Q})\prod_{Q\in\mathcal{Q}}\prod_{J\in\mathcal{P}_n\setminus\mathcal{Q}
}\frac{C}{N^{d_{q}+d_j}}\leq\nonumber\\
&&\leq \sum_{\mathcal{Q}\subset
\mathcal{P}_n}C(\mathcal{Q})\prod_{Q\in\mathcal{Q}}\prod_{J\in\mathcal{P}_n\setminus\mathcal{Q} }\frac{C}{N^{d_k}}\leq
\frac{C}{N^{d_k}}.
%%
%%\leq \delta_j(I)\frac{C}{N^{d-1}}+(1-\delta_j(I))\frac{C}{N^{d}}.
\end{eqnarray}
In the end, we have just proven that each term in
(\ref{eq:estAppA6bis}) is bounded by $\frac{C}{N^{d_k}}$. Therefore,
by using this estimate in (\ref{eq:estAppA6}), we find:
\begin{eqnarray}\label{eq:estAppA13}
&& M(t) \leq\frac{C}{N^{d_k}}.
\end{eqnarray}
By (\ref{eq:estAppA13}) and (\ref{eq:estAppA5}), it follows that:
\begin{eqnarray}\label{eq:estAppA14}
&&\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I) x_{i}(t)\right\vert \leq
\int_{0}^{t} \ud s (t-s)
%\frac{1}{N}
\frac{C}{N}\sum_{i=1}^{N}\left\vert D(I)x_{i}(s)\right\vert
+\frac{C}{N^{d_k}}.\nonumber\\
&&
\end{eqnarray}
Therefore, by using the Gronwall lemma, we find:
\begin{eqnarray}\label{eq:estAppA15}
&&\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I) x_{i}(t)\right\vert
\leq\frac{C}{N^{d_k}}.
\end{eqnarray}
As regard to the derivatives of $v_i(t)$ with respect to some initial
velocities $v_{j_1},\dots,v_{j_k}$, an analogous estimate holds and
the proof works in the same way.
% by computing the derivative $D(I)$ of $v_i(t)$ and by doing
estimates through an induction procedure.
Furthermore, this strategy leads to the same estimate for the
derivatives of the function $\frac{1}{N}\sum_{i=1}^{N}z_i(t)$ with
respect to some initial positions $x_{j_1},\dots,x_{j_k}$.
Now, thanks to the estimate we have just proven for the derivatives of
the function $\frac{1}{N}\sum_{i=1}^{N}z_i(t)$, we are able to prove
the claim of Proposition \ref{DERIVATE1}. In fact, we have:
\begin{eqnarray}\label{eq: specificoIV}
&&\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I)z_i(t)
\right\vert=\frac{1}{N}\sum_{\substack{i=1\\i\in D}}^{N}\left\vert
D(I) z_{i}(t)\right\vert+\frac{1}{N}\sum_{\substack{i=1\\i\notin
D}}^{N}\left\vert D(I)z_i(t) \right\vert\leq \frac{C}{N^{d_k}},
%$\nonumber\\
\end{eqnarray}
where $D\subset I$ contains the different indices appearing in the
sequence $I$. Thus, according to our previous notation, $\vert D\vert
=d_k$ and we denote the elements of $D$ by
$\tilde{j}_1,\dots,\tilde{j}_{d_k}$. Then by (\ref{eq: specificoIV})
we find:
\begin{eqnarray}\label{eq: correzME1}
\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I)z_i(t) \right\vert&&=\frac{1}{N}
%\sum_{\substack{i\in \(\tilde{j}_1,\dots,\tilde{j}_{d_k}}\)
\left\vert D(I)z_{\tilde{j}_1}(t) \right\vert+\dots +
\frac{1}{N}\left\vert D(I)z_{\tilde{j}_{d_k}}(t) \right\vert+\nonumber\\
&&+\frac{1}{N}\sum_{\substack{i=1\\i\notin D}}^{N}\left\vert
D(I)z_i(t) \right\vert\leq \frac{C}{N^{d_k}},
\end{eqnarray}
which implies
\begin{eqnarray}\label{eq: correzME2}
&&\left\vert D(I)z_i(t) \right\vert\leq
C\(\frac{\sum_{\ell=1}^{d_k}\delta_{i \tilde{j}_\ell}}{N^{d_k
-1}}+\frac{1}{N^{d_k}}\),
\end{eqnarray}
or
\begin{eqnarray}\label{eq: correzME3}
&&\left\vert D(I)z_i(t) \right\vert\leq \frac{C}{N^{d_k^{(i)}}},
\end{eqnarray}
where $d_k^{(i)}$ is the number of different indices in the sequence
$I$ which are also different from $i$.
%So that, the claim of Proposition 5.1 is proven for any $k$.
\\ \spaz \hspace{1cm} \hfill $\square$ \newline
\section*{Appendix B}
\setcounter{equation}{0}
\def\theequation{B.\arabic{equation}}
{\bf Proof of Proposition \ref{PROP6.1}}\\
Let $U_h(t,s)$ be the two parameters semigroup solution of the linear problem:
\begin{equation}\label{eq: AppB1}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)U_h(t,s)\gamma_0=\left(\nabla
\phi\ast h\right)\ast \nabla_v U_h(t,s)\gamma_0,\\
&U_h(s,s)\gamma_0=\gamma_0.
\end{aligned}
\right.
\end{equation}
The solution of (\ref{eq: AppB1}) is obtained by carrying the initial
datum $\gamma_0$ along the characteristic flow
\begin{equation}\label{eq: AppB2}
\left\{
\begin{aligned}
&\dot{x}=v,\\
&\dot{v}=-\nabla \phi \ast h.
\end{aligned}
\right.
\end{equation}
Next, we consider the problem
\begin{equation}\label{eq: AppB3}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\tilde{\gamma}=L(h)\tilde{\gamma},\\
&\tilde{\gamma}\vert_{t=0}=\gamma_0.
\end{aligned}
\right.
\end{equation}
which can be reformulated in integral form:
\begin{equation}\label{eq: AppB4}
\tilde{\gamma}(t)=U_h(t,0)\gamma_0+\int_0^t\ud s \
U_h(t,s)\left[\left(\nabla \phi\ast \tilde{\gamma}(s)\right)\cdot
\nabla_v h(s)\right].
\end{equation}
The above formula can be iterated to yield the formal solution
\begin{eqnarray}\label{eq: expAppB5}
\tilde{\gamma}(x,v;t)=&& U_h(t,0)\gamma_0(x,v)+\sum_{n\geq
1}\int_0^{t}\ud t_1\int_{0}^{t_1}\ud t_2\dots \int_{0}^{t_{n-1}}\ud
t_n\int \ud x_1\int \ud v_1\dots \int \ud x_n\int \ud v_n\nonumber\\
&&U_h(t,t_1)\left[\nabla_v h(x,v;t_1)\cdot
\nabla_x\phi(x-x_1)\right]\nonumber\\
&&U_h(t_1,t_2)\left[\nabla_{v_1} h(x_1,v_1;t_2)\cdot
\nabla_{x_1}\phi(x_1-x_2)\right]\nonumber\\
&&\dots\nonumber\\
&&U_h(t_{n-1},t_n)\left[\nabla_{v_{n-1}} h(x_{n-1},v_{n-1};t_n)\cdot
\nabla_{x_{n-1}}\phi(x_{n-1}-x_n)\right]\nonumber\\
&&U_h(t_{n},0)\gamma_0(x_n, v_n).
\end{eqnarray}
We remark that $U_h(t_k,t_{k+1})$ acts on the variables $x_k, v_k$
with the convention that $(x_0,v_0)=(x,v)$ and, furthermore, $U_h$ is
multiplicative and preserves the $L^p(\R^3\times\R^3)$ norms
$(p=1,2,\dots,\infty)$.\\
Under the assumptions of Proposition \ref{PROP6.1}, the above series
is bounded in $L^1(\R^3\times\R^3)$ by:
\begin{eqnarray}\label{eq: expUNIFBOUNDAppB6}
&&\sum_{n\geq 0}\frac{t^n}{n!}\left(\text{sup}_{\tau \in
[0,t]}\left\|\nabla_v h(\tau)\right\|_{L^1(\R^3\times\R^3)}\right)^n
\left\|\nabla_x \phi\right\|_{L^{\infty}(\R^3)}^n
\left\|\gamma_0\right\|_{L^{1}(\R^3\times\R^3)},\nonumber\\
&&
\end{eqnarray}
which is converging for each $t$. Now, we denote by
$\Sigma_h(t,s):L^1(\R^3\times\R^3)\rightarrow L^1(\R^3\times\R^3)$,
the two parameters semigroup given by the series (\ref{eq: expAppB5}).
Then, the solution $\gamma$ to the problem (\ref{eq: Prop6.1}) is
given by:
\begin{equation}\label{eq: AppB7}
\gamma(t)=\Sigma_h(t,0)\gamma_0+\int_0^t\ud s \ \Sigma_h(t,s)\Theta(s),
\end{equation}
and, thanks to the assumption we made on $\Theta$ and to the fact that
the above series (\ref{eq: expAppB5}) is converging for any $t$, we
are guaranteed that $\gamma\in
\mathcal{C}^0\(L^1(\R^3\times\R^3),\R^+\)$.
The $\mathcal{C}^k$ regularity of
$\tilde{\gamma}(t)=\Sigma_h(t,0)\gamma_0$ follows by (\ref{eq:
expAppB5}) and
%the $\mathcal{C}^k$ regularity and
the fact that $U_h(t,t_1)$ propagates the $\mathcal{C}^k$ regularity.
\newline \spaz \hspace{1cm} \hfill $\square$ \newline
{\bf Proof of Proposition \ref{PROP6.2}}\\
The proof consists of two steps.\\\\
\emph{Step 1)}:
Let $\gamma_N$ be as in Proposition \ref{PROP6.2}. Then, we show that
$\gamma_N$ solves the problem:
\begin{equation}\label{eq: AppB8}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\gamma_N=L(h)\gamma_N+\Theta_{N}',\\
&\gamma_N\vert_{t=0}=\gamma_{N,0},
\end{aligned}
\right.
\end{equation}
with
\begin{equation}\label{eq: AppB9}
\Theta_{N}'=\Theta_N +R_N,
\end{equation}
and $R_N$ is such that:
\begin{equation}\label{eq: AppB10}
R_N\rightarrow 0 ,\ \ \ \mathcal{C}_{b}^{\infty}-\text{weakly}.
\end{equation}
In proving (\ref{eq: AppB10}), the assumption {\bf ii)} on $\gamma_N$
is crucial.\\\\
\emph{Step 2)}:
By virtue of Step 1), the hypotheses we made on $\nabla_v h$ and
Proposition \ref{PROP6.1}, we find that:
\begin{equation}\label{eq: AppB11}
\gamma_N(t)=\Sigma_{h}(t,0)\gamma_{N,0}+\int_0^t\ud s \
\Sigma_{h}(t,s)\Theta_{N}'(s).
\end{equation}
Then, reminding that:\\
$\circ$ $h(t)\in \mathcal{C}_b^{\infty}(\R^3\times\R^3)$ for any $t$,\\
$\circ$ the flow $\Sigma_{h}$ propagates the $\mathcal{C}^k$ regularity,\\
$\circ$ $R_N\rightarrow 0 ,\ \ \ \mathcal{C}_{b}^{\infty}-\text{weakly}$,\\
and by virtue of
the assumptions on $\gamma_{N,0}$ and $\Theta_N$,
% and to what has been proven in Step 1) as regard to $R_N$,
we can easily show that:
\begin{equation}\label{eq: AppB12}
\gamma_N\rightarrow\gamma,\ \text{as}\ \ N\to\infty,\ \
\mathcal{C}_{b}^{\infty}-\text{weakly},
\end{equation}
where
\begin{equation}\label{eq: AppB13}
\gamma(t)=\Sigma_{h}(t,0)\gamma_{0}+\int_0^t\ud s \ \Sigma_{h}(t,s)\Theta(s).
\end{equation}
Therefore, we recognize that $\gamma$ solves the problem (\ref{eq: Prop6.1})
and, by virtue of Proposition \ref{PROP6.1}, it is uniquely determined
by (\ref{eq: AppB13}) and hence it is in
$\mathcal{C}^0\(L^1(\R^3\times\R^3), \R^+\)$.\\\\
\emph{Proof of Step 1)}:\\
We have:
\begin{equation}\label{eq: Prop6.2APP}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot
\nabla_x\right)\gamma_N=L(h)\gamma_N+\Theta_N+L(h_N - h)\gamma_N\\
&\left. \gamma_N(x,v;t)\right\vert_{t=0}=\gamma_{N,0}(x,v),
\end{aligned}
\right.
\end{equation}
where
\begin{equation}\label{eq: AppB15}
R_N=R_N(x,v;t):=L(h_N - h)\gamma_N.
\end{equation}
We want to show that $R_N\to 0$,\ \ $\mathcal{C}_b^{\infty}$-weakly.
According to the definition of the operator $L$, we have:
\begin{equation}\label{eq: AppB16}
R_N=\(\nabla_x\phi\ast (h_N -h)\)\nabla_v\gamma_N+\(\nabla_x\phi\ast
\gamma_N\)\nabla_v(h_N -h),
\end{equation}
thus, we have to show that
\begin{equation}\label{eq: AppB17}
\(u,\(\nabla_x\phi\ast (h_N -h)\)\nabla_v\gamma_N\)\rightarrow 0,\
\text{as}\ \ N\to\infty,\ \ \forall\ u\in
\mathcal{C}_b^{\infty}(\R^3\times\R^3),
\end{equation}
and
\begin{equation}\label{eq: AppB18}
\(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N -h)\)\rightarrow 0,\
\text{as}\ \ N\to\infty,\ \ \forall\ u\in
\mathcal{C}_b^{\infty}(\R^3\times\R^3).
\end{equation}
We show only (\ref{eq: AppB18}) in detail because (\ref{eq: AppB17})
will follow the same line.
We have:
\begin{eqnarray}\label{eq: AppB19}
&&\(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N -h)\)=\int \ud x \ud
v\int \ud y\ud w\ u(x,v)\nabla_x\phi(x-y)\gamma_N(y,w;t)\cdot\nonumber\\
&&\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\nabla_v(h_N(x,v;t) -h(x,v;t))=\nonumber\\
&&\qquad\qquad\qquad\qquad=-\int \ud x \ud v\int \ud y\ud w\ \nabla_v
u(x,v)\nabla_x\phi(x-y)\gamma_N(y,w;t)\cdot\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (h_N(x,v;t)
-h(x,v;t))=\nonumber\\
&&\qquad\qquad\qquad\qquad=\int \ud x \ud v\int \ud y\ud w\ \nabla_v
u(x,v)\(\nabla_x\phi\ast\gamma_N\)(x,v;t)(h-h_N)(x,v;t).\nonumber\\
&&
\end{eqnarray}
Setting
\begin{eqnarray}\label{eq: AppB20}
\zeta_N(x,v):=\nabla_v u(x,v)\int \ud y\ud w\nabla_x\phi(x-y)\gamma_N(y,w;t),
\end{eqnarray}
we can write (\ref{eq: AppB19}) as:
\begin{eqnarray}\label{eq: AppB21}
&&\(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N -h)\)=\int \ud x\ud v\
\zeta_N(x,v)(h(x,v;t) -h_N(x,v;t))=\nonumber\\
&&\qquad\qquad\qquad\quad=\int \ud x\ud v\int \ud x'\ud
v'\(\zeta_N(x,v)-\zeta_N(x',v')\)P_N(x,v;x',v';t),\nonumber\\
&&
\end{eqnarray}
where $P_N$ is a coupling of $h$ and $h_N$, namely a probability
density in $\R^6\times \R^6$ with marginals given by $h$ and $h_N$.
Now we observe that:
\begin{eqnarray}\label{eq: AppB22}
\nabla_{x,v}\zeta_N(x,v):=\int \ud y\ud w\nabla_{x,v}\left[\nabla_v
u(x,v)\nabla_x\phi(x-y)\right]\gamma_N(y,w;t),\nonumber\\
&&
\end{eqnarray}
and, thanks to the assumption {\bf ii)} we made on $\gamma_N$, we know
that there exists a constant $C=C(u,\phi)>0$ such that:
\begin{eqnarray}\label{eq: AppB23}
\sup_{x,v}\left\vert \nabla_{x,v}\zeta_N(x,v)\right\vert
=\left\|\nabla\zeta_N\right\|_{L^{\infty}(\R^3\times\R^3)}0$, let us define the
operator $\hat{T}_N^{(n)}(\tau)$ as follows:}
$$
\hat{T}_N^{(n)}(\tau):=S_N(-\tau)\hat{T}_N^{(n)}S_N(\tau).
$$
\emph{Then, for each $m\geq 0$ and for each
$u\in\mathcal{C}_b^{\infty}(\R^3\times\R^3)$, there exists a constant
$C>0$, not }
\emph{depending on $N$, such that:}
\begin{eqnarray}
&&\text{{\bf i)}}\left\vert \left(u,\hat{T}_N^{(r_m)}(t_m)\dots
\hat{T}_{N}^{(r_1)}(t_1)\mu_N(t)\right)\right\vert0$ being the case $m=0$ obvious.
\\
By using the notations:
\begin{eqnarray}\label{eq: notazSTRINGA}
&& S({\bf \underline{r}}_m,{\bf
\underline{t}}_m):=T_N^{(r_m)}(t_m)\dots T_{N}^{(r_1)}(t_1)
\end{eqnarray}
and
\begin{eqnarray}\label{eq: notazSTRINGAdiag}
&&\hat{S}({\bf \underline{r}}_m,{\bf
\underline{t}}_m):=\hat{T}_N^{(r_m)}(t_m)\dots \hat{T}_{N}^{(r_1)}(t_1),
\end{eqnarray}
we have (see the first term in the right hand side of (\ref{eq:
opTtemposuG})):
\begin{eqnarray}\label{eq: azSTRINGAdiag}
\hat{S}({\bf \underline{r}}_m,{\bf
\underline{t}}_m)\mathcal{U}(Z_N(t))&&=\frac{C}{N^m}\sum_{j_1\dots
j_m}\sum_{l_1\dots l_m}D_x^{r_m+1}\phi(x_{j_m}(t_m)-x_{l_m}(t_m))\cdot
D_{v_{j_m}}^{r_m+1}(t_m)\nonumber\\
&&D_x^{r_{m-1}+1}\phi(x_{j_{m-1}}(t_{m-1})-x_{l_{m-1}}(t_{m-1}))\cdot
D_{v_{j_{m-1}}}^{r_{m-1}+1}(t_{m-1})\nonumber\\
&&\dots\nonumber\\
&&D_x^{r_1+1}\phi(x_{j_1}(t_1)-x_{l_1}(t_1))\cdot
D_{v_{j_1}}^{r_1+1}(t_1)U(Z_N(t)),
\end{eqnarray}
$C$ depending on ${\bf \underline{r}}_m$. By setting:
\begin{eqnarray}
&&\Phi_{j_n}(Z_N(t_n)):=\frac{1}{N}\sum_{l_n=1}^{N}D_x^{r_n+1}\phi(x_{j_n}(t_n)-x_{l_n}(t_n))\label{eq:
PHI}\\
&& \forall\ \ n=1,2,\dots, m\nonumber
\end{eqnarray}
(\ref{eq: azSTRINGAdiag}) can be rewritten as
\begin{eqnarray}\label{eq: azSTRINGAdiagII}
\hat{S}({\bf \underline{r}}_m,{\bf
\underline{t}}_m)\mathcal{U}(Z_N(t))&&=C\sum_{j_1\dots
j_m}\Phi_{j_m}(Z_N(t_m))\cdot D_{v_{j_m}}^{r_m+1}(t_m)\nonumber\\
&&\Phi_{j_{m-1}}(Z_N(t_{m-1}))\cdot
D_{v_{j_{m-1}}}^{r_{m-1}+1}(t_{m-1})\nonumber\\
&&\dots\nonumber\\
&&\Phi_{j_{1}}(Z_N(t_1))\cdot D_{v_{j_1}}^{r_1+1}(t_1)U(Z_N(t)).
\end{eqnarray}
We observe that, thanks to the smoothness of the potential $\phi$,
$\Phi_{j_n}$ (for each $n$) is a uniformly bounded function of the
configuration $Z_N$, together with its derivatives.\\
Performing the derivatives in (\ref{eq: azSTRINGAdiagII}), we realize
that $\hat{S}({\bf \underline{r}}_m,{\bf
\underline{t}}_m)\mathcal{U}(Z_N(t))$ is a linear combination of terms
of the following type:
\begin{eqnarray}\label{eq: azSTRINGAdiagIII}
&&\sum_{j_1\dots j_m}\Phi_{j_m}(Z_N(t_m))\cdot
D_{v_{j_m}}^{a_{m,1}}(t_m)\dots
D_{v_{j_2}}^{a_{2,1}}(t_2)D_{v_{j_1}}^{a_{1,1}}(t_1)U(Z_N(t))\nonumber\\
&& \qquad \quad D_{v_{j_m}}^{a_{m,2}}(t_m)\dots
D_{v_{j_2}}^{a_{2,2}}(t_2)\Phi_{j_1}(Z_N(t_1))\nonumber\\
&&\qquad \quad\dots\nonumber\\
&&\qquad \quad
D_{v_{j_m}}^{a_{m,m-1}}(t_{m})D_{v_{j_{m-1}}}^{a_{m-1,m-1}}(t_{m-1})\Phi_{j_{m-2}}(Z_N(t_{m-2}))\nonumber\\
&&\qquad \quad D_{v_{j_m}}^{a_{m,m}}(t_m)\Phi_{j_{m-1}}(Z_N(t_{m-1})),
\end{eqnarray}
with the constraint
\begin{equation}\label{eq: constraint}
\left\{
\begin{aligned}
& a_{1,1}=r_1 +1\\
& a_{2,1}+a_{2,2}=r_2+1\\
&\dots\\
& a_{m,1}+a_{m,2}+\dots +a_{m,m}=r_m+1.
\end{aligned}
\right.
\end{equation}
For a fixed sequence $a_{\ell,s}$, we have to compensate the
divergence arising from the sum $\sum_{j_1\dots j_m}$, which is
$O\(N^m\)$, by the decay of the derivatives as given by Proposition
\ref{DERIVATE1} and Corollary \ref{PROP5.2}. Indeed we have:
\begin{eqnarray}\label{eq: correzMARIO1}
&&\left\vert D_{v_{j_m}}^{a_{m,1}}(t_m)\dots
D_{v_{j_2}}^{a_{2,1}}(t_2)D_{v_{j_1}}^{a_{1,1}}(t_1)\mathcal{U}(Z_N(t))\right\vert\leq
\frac{C}{N^d},
\end{eqnarray}
where $d$ is the number of different indices in the sequence
$j_1,j_2,\dots,j_m$ for which $a_{m,1},\dots, a_{2,1},a_{1,1}$ are
strictly positive. Note that the fact that the derivatives are not
computed at time $t=0$ but at different times $t_1,t_2,\dots,t_m$,
does not change the estimate in an essential way.\\
An analogous estimate holds when we replace $\mathcal{U}$ by some
$\Phi_{j_s}$, namely
\begin{eqnarray}\label{eq: correzMARIO2}
&&\left\vert
D_{v_{j_m}}^{a_{m,k}}(t_m)D_{v_{j_{m-1}}}^{a_{m-1,k}}(t_{m-1})\dots
D_{v_{j_k}}^{a_{k,k}}(t_k)\Phi_{j_{k-1}}(Z_N(t_{k-1}))\right\vert\leq
\frac{C}{N^{d_{k-1}}},
\end{eqnarray}
where $d_{k-1}$ is the number of different indices in the sequence
$j_k,\dots,j_m$ which are also different from $j_{k-1}$ and from which
$a_{m,k}, \dots,a_{k,k}$ are strictly positive.
As regard to the term in the sum $\sum_{j_1\dots j_m}$ in which all
the indices are different (which is the only one of size $O(N^m)$),
the constraints (\ref{eq: constraint}) together with estimates
(\ref{eq: correzMARIO1}) and (\ref{eq: correzMARIO2}) ensure that the
product of derivatives on the right hand side of (\ref{eq:
azSTRINGAdiagIII}) is bounded by $1/N^m$. Thus this term is of order
one.
Now for each $s=1,\dots, m-1$ consider the $\frac{m!}{s!(m-s)!}$ terms
in the sum $\sum_{j_1\dots j_m}$ in which $s$ indices are equal. The
sum is bounded by $N^{m-s}$. On the other hand, the constraints
(\ref{eq: constraint}) together with (\ref{eq: correzMARIO1}) and
(\ref{eq: correzMARIO2}) ensure that the product of derivatives on the
right hand side of (\ref{eq: azSTRINGAdiagIII}) is bounded by
$1/N^{m-s}$. Thus even these terms are of size one and {\bf i)} is
proven.
To prove {\bf ii)} we observe that:
\begin{eqnarray}\label{eq: correzMARIO3}
&& S({\bf \underline{r}}_m,{\bf \underline{t}}_m)\mathcal{U}(Z_N(t)) -
\hat{S}({\bf \underline{r}}_m,{\bf \underline{t}}_m)\mathcal{U}(Z_N(t))
\end{eqnarray}
can be expanded as in (\ref{eq: azSTRINGAdiag}) and (\ref{eq:
azSTRINGAdiagIII}). However now we have an extra derivative, arising
from the definition of $R_{N}^{(n)}$ (see (\ref{eq: OFFdiagOPt})),
which yields an additional $1/N$. We omit the details of the proof
which follows the same line of {\bf i)}.
\spaz \hspace{1cm} \hfill $\square$ \newline
In the same way we can also prove the following\\
\\
{\bf Lemma C.2}: \emph{For each $m\geq 0$, $k>0$ and
$u\in\mathcal{C}_b^{\infty}(\R^3\times\R^3)$, there exists a constant
$C>0$, not depending on $N$, such that:}
\begin{eqnarray}
&&\left\vert \mathcal{D}^{2k}S({\bf \underline{r}}_m,{\bf \underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert0$. Reminding the structure of the
operator $\mathcal{D}^{2k}$ (see (\ref{eq: opDstortoGEN})), we are led
to consider the term
$D_{G,j}^{2s_j}\hat{S}(\underline{\mathbf{r}}_m,\underline{\mathbf{t}}_m)\mathcal{U}(Z_N(t))$. We remind that $D_{G,j}^{2s_j}$ is a derivation operator with respect to the variable $z_j$ that acts as specified by (\ref{eq: operatorD_G}). By the expansion (\ref{eq: azSTRINGAdiagIII}) we readily arrive to the
bound:
\begin{eqnarray}\label{eq: correzMARIO4}
&&\left\vert D_{G,j}^{2s_j}\hat{S}({\bf \underline{r}}_m,{\bf
\underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert\leq \frac{C}{N}.
\end{eqnarray}
Indeed by applying $D_{G,j}^{2s_j}$ to (\ref{eq: azSTRINGAdiagIII})
either $j\notin (j_1\dots j_m)$ so that we gain $1/N$ by the extra
derivative, or $j\in (j_1\dots j_m)$ so that we reduce the sum
$\sum_{j_1\dots j_m}$ by a factor $1/N$. More generally, by the same
argument we find:
\begin{eqnarray}\label{eq: correzMARIO5}
&&\left\vert \prod_{j\in I}D_{G,j}^{2s_j}\hat{S}({\bf
\underline{r}}_m,{\bf \underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert\leq \frac{C}{N^n},
\end{eqnarray}
where $n=\vert I\vert$.\\
Finally by writing the action of the operator $\mathcal{D}^{2k}$ as in
(\ref{eq: convINDATstruttura}), we obtain
\begin{eqnarray}\label{eq: correzMARIO6}
&&\left\vert \mathcal{D}^{2k} \hat{S}({\bf \underline{r}}_m,{\bf
\underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert\leq
\sum_{n=1}^{N}\frac{N!}{n!(N-n)!}\sum_{\substack{s_1\dots s_n\\1\leq
s_j\leq k\\ \sum_j s_j=k}}\frac{C}{N^n}\leq
B^k\sum_{n=1}^{N}\frac{N!}{n!(N-n)!}\frac{C^n}{N^n}\leq \nonumber\\
&&\leq B^k\(1+\frac{C}{N}\)^N\leq C,
\end{eqnarray}
$B, C$ being positive constants not depending on $N$. Again
$\mathcal{D}^{2k} \hat{S}({\bf \underline{r}}_m,{\bf
\underline{t}}_m)\mathcal{U}(Z_N(t))$ is the leading term of
$\mathcal{D}^{2k} S({\bf \underline{r}}_m,{\bf
\underline{t}}_m)\mathcal{U}(Z_N(t))$ for the same reasons we
discussed in Lemma C.1.
If $m=0$, the estimates (\ref{eq: correzMARIO4}) and (\ref{eq:
correzMARIO5}) follow directly by Proposition 5.2. Thus, even in this
case, the proof is concluded by (\ref{eq: correzMARIO6}).
\newline \spaz \hspace{1cm} \hfill $\square$ \newline
The fact that the error term $E_N^1$ (see (\ref{eq: eqPEReta})) and
hence $E_N^2$ (see (\ref{eq: eqPERnuDEF})) are
$\mathcal{C}^{\infty}_{b}$-weakly vanishing when $N\to\infty$ is an
immediate consequence of the following\\\\
{\bf Lemma C.3}: \emph{
Let ${\bf \underline{r}}_{J}$ and ${\bf \underline{t}}_{J}$ be defined
as in Section 7, for any $J\subset I_n$ with $I_n=\{1,2,\dots,n\}$.
For any
%$k> 0$,
$r
%,n$
\geq 0$
we have:}
\begin{eqnarray}\label{eq: correzMARIO7}
&& \mathcal{D}^{2r} S({\bf \underline{r}}_n,{\bf
\underline{t}}_n)\mu_N(z'_1\vert Z_N(t))\mu_N(z'_2\vert
Z_N(t))=\nonumber\\
&&=\sum_{0\leq \ell \leq r}\sum_{0\leq m\leq
n}\sum_{\substack{I\subset I_n\\ \vert
I\vert=m}}\(\mathcal{D}^{2\ell}S({\bf \underline{r}}_{I},{\bf
\underline{t}}_{I})
\mu_N(z'_1\vert Z_N(t)) \)\(\mathcal{D}^{2(r-\ell)}S({\bf
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus
I})\mu_N(z'_2\vert Z_N(t))\)+e_{r,N}\nonumber\\
&&
\end{eqnarray}
\emph{where}
\begin{eqnarray}\label{eq: correzMARIO8}
&& e_{r,N}\rightarrow 0\ \ as\ N\to\infty\ \ \ \mathcal{C}_b^{\infty}-weakly.
\end{eqnarray}
\newpage
{\bf Proof:}
It is enough to prove (\ref{eq: correzMARIO7}) and (\ref{eq:
correzMARIO8}) replacing each streak $S$ with the corresponding
$\hat{S}$, being the difference $S-\hat{S}$ negligible in the limit.
We start by assuming $r=0$.
%$n>0$.
In that case, testing the left hand side of (\ref{eq: correzMARIO7})
against a product of two test functions $u_1,u_2$, we are led to
consider:
\begin{eqnarray}\label{eq: correzMARIO9}
&&\hat{S}({\bf \underline{r}}_n,{\bf
\underline{t}}_n)\mathcal{U}_1(Z_N(t))\mathcal{U}_2(Z_N(t))
\end{eqnarray}
for which we can apply the expansion (\ref{eq: azSTRINGAdiag}).
Proceeding as in the proof of Lemma C.1 (see (\ref{eq:
azSTRINGAdiagIII})), we have to consider:
\begin{eqnarray}\label{eq: correzMARIO10}
&& D_{v_{j_m}}^{a_{m,1}}(t_m)\dots
D_{v_{j_2}}^{a_{2,1}}(t_2)D_{v_{j_1}}^{a_{1,1}}(t_1)\mathcal{U}_1(Z_N(t))\mathcal{U}_2(Z_N(t)),
\end{eqnarray}
where $a_{1,1}=r_1+1>0$. Now any contribution of the form
\begin{eqnarray}\label{eq: correzMARIO11}
&&
D_{v_{j_1}}^{\alpha}(t_1)\mathcal{U}_1(Z_N(t))D_{v_{j_1}}^{\beta}(t_1)\mathcal{U}_2(Z_N(t)),
\end{eqnarray}
with $\alpha>0$, $\beta>0$, $\alpha+\beta=a_{1,1}$ is
$O\(\frac{1}{N^2}\)$, therefore it is negligible in the limit. The
same argument applies to $D_{v_{j_k}}^{a_{k,1}}(t_k)$ whenever
$a_{k,1}>0$ . This means that each derivative appearing in $\hat{S}$
either applies to $\mu_N(z'_1\vert Z_N(t))$ or to $\mu_N(z'_2\vert
Z_N(t))$ up to an error $e_{0,N}$ vanishing in the limit. This is
exactly what (\ref{eq: correzMARIO7}) and (\ref{eq: correzMARIO8}) say
for $r=0$.
For $r>0$
%and $n>0$
we have to apply $\mathcal{D}^{2r}$ to (\ref{eq: correzMARIO7})
(replacing $S$ by $\hat{S}$) with $r=0$. Clearly $\mathcal{D}^{2r}
e_{0,N}$ vanishes in the limit. Moreover:
\begin{eqnarray}\label{eq: correzMARIO12}
&& D_{G,j}^{2s_j}\[\hat{S}({\bf \underline{r}}_I,{\bf
\underline{t}}_I)\mathcal{U}_1(Z_N(t))\hat{S}({\bf
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus
I})\mathcal{U}_2(Z_N(t))\]=\nonumber\\
&&= \(D_{G,j}^{2s_j}\hat{S}({\bf \underline{r}}_I,{\bf
\underline{t}}_I)\mathcal{U}_1(Z_N(t))\)\hat{S}({\bf
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus
I})\mathcal{U}_2(Z_N(t))+\nonumber\\
&&+\hat{S}({\bf \underline{r}}_I,{\bf
\underline{t}}_I)\mathcal{U}_1(Z_N(t))\(D_{G,j}^{2s_j}\hat{S}({\bf
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus
I})\mathcal{U}_2(Z_N(t))\)+O\(\frac{1}{N^2}\)
\end{eqnarray}
By simple algebraic manipulation we finally arrive to (\ref{eq:
correzMARIO7}) and (\ref{eq: correzMARIO8}).
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\end{document}
ENDBODY