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\title{Fokker-Planck Equations as Scaling Limits of Reversible
Quantum Systems }
\date{September 30, 1999}
\author{Francois Castella \\
CNRS et IRMAR\\
Universit\'e de Rennes 1\\
Campus de Beaulieu\\
35042 Rennes Cedex, France \\ \\
\and
L\'aszl\'o Erd\H os \\
School of Mathematics \\
Georgiatech\\
Atlanta GA-30332, USA \\ \\
\and
Florian Frommlet \quad and \quad Peter A. Markowich\\
Institut f\"ur Mathematik\\
Universit\"at Wien\\
Boltzmanngasse 9\\
A-1090 Wien, Austria}
\maketitle
\begin{abstract}
We consider a quantum particle moving in a harmonic
exterior potential and linearly coupled to a heat bath of quantum
oscillators. Caldeira and Leggett \cite{CL1}
have derived the Fokker-Planck equation with friction for the
Wigner distribution of the particle in the large temperature limit,
however their (nonrigorous) derivation
was not free of criticism, especially since the limiting equation
is not of Lindblad form.
In this paper we recover the correct form of their
result in a rigorous way. We also point out that
the source of the diffusion is physically unnatural under their scaling.
We investigate the model at a fixed temperature and in the
large time limit, where the origin of the diffusion is
a cumulative effect of many resonant collisions.
We obtain a heat equation with a friction
term for the radial process in phase space and we prove the Einstein relation
in this case.
\end{abstract}
\noindent
{\it Keywords:} Fokker-Planck equation, Wigner distribution, scaling limit,
coupled harmonic oscillators.
\tableofcontents
\nsection{Introduction} \label{S1}
In \cite{CL1}, Caldeira and Leggett introduced a Hamiltonian for
a quantum system of a test-particle coupled to an abstract reservoir.
The Schr\"odinger equation for the evolution of the
quantum state can be equivalently
written as a kinetic (phase-space) equation for the associated Wigner
distribution of the test particle-reservoir system.
The goal of \cite{CL1} was to derive (formally) a Fokker-Planck equation
for the Wigner distribution of the test-particle
by taking various limits
which we explain below and by "tracing out" the reservoir coordinates.
The Fokker-Planck equation represents an irreversible
collisional evolution with a diffusive term,
while the Schr\"odinger equation is reversible. Hence
this derivation was expected to shed some light on the origin of diffusion
in the evolution of a small system coupled to an infinite reservoir.
Caldeira and Leggett used a Feynman path integral
approach which has no rigorous mathematical justification
(despite its great successes in formal computations).
More importantly, several other
steps in their derivation admittedly lack precision, both from the mathematical and from
the physical point of view.
Starting from this observation,
the aim of the present paper is threefold.
First, we discuss the origin of the diffusion and
the physical
meaning of the different scalings and limiting
procedures introduced in \cite{CL1}. In particular, we point out in Section
\ref{S2}
that the model introduced in \cite{CL1}
is not physically satisfactory in the regime where they let
the diffusion appear.
Second, in Sections \ref{S3} and \ref{S4} we present
a mathematically rigorous derivation of
the frictionless Fokker-Planck equation from the model
introduced in \cite{CL1}.
Third, in Sections \ref{S4.4} and \ref{Ssmooth}
we show how to recover other types of diffusive
behaviour from the Caldeira-Leggett Hamiltonian, using different,
more realistic
scalings and limiting procedures.
%Before describing the Hamiltonian introduced in \cite{CL1}
We point out
that \cite{CL1} heavily relies on the use of ideas from Feynman, Hibbs, and
Vernon \cite{FH}, \cite{FV}.
%As it is well-known, the Feynman path integrals
%compute, at least formally, the kernel of the time evolution
%operator, $\exp(i t H )$,
%for any reasonable Hamiltonian $H$.
%These integrals are considered with respect to some ``uniform measure''
%on the space of all classical trajectories, however, in general,
%such a measure does not exist as the space of all paths is "too big".
%Nevertheless, for elliptic operators $H$,
%one can rigorously define these path integrals for imaginary time,
%i.e. for the heat kernel $\exp( - t H )$ (Feynman-Kac formula).
%In this case the strongly contractive elliptic part of
%the operator $H$ (usually the Laplacian)
%combined with the formal uniform measure actually
%yields a measure (Wiener measure) concentrated on regular paths.
%The contraction effect of the elliptic part in $\exp(i t H )$ is
%weaker as it is due to oscillations.
%The total variation measure of this complex formal "measure"
%does not exist
%which makes rigorous analysis very hard. This is the main reason
%why there are essentially no rigorous results using this
%very attractive formalism.
%Nevertheless, the Feymann approach has proved to be
%very useful at least from a formal point of view to derive
%various analytic formula.
In particular Feynman and Vernon \cite{FV} considered
a system of the form
$\{$ test ``particle'' (A) $+$ reservoir (R) $\}$.
The Hamiltonian is $H_A+H_R+H_I$, where $H_A$ is the free Hamiltonian for
the test-particle, $H_R$ is the free Hamiltonian for the reservoir,
and $H_I$ is the interaction Hamiltonian.
They integrated out the reservoir variables,
i.e. they computed the time evolution of the
wave function of the test-particle itself,
given by
$Tr_R\{\exp( i t h^{-1}(H_A+H_R+H_I) )\}$, where $Tr_R$ is the partial trace
on the Hilbert space of the reservoir and $h$ is the Planck constant.
Feynman path integral formalism was used which is
particularly
powerful when the total Hamiltonian, or at least $H_R+H_I$,
is quadratic (in particular
the interaction $H_I$ must be linear both in the test-particle and
in the reservoir variables).
In this case, one is led to computing Gaussian integrals,
which, in principle, are explicit. The difficulty stems from
the large (infinite) number of variables.
\ \\
In this context \cite{CL1} takes place. Namely, \cite{CL1} introduces
the following Hamiltonian,
\bea
\label{HCL}
H_{CL}&=&H_A+ H_R + H_I\\
\nonumber
&=&\left(-{ h^2 \over 2 M} \D_x + V(x)\right)
+\sum_{j=1}^{N\Omega} \left( -{ h^2 \over 2 } \D_{R_j} + {1 \over 2}
\om_j^2 |R_j|^2 \right)
+ {1\over \sqrt{N}}\left( \sum_{j=1}^{N\Omega} C_j R_j \right) \cdot x \; .
\eea
\noindent
The first term of (\ref{HCL})
represents the Hamiltonian of the test-particle
with mass $M$ where $x \in \zr^d$ denotes the
test-particle position in dimension $d$.
The abstract reservoir here is a set
of finitely many (say $N\Omega$, which is assumed to be integer)
independent oscillators written in normal
variables $R_j\in \zr^d$, having
frequencies $\om_j \in [0,\Om]$. Here $\Om$ is
the maximum frequency of the oscillators and $N$ is the number
of oscillators per unit frequency.
The typical case is the uniform frequency distribution:
$\om_j = {j\over N}$.
The coupling is linear in $x$ and the $R_j$'s,
with coupling coefficients given by the $C_j$'s.
The normalization factor $N^{-1/2}$ simply stems from
the central limit theorem,
since, roughly speaking, the variables $R_j$'s become
independent random variables with vanishing expectation
in the thermodynamic limit $N \rgt \infty$.
The operator $H$ acts on the Hilbert space $L^2_x(\zr^d)\otimes \Big(
\bigotimes_{j=1}^{N\Om} L^2_{R_j}(\zr^d)\Big)$. The authors
of \cite{CL1} consider only $d=1$ for simplicity, as we shall do as well,
but the method
extends to any dimension.
Caldeira-Leggett
assume that the reservoir is initially in thermal equilibrium
at inverse temperature $\b$, i.e.
the initial density matrix of the system $A + R$ is given by,
\bea
\label{1.2}
\rho^0=\rho_A^0 \otimes \exp{(-\b H_R)} \; ,
\eea
where $\rho_A^0$ is the initial state of the test-particle.
Finally, they choose the coupling coefficients,
\bea
\label{1.3}
C_j := \l \om_j \; ,
\eea
with some $\beta$-dependent
coupling parameter $\l$, and specify,
\bea
\label{1.4}
\l = \l_0 \b^{1/2} \; ,
\eea
for some fixed $\l_0$.
Note that equation (\ref{1.3}) could be interpreted mathematically as a
frequency dependent coupling, whereas in physical applications
the coupling is typically frequency independent.
This apparent difficulty is actually an artefact;
the $\om_j$ prefactor in (\ref{1.3}) stems from the three-dimensionality of
the underlying phonon or photon bath implicitly described
by the abstract reservoir in (\ref{HCL}) (see Section \ref{heatsec}).
{\bf Remark.} Instead of uniformly spaced
oscillator frequencies $\om_j={j\over N}$,
it is sufficient
to assume that the frequency distribution
tends, in the thermodynamic limit ($N \rgt \infty$),
to a uniform distribution on $[0,\Omega]$ with density, say, $c$, i.e.
\bea
\label{3.2}
\lim_{N\to\infty}
{1 \over N} \sum_{k=1}^{N\Omega} h(\omega_k) =
c\int_0^{\Omega}
h(\om) d\om, \quad \forall h \in C[0,\Omega] \; .
\eea
Without loss of generality $c=1$ can be assumed because
changing $c$ to 1 is equivalent to changing
$\lambda\to \sqrt{c}\lambda$ (see Section \ref{heatsec}).
%The reason is that $c$ oscillators with identical frequencies yield
%the same effect on the test-particle
% as one oscillator of the same frequency
%with a $\sqrt{c}$ prefactor in the coupling (see Section \ref{heatsec}).
\ \\
Now the main steps of \cite{CL1} are the following:
\ \\
$\bullet$ {\it First}, using that $H_I+ H_R$ is quadratic
and relying on Feynman path integrals,
Caldeira and Leggett
explicitly compute the evolution of the
test-particle after tracing out the reservoir variables.
The evolution equation of the test-particle
involves a diffusive forcing term and a memory term (friction),
the latter being non-local in time
(see (\ref{1.6}) below, as well as (\ref{3.27})).
These terms translate the effect
of the evolution of the reservoir on the test-particle.
It is very standard
in this context that integrating out the reservoir variables gives rise to a
non-Markovian evolution for the test-particle, despite that the evolution
of the full system is Markovian.
$\bullet$ {\it Second}, they perform the thermodynamical limit
where the number of oscillators (per unit frequency)
in (\ref{HCL}) becomes infinite ($N\to\infty)$.
More precisely, they let the oscillator ensemble tend to a continuous
distribution of oscillators with uniform density
in some finite range of frequency $[0, \Om]$.
%We mention that taking such a continuous limit where
%the reservoir tends to have infinitely many degrees of
%freedom is standard when dealing with
%irreversible limits (see, e.g., \cite{Sp1,2}).
This step operates with apparently ill-defined objects,
but it can easily be made rigorous as we will show.
$\bullet$ {\it Third}, they consider the semiclassical limit $h\to0$,
they perform
the limit $\Om \rgt \infty$, i.e. the frequency range becomes
infinite (removing ultraviolet cutoff), and they let
the inverse temperature $\b$ go to zero. The value $\beta\Om$
is responsible for a potential renormalization (frequency shift).
\medskip
These last two limits allow them
to eliminate all the non-Markovian effects. Caldeira and Leggett
state the Fokker Planck equation
\bea
\label{1.5}
\d_t w+v \cdot \nabla_x w - \nabla_x V(x) \cdot
\nabla_v w =\gamma \nabla_v (vw) + \sigma \D_v w \;
\eea
for the particle's Wigner distribution $w=w(t, x, v)$, which can
be interpreted as a phase space (quasi)density, as a result
of their asymptotic procedures.
The friction coefficient
$\gamma$ is given as $\gamma= \sigma\beta/ M$, which
is the well-known Einstein's relation between friction, diffusivity and
inverse temperature. Depending on the order of limits, $V$ may be
modified to an effective potential $V_{eff}(x)= V(x) - \om_R^2x^2$,
where $\om_R$ is called the frequency shift.
This type of equation is also known under the name
of ``Quantum Brownian motion'', or ``Quantum Langevin equation'',
and received a large interest in the context of interaction
between light and matter (see, e.g. \cite{CTDRG}).
\medskip
We mention that the idea
of formally deriving Fokker-Planck-like
equations from a reservoir of oscillators with linear coupling
has been exploited by many authors, e.g.
\cite{CL2}, \cite{Di1,Di2}, \cite{De}, \cite{HR}, \cite{UZ} (see also \cite{DGHP} for comments on this
equation and the relationship with questions of decoherence).
These authors use similar scalings as \cite{CL1}.
In particular, in \cite{Di1,Di2}, \cite{UZ}, \cite{HR}, corrections
to (\ref{1.5}) are derived when the temperature
is large but finite, and these equations
involve both a diffusive term in velocity
and friction terms in space and velocity.
Mathematically rigorous work on these types of models is slightly less
abundant. A rigorous operator-algebraic approach is given
in \cite{SDLL}, and a path-integral approach is found in
\cite{CLL}. A similar model has also been used in the program
of Jak$\check{\mbox{s}}$i\'c
and Pillet to study thermal relaxation with spectral
methods
(see \cite{JP} and references therein).
Recently an analogous system with an extra white-noise is studied in
\cite{FLM}.
Under different scalings Arai derives ballistic behaviour
for the test-particle \cite{Ar}.
In a different context
and with different scaling assumptions than \cite{CL1} and others,
but still with the assumption of linear coupling, we also mention
\cite{CTDRG}.
The key assumption in all these papers is that the test-particle
is linearly coupled to the infinite bath of
harmonic oscillators, which gives rise to
Gaussian computations,
and many quantities of interest become explicitly computable.
This certainly explains at least part of the interest
that these kinds of models have received.
\ \\
The paper by Caldeira and Leggett
raises several
questions which have to be addressed. The most serious is that
the limiting equation (\ref{1.5}) is not of Lindblad form
(see \cite{ALMS}, \cite{Di2}, \cite{Li}), which is a
generic condition for quantum systems to preserve the complete
positivity of the density operator along the evolution.
Recall that the true quantum evolution preserves this property.
This shortcoming is closely related
to the fact, that the equation itself contains $\beta$
(as the ratio of $\gamma$ and $\sigma$), while $\beta\to0$
limit was actually used along its derivation. This is not
just a mathematical inconsistency.
Either the friction
term should be negligible compared to the diffusion term
in (\ref{1.5}) if $\beta\to0$ limit is really taken;
or there should be an extra term in the equation if $\beta$
is thought of as a small but nonzero number. In this latter case
this extra term should restore the Lindblad form of the equation,
and it is not clear why this term could be considered negligible
compared to the friction. The confusion probably comes from
the unspecified order of limits.
The second most important question is the physical meaning
of the system modelled and the limits taken.
Indeed,
\cite{CL1} relies on the assumptions that
{\bf 1)} the coupling $C_j$ is linear in $\om$
with $\om$ ranging
over the full interval $[0,\infty]$, i.e. high forcing
frequencies are needed ($\Om\to\infty$);
{\bf 2)} the weak coupling is proportional to
$\b^{1/2}$, with $\b \rgt 0$;
{\bf 3)}
the test-particle is {\it linearly} coupled
to the reservoir of {\it oscillators}.
We remark that none of these assumptions can be taken
for granted on physical grounds, in fact quite the opposite,
they pose strong restrictions on the applicability of the result.
Finally, from mathematical point of view, it is desirable to
eliminate the nonrigorous steps in the original derivation;
especially since the proper order of limits actually does influence
the form of the limiting equation. In addition, the systematic
use of the Feynman path integral
should be avoided in a rigorous proof, since it is
a mathematically undefined.
\ \\
The present paper has five parts:
\medskip
\indent
{\bf a)} In Section \ref{diffsec} we explain that the origin of the
diffusion in the original Caldeira-Leggett model is the artificial
$\Om\to\infty$ limit. Then we explain
how to modify the model to obtain
diffusion via a more realistic mechanism using scaling limit.
We also explain how these derivations are related to
other derivations of the Fokker-Planck equation via the Boltzmann
equation.
\indent
{\bf b)}
In Section \ref{S2},
we discuss the
physical implications of the assumptions and scalings considered
in \cite{CL1}.
\medskip
\indent
{\bf c)}
In Section \ref{S4}, we present a rigorous mathematical
convergence result
for the model introduced in \cite{CL1}. Our approach is
very elementary and physically transparent.
\indent
{\bf d)} In Section \ref{S4.4},
we show that one can also recover a diffusive
non-kinetic behaviour (frictionless heat equation)
from the Caldeira-Leggett Hamiltonian using scaling limit
and without assuming
infinite frequency range.
\indent
{\bf e)} In Section \ref{Ssmooth}, under a different
scaling limit, we derive a Fokker-Planck equation
with friction but without convective terms. The temperature is finite.
Einstein relation is valid in a modified form which
takes into account the ground state quantum fluctuations of the heat
bath.
\medskip
Our main results are Theorem \ref{T4.1}, \ref{T4.2} and \ref{T4.3}.
\medskip
{\bf Remark.} The equation derived in Section \ref{S4}
is of Lindblad form (see \cite{ALMS}). Since there is no rescaling in the
variables, one can reconstruct
the quantum (restricted) density matrix from the evolved
Wigner distribution, hence the equation must preserve
the positivity of the corresponding density matrix.
The Wigner distribution itself is typically not positive.
On the other hand, the heat equations in Sections \ref{S4.4}
and \ref{Ssmooth} are positivity preserving equations
in pointwise sense. After rescaling the variables,
the weak limit of the Wigner distribution
is a nonnegative phase space density, hence the
equation must preserve this property. The time dependent quantum states
(density matrices) cannot be reconstructed, but
the heat equation determines their rescaled weak limits at any
time.
\nsection{Source of diffusion in various kinetic models}\label{diffsec}
In order to explain the origin of diffusion in \cite{CL1}, we have
to analyze the effects of the limits introduced there. To avoid Feynman
path integrals, we present the basic idea of \cite{CL1}
in the mathematical language we will use in our proofs.
\subsection{Eliminating the semiclassical parameter}
We take the Hamiltonian
as in \cite{CL1} (see (\ref{HCL})) with $M=1$
and specify the choice $V(x)=\frac{1}{2}x^2$
(harmonic oscillator),
in the spirit of \cite{De}, \cite{Ar}, \cite{HR}, \cite{UZ},
\cite{CTDRG}.
We use the fact that, for Gaussian Hamiltonians,
the evolution equation for the Wigner transform
of the density matrix is a first order linear partial differential
equation (\cite{W}, \cite{LP}, \cite{GMMP}),
which can be solved by the method
of characteristics (see also \cite{UZ} for a similar
observation).
In the quadratic case, we can scale $h$ out of the equation (\ref{HCL}).
Let
\bea\label{1.1}
H : = \frac{1}{2}\Big( -\Delta_x + x^2\Big)
+ \frac{1}{2}\sum_{j=1}^{N\Om}\Big( -\Delta_{R_j}
+ \om_j^2R_j^2\Big)
+ {1\over \sqrt{N}}\Big( \sum_{j=1}^{N\Om}
C_j R_j\Big)\cdot x ,
\eea
then $\exp{(-ith^{-1}H_{CL})}$ and $\exp{(-itH)}$ are
unitarily equivalent under the rescaling of variables
$x\to xh^{-1/2}$, $R_j\to R_j h^{-1/2}$.
Hence {\it mathematically} (\ref{HCL}) is the same as (\ref{1.1})
and we will prefer to work with (\ref{1.1}).
However, the {\it physical} interpretations are different. The Hamiltonian
(\ref{HCL}) is written in macroscopic coordinates, i.e. the particle
position $x$ is measured
in meters and time $t$ is measured in seconds.
The Hamiltonian (\ref{1.1}) is on microscopic (atomic) scales, where $x$ is
measured in Angstr\"oms, i.e. the particle is confined
to a lengthscale of a few \AA . The time is also measured
in atomic time units.
Hence the interpretation affects not only the lengthscale of the
particle confinement, which depends on the size of the actual physical device,
but more importantly the timescale of the evolution.
Naturally, one prefers the macroscopic interpretation for applications.
However, we shall point out that in realistic models the linear
coupling assumption is questionnable on macroscopic
scales and even on microscopic scales it is a serious restriction
(see Section \ref{linsec}).
In order to assume the linearity only on microscopic scales,
which we consider more realistic assumption for applications of these models,
but still to be able to follow the evolution on larger (than atomic)
time scales, we will introduce {\it scaling limits} of these
models in Sections \ref{S4.4} and \ref{Ssmooth}. In these
sections the word "macroscopic" will refer to the scaling
to be introduced there and it should not be confused with
the scale separation provided by the semiclassical limit.
Hence our point of view is microscopic, we
use the Hamiltonian (\ref{1.1}),
in our units $h=1$, and we assume
linear coupling on microscopic scales. In Section \ref{S4}
we prove rigorous convergence to the Fokker-Planck equation
for the Wigner distribution on atomic scales. Due to the
unitarity equivalence, this gives immediately the same
Fokker-Planck equation on macroscopic scales if (\ref{HCL})
can be used (with $V(x)=\frac{1}{2}x^2$), i.e. if the linear
coupling is considered valid on macroscopic scales.
Moreover, in this case the potential $\frac{1}{2}x^2$ can be
replaced by an arbitrary potential $V(x)$, as (\ref{HCL}) stands.
Recall that any potential $V(x)$, apart
from the quadratic ones, gives rise to a genuine pseudodifferential
operator in the Wigner equation. In the semiclassical limit $(h\to0)$
this converges to the differential operator term
$\nabla_xV\cdot \nabla_v w$ in (see also (\ref{1.5})).
This fact is well-known for general semiclassical Wigner equations
\cite{LP}, \cite{MRS}, \cite{H}, \cite{Ni1}.
We will not prove Theorem \ref{T4.1} for a
general potential because our main goal is to find the
origin of diffusivity which is independent of the confining
potential. We restrict ourselves to the most convenient quadratic case.
More importantly, we present two different scaling limits
starting from (\ref{1.1}) which allows one to follow
the dynamics up to macroscopic times.
However, we believe that not just our result on the original Caldeira-Leggett
model (in Section \ref{S4}) can be extended to include general
potential, but also the resonance effect in Sections \ref{S4.4} and
\ref{Ssmooth}. Due to the lack of explicit solutions, this
requires extra analysis which we leave to further works.
\subsection{Diffusion in the original model}\label{origdiff}
After integrating out the reservoir
variables in the equations for the characteristics,
it eventually reduces to the following ODE for
the particle's position variable $X(t)$
(see (\ref{3.27}) for the exact result),
\bea
\label{1.6}
X^{''}(t) + X(t) = f(t) + \lambda^2 \int_0^t S(t-s) X(s) \; ds \; .
\eea
Here $\l$ is as in (\ref{1.3})-(\ref{1.4}), $S$ is an explicit
function corresponding to the memory effects,
and the forcing term $f$ is,
\bea
\label{1.7}
f(t) = - {\lambda\over \sqrt{N}} \sum_{j=1}^{N\Om} \om_j
\Big[ R_j \cos \om_j t + P_j {\sin \om_j t\over \om_j}\Big] \; ,
\eea
where
$R_j$, $P_j$ are the initial position and momentum variables
of the oscillators.
Let $R_j^* : = \sqrt{2\beta}
\om_jR_j$ and $P_j^*: = \sqrt{2\beta} P_j$ be their rescaled versions.
In the high temperature
limit these become standard
Gaussian variables since the classical Gibbs distribution is
given by,
$$
\prod_j e^{-\beta (P_j^2 + \om_j^2 R_j^2)} =
\prod_j e^{-\frac{1}{2}[(P_j^*)^2 + (R_j^*)^2]}\; ,
$$
and at high temperature the quantum Gibbs distribution converges to the
classical one (for the precise formulas, see (\ref{3.28})-(\ref{3.30})).
Hence the choice (\ref{1.3}) for $C_j$, together with (\ref{1.4}) gives that,
\bea
\label{1.8}
f(t) =- {\l_0\over \sqrt{2}}
\sum_{j=1}^{N\Om} \Big[ {R_j^*\over \sqrt{N}} \cos(\om_jt)
+ {P_j^*\over \sqrt{N}} \sin(\om_jt)\Big] \; ,
\eea
as $\beta\to0$ with $R_j^*, P_j^*$ being standard Gaussians.
After integration by parts in the memory term in (\ref{1.6}) we obtain
(see (\ref{3.34}))
\bea\label{xeq}
X''(t) + X(t)
= f (t) +\lambda^2\Om X(t)- (M\star X')(t) - x M(t)
\eea
where $M$ is an approximate Dirac delta function
$M(t) \sim \lambda^2\delta_0(t)$ in the limit $\Om\to\infty$.
Here $\star$ stands for convolution.
The term $\lambda^2\Om$ is the frequency shift of the test-particle
oscillator. The friction term $M\star X'$ has a main Markovian part
$\lambda^2X'$ and a non-Markovian part which is negligible
as $\Om\to\infty$.
\medskip
The effect of the limits introduced in \cite{CL1} are as follows
$\bullet$ The high temperature limit ($\beta\to0$) plays two roles.
First, it ensures that the full friction term vanishes (recall $\lambda\sim
\beta^{1/2}$). Second, it makes the rescaled initial data $R_j^*, P_j^*$
standard Gaussians.
$\bullet$ In the thermodynamic limit ($N\to \infty$) the
sum in (\ref{1.8}) becomes the sum of the real and
imaginary parts of the truncated
complex white noise,
$$
dW^{(\Om)}(t) : = \int_0^\Om e^{i\om t} g(d\om) \; ,
$$
where $g(d\om)$'s are independent centered Gaussian random variables with
variance $\bE \Big[ g(d\om)^2 \Big]= d\om$
(for precise definition see Section \ref{stochint}).
$\bullet$ Removing the ultraviolet cutoff ($\Om\to\infty$)
gives the (complex) white noise,
\bea
\label{1.8.1}
dW(t) = \int_0^\infty e^{i\om t} g(d\om) \;
\eea
for the forcing term. Moreover, the simultaneous limit $\beta\to0$,
$\Omega\to\infty$ may lead to a constant phase shift $\lambda^2\Om \sim\beta\Om$.
\medskip
In summary, the solution $X(t)$ to (\ref{1.6}) converges to the solution
of a pure harmonic oscillator with a white noise forcing, i.e.
$\theta X(t) + \sigma X'(t)\sim (\eta\star dW )(t)$, where
$\eta (s) = \theta \sin s + \sigma \cos s$
is the harmonic oscillator trajectory (with initial condition
$\eta(0)=\sigma$, $\eta'(0)=\theta$).
In particular the mean square displacement (both in space and velocity)
\bea\label{meansq}
\bE \Big| \theta X(t)+\sigma X'(t)\Big|^2
\sim \bE \Big|\Big( \eta\star dW^{(\Om)}\Big)(t)\Big|^2
= \int_0^\Om \Big| \int_0^t \eta (t-s)e^{-i\om s}ds\Big|^2 d\om
\eea
behaves {\it quadratically} in $t$ for small $t$ for every finite
$\Omega$, hence it is {\it not} diffusive. The diffusive behavior
(linear mean square displacement) is regained only {\it after}
the $\Om\to\infty$ limit or after long times.
We emphasize that, from this point of view, the
irreversibility (the $v$-Laplacian)
in the CL model immediately stems from the particular
asymptotic distribution
of the frequencies (uniform from zero to infinity) in the forcing term.
In other terms this model demonstrates diffusion in a setup where
a plain diffusive forcing mechanism was essentially put in by hand.
Diffusion appears already in the microscopic (atomic) time scale
as a result of high frequency oscillators. This means that
there is a shorter, unexplored time scale on which most
of the oscillators live, hence the initial Hamiltonian
with the Caldeira-Leggett limits
should not be considered microscopic, rather mesoscopic.
This problem is especially transparent if the heat bath
is provided by phonons (crystal lattice vibrations)
which have an ultraviolet cutoff.
We emphasize that introducing the Planck constant
does not eliminate this problem, since large frequency
{\it and} large wavelength (which is necessary for linear
approximations) mean large propagation speed
and the
sound speed in metals ($10^3-10^4 {m\over s}$)
cannot be considered a very huge
number in macroscopic units.
In case of photons (electro-magnetic field), huge $\Omega$ is more realistic.
We return to this point in
see Section \ref{linsec}.
In contrast to this diffusive mechanism,
the source of the diffusion in more realistic models
dealing with a moving test-particle
interacting with many degrees of freedom is the {\it scaling limit}.
This means that in these models the full frequency spectrum of the diffusion
is collected over a long time from the cumulative
effects of interactions with bounded frequency,
and the diffusive behaviour
% (especially its full frequency distribution)
is visible only on a much larger time
(and sometimes space) scale than that of the
microscopic interaction (collision) mechanism.
This makes a key difference between the present
model and other works dealing, for instance, with collisional
models as scaling limits of microscopic dynamics, i.e.
macroscopic long time behaviour of Schr\"odinger equations
(see e.g. \cite{Sp2}, \cite{Sp3}, \cite{La}, \cite{HLW}, \cite{EY1},
\cite{EY2}, \cite{EY3},
\cite{EPT}, \cite{Ni1}, \cite{Ni2}, \cite{Ca}, \cite{CD},
\cite{KPR} or also \cite{BD}).
We remedy this drawback of the CL scaling
in Sections \ref{S4.4} and
\ref{Ssmooth}, as we indicate now.
\subsection{Diffusion from resonances in the scaling limit}
\indent
In Section \ref{S4.4},
we show that one can also recover a diffusive
non-kinetic behaviour
from the Caldeira-Leggett Hamiltonian
under a more realistic space-time scaling limit. Namely,
for a {\it fixed } cutoff in frequency $\Om$,
and after the high-temperature limit, we consider the resulting dynamics
for the test-particle for large time $t \sim \a^{-2}$
and large space and velocity variables $x$, $v \sim \a^{-1}$.
Here $\a \to 0$ is a scaling parameter. We prove that the phase
space density is subject to a heat equation both in velocity and position
variables. In particular, the energy of the test-particle increases
up to $\a^{-2}$ due to the resonances with bath particles of
high energy (but bounded frequency). Recall that the temperature
of the heat bath is $\beta^{-1}\to\infty$, hence bath particles
can have large energy even with bounded frequency.
In this case the diffusion indeed comes from the cumulative effect of
bounded frequency interactions via a change of scale
(microscopic to macroscopic behaviour). This is in fact a high
energy diffusion in phase space; the test-particle is heated up.
The forcing frequency distribution can be quite arbitrary, the only condition
is that it has to carry energy at the resonant frequency.
The diffusion comes from a pure resonance effect, and this seems to be
a more universal physical feature in
this context (see \cite{CTDRG}). However, the high temperature
limit is still essential in this derivation. \\
\indent
In Section \ref{Ssmooth}, we keep the temperature fixed and we rescale
only time, $t = T\delta^{-1}$ (where $\delta \to 0$ plays to role
of $\alpha^2$ above), space and velocity remain unscaled.
The reason is that the bath temperature is finite, hence the typical
energy ("temperature") of the test-particle remains finite as well.
Since the particle Hamiltonian is confining (energy level sets are
compact in phase space), the particle remains effectively localized.
As a result we
get a small scale diffusion in phase space with friction, after
integrating out the fast circular motion. Again the diffusion
comes from resonance and is developed over a long time period,
and the contributing bath frequencies are bounded.
\subsection{Comparison of the three models}\label{compmod}
The main goal in all these Caldeira-Leggett type models
is to derive diffusion. The time dependence of the mean
square displacement of the characteristics
(\ref{meansq}) is quadratic for small time (unless $\Om\to\infty$)
and is linear for large time. To see diffusion on {\it all}
times considered, there are two alternatives: either we
take $\Om\to\infty$ or we rescale time.
\bigskip
{\bf I.)} If $\Om\to\infty$, then $\lambda$ must go to zero
to keep the frequency shift $\lambda^2\Om$ finite.
Up to a positive time $t$, the total effect of the friction term is
of order $\lambda^2t$, while the diffusive (forcing) term
is roughly of order $\lambda^2 t/\beta$ for larger times, see (\ref{Qest}),
however for short times it is only quadratic in $t$.
Hence for finite times $\lambda^2t\to0$, the friction term
vanishes. Moreover, the diffusive term vanishes as well, unless
$\beta\to0$ is chosen such that $\l^2\sim \beta$.
The frequency shift is of order $\beta\Om$ and its actual size
depends on the simultaneous limits $\beta\to0$, $\Om\to\infty$.
If $\beta\to0$ is taken first, then $\Omega\to\infty$, then there
is no frequency shift.
If $\beta\Om$ is kept at a positive constant along the
limits, then we see a frequency shift. These two cases
are described in Theorem \ref{T4.1}, where frictionless
Fokker-Planck equations are derived on the microscopic time scale.
\bigskip
{\bf II.)} If we consider long times, i.e. $t\sim \a^{-2}T$,
$\alpha\to0$ and $T$ is fixed, then the size of the diffusive term
is roughly $\lambda^2\alpha^{-2}T/\beta$ for {\it all} $T$.
To compensate for the blowup $\a^{-2}$, we can either
rescale space and velocity ($x= \a^{-1}X$, $v=\a^{-1}V$)
or we set $\lambda^2\sim \a^2$.
\medskip
{\bf II/a.} If we rescale space and velocity as well, then
the friction term has a size $\l^2T$ and the diffusion term
is of order $\l^2T/\beta$ (in the new variables). One would
like to keep $\l$ and $\beta$ fixed to see both friction
and diffusion. But since the phase shift, $\l^2\Om$, has to
be kept finite, it forces keeping $\Omega$ finite as well.
This is the most realistic physical situation. However, the
friction has a non-Markovian part, whose size is $\l^2T$
if $\Om$ is fixed (and it goes to zero only if $\Om\to\infty$).
Hence the limiting equation must have a term which is
nonlocal in time. This is the extra term which is missing in
(\ref{1.5}), but its inclusion would lead to an integro-differential
equation and not to Fokker-Planck.
To derive a differential equation,
%like Fokker-Planck,
the non-Markovian friction part has to be killed. With finite
$\Om$ it is possible only if $\l\to0$, and then the full friction
is eliminated. In order not to eliminate the diffusive term
as well, $\beta\sim\l^2$ is necessary. This again leads
to the high temperature limit, but now $\Om$ is fixed
and the diffusion comes from long-time cumulative resonance effects.
The fast oscillator motion on the microscopic time scale has to be
integrated out; either in time or by a radial averaging.
This is the model in Section \ref{S4.4}.
\medskip
{\bf II/b.} If we set $\lambda^2\sim \alpha^2$ and keep $\beta$
finite, then we see a finite diffusion on a microscopic
space and velocity scale. The friction term $\l^2 t $ remains positive
and the ratio of the friction to the diffusion is $\beta$,
which gives Einstein relation. Hence $\Om$ could be kept
fixed to see the diffusion mechanism.
However, the non-Markovian part of the memory does not vanish
unless $\Om\to\infty$. The qualitative analysis of Section
\ref{Ssmooth} shows that $\Om$ can grow very slowly
(like $|\log\alpha|^7$), i.e. the non-Markovian part of the friction
is weak for large times and moderately large $\Omega$. This
was probably the heuristic idea of Caldeira and Leggett
to neglect this term. However, this effect shows up only
after time rescaling; for finite microscopic times $t$ this
term is not negligible.
Hence we let $\Om\to\infty$, and assume that $\l^2\Om$
converges to a fixed number (possibly zero). This number
gives the frequency shift. Again, we see that the size
of the frequency shift delicately depends on the simultaneous
limiting procedure.
This is the model of Section \ref{Ssmooth} (where $\delta: = \a^2$
is introduced for brevity).
\bigskip
We point out that in models II/a and II/b the origin of the
diffusion is the time rescaling. Since the forcing frequencies
are kept finite, there is no diffusion on the microscopic scale;
it becomes visible only after the large time rescaling.
Hence the physically questionnable limits, $\beta\to0$,
$\Om\to\infty$ (see Section \ref{S2})
have nothing to do with the emergence of the diffusion
in these models.
However, at least one of these limits is necessary to arrive
at a differential equation instead of an integro-differential
equation with time delayed memory term. In model II/a. (Section
\ref{S4.4}) we used $\beta\to0$ and kept $\Omega$ fixed,
while in II/b. (Section \ref{Ssmooth}) we let $\Om\to\infty$
and kept $\beta$ finite.
\medskip
We always consider nonnegative times $t\ge0$. However, most
of our computations are valid for {\it any} time, except those
which are directly responsible for the emergence of
the diffusion (Laplacian, or linear mean square displacement).
We shall point out these steps. If time were evolved backward, $t<0$, then
the same argument would yield an opposite sign of the Laplacian
(so that along the evolution it is regularizing) in the final limiting
equations. This is the usual phenomenon of irreversibility of the
parabolic equations.
\subsection{Derivation of Fokker-Planck equation via
Boltzmann equation}\label{boltzder}
In the Caldeira-Leggett type models we assumed that
the test-particle is localized and is subject to a
harmonic heat bath with linear interaction. This
usually describes particles trapped in a microscopic cavity.
For transport phenomena it is more natural to
consider a free test-particle subject to a collision
mechanism. In these models the
collisions are provided by impurities (Lorenz gas)
or by a system of many noninteracting particles (Rayleigh gas
or phonon models) and one focuses only on the dynamics
of the test-particle. The goal is to derive an equation
for the reduced phase space distribution from
the Hamiltonian dynamics with many degrees of freedom.
A scaling limit is necessary to eliminate the details
of the single collisions and to keep only their cumulative
long-time effects. The effect of a single collision
is weakened. One can introduce a
weak coupling parameter $\l\to0$; one can
consider a gas at low density $\varrho\to0$
or, in the Rayleigh gas case, one can let the
mass ratio of the gas particle and test-particle $m/M $
go to zero. In all cases the
time is rescaled as $t=T\delta^{-1}$.
The first scale on which collision effects are visible is
$\delta\sim \l^2$ (weak coupling or van-Hove limit) or $\delta\sim\varrho$
(low density or Grad limit) and $\delta\sim m/M$ (heavy test-particle
limit).
In classical mechanics, the limiting equation
is the linear Landau equation (or: diffusion on the
energy surface) for the van-Hove limit \cite{KP};
the linear Boltzmann equation
for the low density case (\cite{G}, \cite{Sp1},
\cite{BBS}); and the Fokker-Planck equation for
the heavy test-particle case (\cite{DGL}).
The Fokker-Planck equation can be obtained in a two step
limit as well: first one obtains a linear Boltzmann equation
via a low density limit, then a Fokker-Planck equation
from a mass rescaling (for an excellent review see \cite{Sp2}).
In quantum mechanics the limiting equation
is the linear Boltzmann equation both in the case
of the Lorenz gas (see \cite{EY1} for the low density case
and \cite{EY2} for the weak coupling case)
and in the case of the weakly coupled phonons \cite{EY3}.
In the model of \cite{EY3} a more realistic nonlinear
phonon coupling is considered.
In all cases when the first nontrivial limiting equation is
Boltzmann, one needs an extra limiting procedure to derive
a diffusive equation. For example if the momentum change
in the collisions is small (e.g. the mass ratio $m/M$ is small),
then a Taylor expansion in the Boltzmann collision operator
gives the Fokker-Planck equation in the first nontrivial
order (see \cite{LL}, for rigorous proof \cite{IK}).
The smallness of the collisions
has to be compensated by an extra time rescaling.
However, the two step time rescaling cannot be considered
as a fully satisfactory derivation since in the first (Boltzmann)
limit correlations are neglected which could become relevant
on a larger time scale. The proper (but much harder) procedure is
to follow the Hamiltonian dynamics up to the desired (larger)
time scale.
\medskip
We remark that a considerably more difficult collision mechanism
is when all particles interact, they are identical, and we
are interested in the evolution of the one particle marginal
distribution (or density matrix). In this case,
the limiting equation is expected to be a nonlinear
Boltzmann equation and in classical mechanics it was proven by Lanford
\cite{L}.
In quantum mechanics the correlation structure
is complicated and even the first nontrivial (Boltzmann) time
scale is not understood rigorously.
\medskip
Finally, we compare our model II/b to these free kinetic models with
collisions. The closest related model is a free electron
subject to a weakly coupled phonon interaction considered in \cite{EY3},
where a (linear) Boltzmann equation was derived.
In both models the time scale is the van Hove scale $t\sim \l^{-2}$,
where $\l$ is the coupling constant. In case of the realistic
(nonlinear) electron-phonon coupling in \cite{EY3}, each
phonon mode contributes equally to the collision mechanism.
However, in the model II/b the source of the diffusion is
resonance which originates merely in the test-particle confinement,
however for the rigorous proof we need to use the special form of the linear
coupling and test-particle Hamiltonian. Phonons with
frequencies away from the base frequency of the test-particle
Hamiltonian do not contribute, while phonons near the resonance
frequency have a strong long time effect.
In particular, it is easy to see that the Duhamel expansion
used in \cite{EY3} diverges for the model II/b, which is
also an indication that there is no Boltzmann equation
behind the Fokker-Planck equation derived in Section \ref{Ssmooth}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nsection{The assumptions of the Caldeira-Leggett model}\label{S2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We discuss the three key assumptions of
\cite{CL1} mentioned in the Introduction. As we have explained
in Section \ref{origdiff}, all these assumptions play
an important {\it mathematical} role in the Caldeira-Leggett
argument, but their physical content has not been explained.
We do not wish to enter long physical speculations in this section,
just we would like to point out that the type of
assumptions needed for the mathematics of the problem
cannot be taken for granted physically.
\subsection{Heat bath and its frequencies}\label{heatsec}
Caldeira and Leggett make two assumptions on the frequencies:
the frequency distribution is
uniform on $[0, \infty]$ and the
coupling coefficients are proportional
to $\om$ (see (\ref{1.3})).
In order to see their validity,
the first physical question is the source of the harmonic heat bath.
We note that it is not so easy to realize
a harmonic heat bath physically; the standard gases or fluids,
which often serve as an environment, are not
harmonic oscillators, but phonons and photons are.
However phonons and photons have a quite specific
frequency distribution.
We refer, e.g. to \cite{Re}, or also \cite{CTDL} for an account
on phonon- or photon- models.
The infinite ultraviolet
cutoff $\Om\to\infty$ is particularly questionnable
in the phonon model, where the lattice spacing is
a natural shortest lengthscale and typically the
Schr\"odinger equation for electrons lives on the same lengthscale.
In photon models removing the ultraviolet cutoff is more
acceptable since it is not related to other physical quantities
on atomic scales. However in Section \ref{linsec} we
will see that the linear coupling assumption sets an upper
limit on the frequency cutoff.
We mention that
in a photon picture one has to consider a velocity coupling
instead of a position coupling in (\ref{1.1}) ($R_j {\d \over \d x}$
instead of $R_j x$),
however the present method goes through for
velocity coupling as well.
\medskip
Setting frequency dependent coupling coefficient (\ref{1.3})
is equivalent to having
a {\it frequency independent} coupling and a quadratic
distribution of oscillator frequencies.
Indeed, we can
construct a Hamiltonian, similar to (\ref{1.1}) which gives the same
dynamics for the test-particle, and the coupling is frequency independent.
To do that, we consider $\om_j^2$ copies of independent harmonic oscillators
with the same frequency $\om_j$
and variable $R_j^{(i)}$, $i=1, 2, \ldots, \om_j^2$
(assuming for the moment that $\om_j^2$
is an integer). Then for all $j$
the $j^{th}$ mode of the reservoir Hamiltonian
$-{1\over 2} \D_{R_j} + {1 \over 2} \om_j^2 |R_j|^2$ is replaced
with,
$$
\sum_{i=1}^{\om_j^2}
\Big( -{1\over 2} \D_{R_j^{(i)}} + {1 \over 2} \om_j^2
|R_j^{(i)}|^2\Big) \; ,
$$
and
$C_jR_j = \l\om_j R_j$ is replaced with $\l \sum_{i=1}^{\om_j^2}
R_j^{(i)}$ in the interaction term. This modification does not change
the marginal evolution of the test-particle as it is seen by
changing the variable,
$$
R_j \to {1\over \om_j}\sum_{i=1}^{\om_j^2} R_j^{(i)} \; ,
$$
and integrating out the irrelevant relative differences of the $R_j^{(i)}$'s.
This explains how to change the problem with a coupling constant
that depends linearly on the frequency into an equivalent problem of
a test-particle coupled to a reservoir in a frequency independent way,
at the expense of considering quadratically distributed oscillators,
i.e. the number of oscillators with frequency $\om_j$ is proportional to
$\om_j^2 \;$.
If the $\om_j^2$'s are not integers, then this equivalence
becomes exact only after the thermodynamic limit, yet to be described.
We will not prove this fact rigorously here. The proof is not complicated, but
it is unnecessary for the present discussion; we start our rigorous
work from (\ref{1.1}); other possible representations are used
only for physical explanations.
Using this observation, the assumption (\ref{1.3})
suggests that the abstract reservoir of oscillators
in (\ref{1.1}) comes from phonons or photons
in the three-dimensional space.
In general, in the $d$-dimensional physical
space, the number of photons/phonons
having a given frequency $\om$ is proportional to
$\om^{d-1}$ if the dispersion relation is massless.
Hence
the factor $\om_j$ in (\ref{1.3}) should not enter the coupling parameter,
but rather should be considered as the
square root (by the standard normalization) of the number
of photons/phonons with given frequency $\om_j$
in ${\Bbb R}^3$.
\subsection{High temperature}
Choosing temperature dependent coupling and letting the
temperature go to infinity is quite convenient in
the Caldeira-Leggett argument (Section \ref{origdiff}), however
both assumptions are slightly suprising and should not be
needed to detect diffusion. Naturally, the Einstein relation
(if valid) in a frictionless diffusive equation
forces $\beta\to0$. But as we mentioned in Section \ref{boltzder},
Fokker-Planck equation can be obtained
in the $m/M\to0$ limit of the classical Rayleigh gas
for {\it any} value of the temperature of the external bath
and one obtains the Einstein relation as well. One would
like to prove a similar result starting from quantum mechanics
with (massless) phonons at finite temperature;
this was the main motivation for introducing a new scaling limit in
Section \ref{Ssmooth} (model II/b in Section \ref{compmod}).
In conjunction of the $\beta\to0$ limit we also mention that
if we think of the oscillators as phonons,
then large temperature ($>10^3 K$) destroys the lattice structure of
the metal by melting. Large temperatures
could be more realistic in a photon picture.
\subsection{Linear coupling}\label{linsec}
The linear coupling between the heat bath and the test-particle
is the key assumption in \cite{CL1},
as well as in many other works in this field,
and also in this article. By linear coupling we mean an interaction
that is linear in {\it both} the reservoir and the particle variables.
Linearity in the reservoir variables is widely used and accepted since
in typical physical situations
these variables are small (like ion displacement in the lattice),
hence nonlinear interactions can be approximated by linearization.
Linearity in the particle variable is a serious assumption, since it
implicitly assumes microscopically
localized particles. It is especially questionnable
in models for electron transport. However, we readily admit
that our work heavily relies on this assumption, because
this makes the model almost explicitly computable.
Without this assumption only the Boltzmann equation has been
proved in the case of a free test-particle coupled to a phonon field
with different, more involved methods \cite{EY3}.
We briefly discuss the linearity assumption below in the
phonon and photon case.
\medskip
\indent
{\it The case of phonons.}
\medskip
Let $\Lambda$ be a crystal lattice in ${\Bbb R}^d$,
and $\Lambda^*$ be its dual lattice. Written in normal variables
(see e.g. \cite{Re}) and after linearization in the phonon variables
the interaction of an electron with the crystal lattice is,
\bea
\label{2.1}
H_I=\sum_{k \in \Lambda^*}
C_\om \cdot R_\om \exp(i k \cdot x) \; ,
\eea
where $\om=\om(k)=|k|$ from the usual dispersion
relation (more precisely, we have
$H_I=\sum_{k \in \Lambda^*}
D(k) \cdot\tilde{R}_k \exp(i k \cdot x) $ where the $\tilde{R}_k$
is the normal mode of the lattice vibration with wave vector $k$,
and $D(k)$ is the $k$-th Fourier component of the electron-photon interaction,
but we can assume (\ref{2.1}) at least for radial coupling).
The essential point in (\ref{2.1}) is that this interaction is a priori
highly non-linear in $x$. One can reach linear coupling by assuming
that the quantity $k \cdot x$ in (\ref{2.1}) remains small
during the full evolution of the system, and linearize the exponential
accordingly. This means that the wavelength
($= O(|$wavevector$|^{-1})= O(|k|^{-1})$)
of the crystal oscillation should be bigger than the displacement of
the particle ($x$) during its full evolution. Furthermore,
in the original Caldeira-Leggett model (as well as
in Section \ref{S4.3}) the ultraviolet cutoff was removed
($\Om\to\infty$) in order to obtain diffusion (see Section \ref{origdiff}).
Therefore, we are led
to assume huge frequencies together with big wavelengths,
whereas the typical sound speed in metals ($=$
frequency $\times$ wavelength) is
bounded. One can take the idealized "infinite sound speed",
or, equivalently, "infinitely stiff lattice" limit, but given the
actual size of the physical parameters this is a physically questionable
procedure.
On the level of the Hamiltonian, notice that $\sum \om_j R_j \sim \sum_j\sum_i
R_j^{(i)} =\sum_{k\in\Lambda^*}R_k$,
to which the particle coordinate is coupled (\ref{1.1}),
is just the displacement of the ion
at the origin as the normal modes are the Fourier transforms
of the displacement vectors. In other words,
the test-particle is assumed to remain in the vicinity of the origin
(on atomic scales), hence
it interacts with one single ion
of the crystal lattice for all its dynamics
(see e.g. \cite{SDLL}).
On the other hand, we wish to derive
a diffusive equation for the electron, i.e. for
large values of time it is expected to move away from
the origin. Even if the diffusion appears only in the velocity
(see (1.5)), the large velocity implies large fluctuation in
the configuration variable as well.
In summary, the linear model effectively involves an implicit
mean-field assumption by requiring that the test-particle
is coupled to the same mode for all its evolution, which
seems incompatible with the finite sound speed of the metals
along with the removed UV cutoff.
This leaves a serious doubt on the applicability
of the linear coupling assumption for diffusion models
for electron propagation in an ionic lattice
(see also \cite{Ar} for a brief
criticism of this assumption).
\medskip
\indent
{\it The case of photons.}
\medskip
As mentioned before, one can interpret (\ref{1.1})
as the interaction of an electron with an external
electro-magnetic field (up to coupling with velocities - see above).
In this case, the electron interacts
with the field through (see, e.g., \cite{CTDRG}),
\bea
\label{2.2}
A(x)=\int_k \sum_e (\hbar / \om)^{1/2}
\Big[ e a_e(k) \exp(i k x )+ e a_e^+(k) \exp(-i k x)\Big] \; ,
\eea
where $k$ is the momentum of the photon, $e$ its polarization,
$\om=c |k|$, $c$ is the speed of light, and $a_e(k)$,
$a_e^+(k)$ are the annihilation and creation operators.
To be more precise, the interaction potential in this case
is $V(x)=-(q/M) A(x) \cdot p$,
where $q$ is the electric charge
of the electron, $M$ its mass, and $p=-i {\d \over \d x}$ its momentum.
In this case, one can argue that the speed of light is large,
so that, contrary to the phonon picture, one can reach simultaneously
high frequencies and large wavelengths. But again, the linearization
needed for our purposes means that we couple the electron
to the homogeneous field $A(0)$ instead of $A(x)$ during the full
evolution of the electron, and this assumption
is questionnable for large values of time in the kinetic regime,
when the electron evolves far away from the origin.
It is more realistic in models to study energy dissipation
of a localized electron coupled to
a radiation field with bounded frequency.
This is the main reason why we wanted to eliminate
the $\Om\to\infty$ limit from the derivation (Section \ref{Ssmooth}).
Note however that the replacement of $A(x)$
by $A(0)$ is fairly standard (at least for small
displacements of the electron): it is found
in all the above mentioned works, it is also the basic approximation
in the so-called Pauli-Fierz-Kramers transform
(see \cite{CTDRG}).
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\nsection{Preliminary results} \label{S3}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Reducing the number of physical parameters} \label{S3.1}
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%In the setup described in the introduction,
%the most general Hamiltonian we can consider in principle is of the form,
%\bea
%\label{3.1}
% H=-\Delta_x + a^2|x|^2 + \hat\lambda
% \sum_{k=1}^{\hat N\hat \Omega} \om_k
% R_k \cdot x
% + \sum_{k=1}^{\hat N\hat \Omega}
% \Big( -\hat\alpha \Delta_{R_k} + \hat\gamma\om_k^2 |R_k|^2\Big) \;
% ; \qquad x, R_k \in \zr^d \;,
%\eea
%with frequencies $\om_k \in [0,\hat \Om], k=1,\dots,\hat N\hat \Omega$,
%$\om_1\leq \om_2 \leq \ldots$.
%Since different coordinate directions
%completely decouple, the $d$-dimensional Hamiltonian is really the
%sum of $d$ identical operators.
%Therefore it is no restriction if we look
%in the sequel only at the one dimensional model,
%thus $x$ and $ R_k$ are henceforth real valued variables.
%First we are going to reduce the number of parameters by using the
%freedom in scaling.
%Setting $\tilde x =\sqrt{a} x$ and afterwards
% $\tilde R_k : = \sqrt{a\over \hat\alpha} R_k$ leads to,
%\bea
%\label{3.10}
%H = a\Bigg( -\Delta_{\tilde x} + \tilde x^2
% + \hat\lambda a^{-2}\sqrt{\hat\alpha}
% \sum_{k=1}^{\hat N\hat \Omega} \om_k \tilde R_k \tilde x +
% \sum_{k=1}^{\hat N\hat \Omega} \Big( - \Delta_{\tilde R_k}
% + \Big(\sqrt{{\hat\gamma\hat\alpha\over a^2}}\om_k\Big)^2
% \tilde R_k^2\Big)\Bigg) \; .
%\eea
%Hence the new frequency $\tomega_k = \sqrt{\hat\gamma\hat\alpha}a^{-1}\om_k$
%runs from 0 to
%$\Omega: = \sqrt{\hat\gamma\hat\alpha}a^{-1}\hat\Omega$.
%Rewriting in terms of $\tomega_k$ and defining
%$\tilde\lambda = (1/2) \hat\lambda a^{-1}\hat\gamma^{-1/2}$
%yields,
%\bea
%\label{3.13}
%a\Bigg( -\Delta_{\tilde x} + \tilde x^2 + 2 \tilde\lambda
% \sum_{k=1}^{N\Omega} \tomega_k \tilde R_k \tilde x
% +\sum_{k=1}^{N\Omega} \Big( - \Delta_{\tilde R_k}
% + \tomega_k^2\tilde R_k^2\Big)\Bigg)
%\eea
%with $N=\hat N\hat\Omega/\Omega$.
%By choosing the energy unit, or rescaling the time, we can also assume that
%$a=1/2$. The detailed analysis in
% Section \ref{S3.3} shows that one has to take
%$ \tilde \lambda = {\lambda
%\over \sqrt{N}}$, with $\l$ independent of $N$. This scaling is actually
%a consequence of the central limit theorem.
%Therefore
%the most general
%Hamiltonian in this context is of the form,
%\bea
%\label{3.15}
%H ={1\over 2}\Big( -\Delta_x + x^2\Big)
% + {\lambda\over \sqrt{N}} \sum_{k=1}^{N\Omega}
% \om_k R_k x
% + {1\over 2}
% \sum_{k=1}^{N\Omega} \Big( -\Delta_{R_k} + \om_k^2 R_k^2\Big)
%\; .
%\eea
%\ \\
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\subsection{The Wigner formalism} \label{S3.2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The density matrix,
\bea
\label{3.6}
\rho^{N,\eps} := \rho^{N,\eps}(t,x,y, R, Q) \; ,
\eea
which is the solution of,
\bea
\label{3.16}
i \d_t \rho^{N,\eps} = [H,\rho^{N,\eps}] \; ,
\eea
represents the state of the system "particle $+$ reservoir"
at time $t$ with the reservoir
variables $R=(R_1,\dots,R_{N\Omega})$, $Q=(Q_1,\dots, Q_{N\Omega})$.
We index the density matrix by $N$
and the superscript $\eps = (\b,\Om,\l)$ stands for all the other
scaling parameters; recall that $\b$ is
the inverse temperature,
$\Omega$ is the frequency range
and $\lambda$ is the coupling strength in the Hamiltonian (\ref{1.1}).
%Considering (\ref{3.15}) shows that since we shall take partial traces
%of the reservoir modes, in the case of a cutoff frequency
% the only relevant parameters are the
%frequency range $\Omega$, the coupling strength $\lambda$,
%and the inverse temperature $\b$.
%In other words, the superscript $\eps$
%really stands for the triple $(\b,\Om,\l)$.
We take the initial data (independent of $\eps$
for simplicity),
\bea
\label{3.17}
\rho_A^0 \otimes e^{-\beta H_{R}} \; ,
\eea
with $\rho_A^0 := \rho_A^{N,\eps}(t=0)$.
Here $ H_{R} := {1\over 2}\sum_{k=1}^{N\Omega}
\Big( -\Delta_{R_k} + \om_k^2 R_k^2 \Big)$
is the reservoir Hamiltonian and
$\rho^{N,\eps}_A(t,x,y)$ is the density matrix
at time $t$ of the test-particle.
It is defined by
\bea
\label{3.7}
\rho_A^{N,\eps}(t,x,y):=
\int_{\zr^{N\Omega}}
\rho^{N,\eps}(t,x,y, R, R) \;
d R \; ,
\eea
with the obvious notation $dR = dR_1 \dots dR_{N\Omega}$.
As usual, we do not distinguish between operators
and their kernels in the notation.
Following \cite{CL1}, we have to compute,
\bea
\label{3.4}
Tr_{R}\Big( e^{-itH}\big( \rho_A^0 \otimes e^{-\beta H_{R}}
\big)e^{itH}\Big) \; ,
\eea
where $Tr_{R}$ is the
partial trace over the reservoir variables.
We observe that the Hamiltonian (\ref{1.1})
is quadratic, so that equation (\ref{3.16}) can actually be transformed
into a first order transport partial differential equation
by using the Wigner transform.
Indeed, let us define the Wigner
transform $w^{N,\eps}(t)$ of $\rho^{N,\eps}(t)$ by,
\bea
\label{3.18}
w^{N,\eps}(t,x,v, R, P):=
\eea
$$
:=\disp
\int_{\zr^{N\Omega +1}}
\rho^{N,\eps}\Big( t,x+{y \over 2},x-{y \over 2}, R+{Q \over 2},
R-{Q \over 2}\Big) \times \,\, \exp\Big(-i [ y v
+ \sum_{k=1}^{N\Omega} Q_k P_k ]\Big) \; dy \; dQ \; .
$$
Also, let us define the Wigner transform of $\rho_A^{N,\eps}$ by,
\bea
\label{3.19}
w^{N,\eps}_A(t,x,v):=
\int_{\zr}
\rho^{N,\eps}_A\Big(t,x+{y \over 2},x-{y \over 2}\Big) \; \exp(-i y v ) \;
dy \; .
\eea
We have the well-known property,
\bea
\label{3.20}
w_A^{N,\eps}(t,x,v):=\int_{\zr^{2N\Omega}}
w^{N,\eps}(t,x,v, R, P) \;
dR \; dP
\; ,
\eea
and the initial datum for $w^{N,\eps}$ is easily computed from (\ref{3.17})
and the Mehler kernel,
\bea
\label{3.21}
w^{N,\eps}(t=0,x,v, R, P) =
w_0(x,v) W_0 ^{N,\eps} (R,P)
\eea
with
\bea\nonumber
W_0 ^{N,\eps} (R,P): & = &\prod_{k=1}^{N\Omega}
\Bigg[\;
4\pi \Big( {\cosh(\b \om_k) - 1 \over \cosh(\b \om_k) + 1}
\Big)^{1/2}
\\
\nonumber
& &
\times\exp\Big(-\{ { \om_k (\cosh(\b \om_k) -1) \over \sinh(\b \om_k)}
R_k^2 \}\Big)
\;
\exp\Big(-\{ { \sinh(\b \om_k) \over \om_k (\cosh(\b \om_k) +1)}
P_k^2 \}\Big) \; \Bigg]
\; .
\eea
Here, $w_0(x,v)$ is the initial datum
for the test-particle, i.e. it is the Wigner transform
of $\rho_A^0(x,y)$.
%This comes from (\ref{3.20})
%together with the normalization of the Gaussian in (\ref{3.21}).
Here and in the sequel, we shall assume the following regularity for $w_0$,
\bea
\label{3.22}
{\wh w}_0(\xi,\eta):=\int_{\zr^2} w_0(x,v) \exp(-i [x \xi + v \eta]) \;
dx \; dv \; \; \; \in L^1({\Bbb R}_\xi \times {\Bbb R}_\eta ) \; .
\eea
It is well known that, if $\rho^{N,\eps}$ satisfies the Von-Neumann equation
(\ref{3.16}) with Hamiltonian given by (\ref{1.1}),
then its Wigner transform (\ref{3.18}) satisfies the following
partial differential equation,
\bea
\label{3.23}
\d_t w^{N,\eps} + v \; \d_x w^{N,\eps} - x \; \d_v w^{N,\eps}
+ \sum_{k=1}^{N\Omega} \Big( P_k \; \d_{R_k} w^{N,\eps} -
\om_k^2 R_k \; \d_{P_k} w^{N,\eps} \Big)\\
\nonumber
- {\l\over \sqrt{N}} \Big(\sum_{k=1}^{N\Omega} \om_k R_k\Big) \; \d_v w^{N,\eps}
-{\l\over \sqrt{N}}
\Big(\sum_{k=1}^{N\Omega} \om_k x \; \d_{P_k} w^{N,\eps} \Big) = 0
\; .
\eea
As a conclusion
we can now rephrase our original problem
in the Wigner formalism:
following \cite{CL1}, we want to derive a diffusive behaviour
for $w_A^{N,\eps}(t)$, the trace of $w^{N,\eps}(t)$,
%(see (\ref{3.20})),
in the thermodynamic limit ($N \rgt \infty$)
and in certain limiting regimes of
$\eps$. Here, $w^{N,\eps}$
satisfies (\ref{3.23})
with initial datum (\ref{3.21}).
\subsection{Solution by characteristics} \label{S3.35}
Equation (\ref{3.23}) can easily be solved by the method of characteristics.
In fact, for all values of time $t$, and for all smooth, compactly
supported test functions
$\phi(x,v)$, we have,
\bea
\label{3.24}
\int_{\zr^2} w^{N,\eps}_A(t,x,v) \overline{\phi}(x,v) \; dx \; dv
=
\int_{\zr^{2N\Omega +2}}
w(t=0,x,v,R,P)
\; \overline{\phi}(X(t),V(t)) \; dx \; dv \; dR \; dP \qquad
\eea
$$
=
\disp
\int_{\zr^{2N\Omega +6}}
\hat w_0(\xi, \eta)
\overline{\hat \phi(\theta, \sigma)} e^{i(x\xi + v\eta)}
e^{-i(X(t)\theta + V(t)\sigma)}W_0 ^{N,\eps}(R,P)
\; dx \; dv \; dR \; dP \; d\xi \; d\eta \; d\theta \; d\sigma ,
$$
where we have introduced
the (forward) characteristics,
\bea
\label{3.25}
X'(t) = V(t) \; , \; \; \; \;
V'(t) = -X(t) -{\lambda \over \sqrt{N}} \sum_{k=1}^{N\Omega}\om_k R_k(t)\\
\nonumber
R_k'(t) = P_k (t) \; , \; \; \; \;
P_k'(t) = -\om_k^2 R_k(t) - {\lambda\over \sqrt{N}} \om_k X(t) \; ,
\eea
with initial data
$X(0)=x$, $V(0)=v $,
$R_k (0)= R_k$ and $ P_k(0)=P_k$.
Here we used that the flow
(\ref{3.25}) preserves the Lebesgue measure over ${\Bbb R}^{2(N\Omega +1)}$.
For simplicity,
we did not index the characteristics by $N$, $\eps$, but $X(t), V(t)$
in (\ref{3.24}) depend on $N, \eps$. However, sometimes we will use
$X_N(t)$ for special emphasis.
\\
Integrating with respect to $R_k(t)$ in (\ref{3.25}) and inserting
the result in the equation for $X(t)$ gives,
\bea
\label{3.27}
X''(t) + X(t) &=& - {\lambda\over \sqrt{N}} \sum_{k=1}^{N\Omega} \om_k
\Big[ R_k \cos \om_k t + P_k {\sin \om_k t\over \om_k}\Big] \\
\nonumber
&& + {\lambda^2\over N}\sum_{k=1}^{N\Omega}
\int_0^t \om_k \sin \om_k(t-s) X(s) ds \; .
\eea
The right-hand-side of (\ref{3.27}) is of the form
'forcing term $+$ memory term'.
% as we mentioned in the introduction.
%{F}rom now on we are going to use this terminology.
In view of (\ref{3.21}) and (\ref{3.24}), the partial trace over the
oscillators is an integral with respect to
a Gaussian distribution in $R_k$, $P_k$
with (unnormalized) density,
\bea
\label{3.28}
\exp\Big[ -{\om_k(\cosh \beta\om_k -1)\over \sinh \beta \om_k}R_k^2
- {\sinh \beta\om_k\over \om_k(\cosh \beta\om_k +1)} P_k^2\Big] \; .
\eea
Changing variables such that,
\bea
\label{3.29}
r_k=\sqrt{2\om_k(\cosh \beta\om_k -1)\over \sinh \beta \om_k}R_k
\; , \; \; \; \;
p_k = \sqrt{2\sinh \beta\om_k\over \om_k(\cosh \beta\om_k +1)}P_k \; ,
\eea
we obtain (after normalization) the standard Gauss measure,
\bea
\label{3.30}
d \mu_{N} = \prod_{k=1}^{N\Omega} {1\over 2\pi}
e^{-{1\over 2}(r_k^2+p_k^2)} dr_k dp_k \; ,
\eea
i.e. $r_k$, $p_k$ are independent
standard
Gaussian variables. The integration with respect to this probability measure
will be denoted by $\bE_N$.
Using these new variables and integration by parts with respect to $s$,
the equation (\ref{3.27}) for $X_N(t)= X(t)$ becomes,
\bea
\label{3.27M}
X_N''(t) + X_N(t) = f_N(t)
+ \lambda^2\Omega X_N(t) - (M_N\star X_N')(t)
- xM_N(t) \; ,
\eea
with,
\bea
\label{fN}
f_N(t):=-{\lambda \over \sqrt{N}}\sum_{k=1}^{N\Omega} A_\beta(\om)
\Big[ r_k \cos \om_k t + p_k \sin \om_k t\Big]
\; ,
\eea
and,
\bea
\label{MN}
M_N(t): = {\lambda^2\over N}\sum_{k=1}^{N\Omega} \cos \om_kt \; .
\eea
Here we defined,
\bea
\label{Adef}
A(\om) = A_\beta(\om) : = \sqrt{\om(\cosh\beta\om +1)
\over 2\sinh \beta \om}\; .
\eea
We see that the memory term is split into three parts. The term $\l^2\Om X_N$
induces a frequency shift of the test-particle oscillator,
$M_N\star X_N'$ is the friction term and the last inhomogeneous
term will be irrelevant.
We define
$$
a^2=a_\eps^2 : = 1-\lambda^2\Omega
$$
(recall that $\eps$ stands for the triple $ (\beta, \Omega, \lambda)$),
and we always assume that $a_\eps$ is uniformly separated from zero, e.g.
$\frac{1}{2}\leq a_\eps \leq 1$.
We can rewrite (\ref{3.27M}) as
\bea
\label{3.27N}
X_N''(t) + a^2 X_N(t) = f_N(t)
- (M_N\star X_N')(t)
- xM_N(t) \; .
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The thermodynamic limit} \label{S3.3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We now perform the limit $N \rgt \infty$.
A possible way is to solve (\ref{3.27}) (iteratively), and compute
the limit in the corresponding formulae (see (\ref{3.38}) later).
This rigorously gives the thermodynamic limit
but we present an alternative
approach which is more
illuminating to explain the asymptotic diffusion that
we shall recover in Section \ref{S4.3}. We first need an a priori
bound.
\begin{lemma}\label{apriorilemma}
Let $X_N(t)$ solve (\ref{3.27N}) with initial conditions $X(0)=x$,
$X'(0)=v$, and let
\bea
\label{4.2N}
F_N(t) : = \sup_{s\leq t} \bE_N |X_N(t)| +\sup_{s\leq t}
\bE_N |X_N'(t)| \; .
\eea
Then there is a constant $C>0$ such that
\bea
\label{4.13}
F_N(t)
\leq C e^{Kt}\Bigg( |x|+|v|+
K|x| + \sup_{s\leq t} \Big\{ se^{-Ks}\Big\}
\Big[\lambda^2\Omega \Big(\beta^{-1} +
\Omega\Big)\Big]^{1/2}\Bigg) \; .
\eea
uniformly in $N$, where
\bea
\label{Kdef}
K=K(\lambda, \Omega) : = C \lambda^2 \Big( 1
+ {1\over |\Omega -a|}\Big) \; .
\eea
and $a^2 = 1-\lambda^2\Omega\in [\frac{1}{4}, 1]$.
\end{lemma}
\noindent
{\bf Proof.}
{F}rom the fundamental solution of (\ref{3.27N}), one has
\bea
\label{4.3}
X_N(t) &=& x\cos at + va^{-1}\sin at \\ \nonumber
&& + \int_0^t a^{-1}\sin a(t-s)\Big[ f_N(s)-
(M_N\star X_N')(s) - xM_N(s)\Big] ds \; ,\\
\nonumber
X_N'(t) &=& -x a\sin at + v\cos at \\
\nonumber
&& + \int_0^t \cos a(t-s)\Big[ f_N(s)-
(M_N\star X_N')(s) - xM_N(s)\Big] ds \; .
\eea
{\it First step.} To estimate the memory term in (\ref{4.3}), we write,
\bea
\label{4.4}
\int_0^t \sin [a(t-s)] (M_N\star X_N')(s) ds
&=& \Big( \sin(a \; \cdot\; ) \star M_N \star X_N'\Big)(t)\\
\nonumber
&=&
\int_0^t \Big( \int_0^s \sin [a(s-u)] M_N(u) du
\Big) X_N'(t-s) ds \; ,
\eea
An easy calculation shows that the inner integral is bounded by
\bea
\label{4.7}
\Big| \int_0^s \sin [a(s-u)] M_N(u) du
\Big| =
\Big|\Big(M_N\star \sin (a \; \cdot \;) \Big)(s)\Big|
\leq k \lambda^2\Big(1+ {1 \over |a-\Omega|}\Big) \; ,
\eea
with a universal constant $k$ uniformly in $N$.
Indeed, notice that,
\bea
\label{Mconv}
\lim_{N\to\infty} M_N(s) = \lambda^2 {\sin\Omega s\over s} =:
M(s) \; ,
\eea
uniformly for $s\in [0, t]$. Moreover $\int_0^s \sin [a(s-u)] M(u) du$
can be estimated by splitting the integration into two regimes
$u\leq 1$ and $u\ge 1$ (or $u\leq s$ regime only if $s\leq 1$)
and both regimes can be estimated by elementary integration by parts
to obtain (\ref{4.7}).
%For the first part we have
%\bea\nonumber
% \Big|\int_0^1 \sin[a(s-u)]{\sin\Omega u\over u}\; du\Big|
% &\leq& \Big|\int_0^1
% \big[ \sin a(s-u)-\sin as\big]{\sin\Omega u\over u}\; du\Big|
% + \Big|\int_0^1 {\sin\Omega u\over u}\; du\Big|
% \\
%\nonumber
% & \leq & \int_0^1 \sup_{z\in [s-u, s]}\Big|{d\over dz}\sin az
% \Big|
% du + \sup_z \Big|\int_0^z
% {\sin u\over u} \; du \Big|
%\eea
%which is bounded a uniform constant (recall that $a\in [\frac{1}{2}, 1]$).
%For the second part we have,
%\bea
%\label{4.6}
%\int_1^s \sin[a(s-u)] {\sin \Omega u\over u} \; du &=&
%%{1\over 2}\int_{1}^s
%% {d\over du}\Bigg[ -{\sin (as - u(a+\Omega))\over a+\Omega}
%% + {\sin (as - u(a-\Omega))\over a-\Omega}\Bigg] {1\over u} du\\
% {1\over 2}\Bigg[ -{\sin (as - u(a+\Omega))\over a+\Omega}
% + {\sin (as - u(a-\Omega))\over a-\Omega}\Bigg] {1\over u}
% \Bigg|_{1}^s\\
%\nonumber
% && + {1\over 2}
% \int_{1}^s \Bigg[ -{\sin (as - u(a+\Omega))\over a+\Omega}
% + {\sin (as - u(a-\Omega))\over a-\Omega}\Bigg] {1\over u^2} du \; .
%\eea
%which is clearly bounded by $k |a-\Omega|^{-1}$, completing
%the proof of (\ref{4.7}).
Hence the expected value of the integral of the memory terms
in (\ref{4.3}) is estimated by,
\bea
\label{4.8}
\bE_N \Bigg|\int_0^t a^{-1}\sin a(t-s) \Big[
- (M_N\star X_N')(s) - xM_N(s)\Big] ds\Bigg| \\ \nonumber
\leq a^{-1}k \lambda^2\Big(1+ {1 \over |a-\Omega|}\Big)
\Big[|x|+ \int_0^t F_N(s) \; ds \Big] \; ,
\eea
and similarly for the cosine term in (\ref{4.3}).\\
\noi
{\it Second step.}
For the forcing term one computes,
\bea
\label{4.9}
\bE_N \Big| \int_0^t \sin [a(t-s)]f_N(s) ds\Big|
& \leq& t \; \sup_{s\leq t}\Big( \bE_N |f_N(s)|^2 \Big)^{1/2}\; .
\eea
We have,
\bea
\label{4.10}
\bE_N |f_N(s)|^2 = {\lambda^2 \over N}\sum_{k=1}^{N\Omega}
A_\beta^2 (\om)
\leq \hat{k} \lambda^2 \Omega\Big(\beta^{-1} + \Omega\Big) \; ,
\eea
where $\hat k$ is again some positive constant, independent of $N$.
Indeed, this sum is an approximating Riemann sum for the integral,
$$
{\lambda^2}\int_0^\Omega A_\beta^2 (\om) d\om =
{\lambda^2}\int_0^\Omega
{ \om(\cosh \beta\om +1)\over 2\sinh \beta \om} \; d\om\; ,
$$
which satisfies the estimate (\ref{4.10}).
Hence we obtain,
\bea
\label{4.11}
\bE_N \Big[|X_N(t)|+|X_N'(t)|\Big] & \leq& |x|+|v|
+ k \lambda^2\Big(1 + {1 \over |a-\Omega|}\Big)
\Big[|x|+\int_0^t F_N(s) \; ds\Big] \\
\nonumber
&& + t \Big[ \hat{k} \lambda^2\Omega
\Big(\beta^{-1} + \Omega\Big)\Big]^{1/2}
\; .
\eea
By a standard Gronwall-type argument we conclude (\ref{4.13}). \qed
\subsection{Digression on stochastic integrals}\label{stochint}
Stochastic integration is integration with respect to
a random measure. Once the measure is specified, the
integrals are defined as limits of
integrals of stepfunctions. We do not develop this
notion here, just indicate how it is related to the present
problem.
\begin{definition} The ensemble of random variables $g(A)$, $A$ running
over the Borel sets of $\zr$, is called {\it standard Gaussian
random measure} if $g(A)$ is a centered real Gaussian random variable
for all $A$ and $\bE g(A)g(B) =
|A\cap B|$ where $|\, \cdot \, |$ is the Lebesgue measure.
\end{definition}
In the thermodynamic limit $N \rgt \infty$,
the forcing term (\ref{fN}) converges
in an $L^2(d\mu_N)$ sense towards the stochastic integral,
\bea
\label{3.32}
f(t):= -\lambda \int_{0}^{\Om} A_\beta(\om)
\Big[ r (d\om) \cos \om t + p (d\om) \sin \om t\Big] \; ,
\eea
where $r(d\om)$, $p(d\om)$ are independent standard
Gaussian random measures.
The expectation with respect to their joint measure is denoted by $\bE$.
Clearly $f_N(t)$ is a Riemann sum approximation of $f(t)$ by
choosing $r_k : = N^{1/2}r\Big(\Big[ {k-1\over N}, {k\over N}\Big]\Big)$
and $p_k : = N^{1/2}p\Big(\Big[ {k-1\over N}, {k\over N}\Big]\Big)$,
since their distribution is $d\mu_N$ (see (\ref{3.30})).
In particular
we can realize all $f_N$'s and $f$ on a common probability space.
Note that $f(t)$ is formally
a white noise (see (\ref{1.8.1})) when the 'hyperbolic
factor' $A_\beta(\om)$ is replaced by one and $\Om=\infty$.
\begin{lemma}\label{stoch}
For $1<\Omega <\infty$
there exist a random function $X(t)$ such that,
\bea
\label{xn-x}
\lim_{N\to\infty} \Big(\sup_{s\leq t}\bE
| X_N(s) - X(s)| +\sup_{s\leq t}\bE | X_N'(s) - X'(s)|\Big)
=0 \; ,
\eea
and $X(t)$ almost surely satisfies the equation,
\bea
\label{3.34}
X''(t) + a^2X(t)
= f (t) - (M\star X')(t) - x M(t) \; ,
\eea
with initial conditions $X(0)=x$, $X'(0)=v$.
Moreover,
$$
F(t): = \sup_{s\leq t}\bE |X(s)| + \sup_{s\leq t}\bE |X'(s)|
\; ,
$$
satisfies the same estimate as $F_N(t)$ (see (\ref{4.13})),
\bea
\label{Fest}
F(t)\leq C e^{Kt}\Bigg( |x|+|v|+
K|x| + \sup_{s\leq t} \Big\{ se^{-Ks}\Big\}
\Big[\lambda^2\Omega \Big(\beta^{-1} +
\Omega\Big)\Big]^{1/2}\Bigg) \; .
\eea
\end{lemma}
\noindent
{\bf Proof.} Let us define $X(t)$ by the integral equation,
\bea
\label{Xdef}
X(t) &=& x\cos at + va^{-1}\sin at \\ \nonumber
&& + \int_0^t a^{-1}\sin [a(t-s)]\Big[ f(s)-
(M\star X')(s) -xM(s)\Big] ds \; ,
\eea
Since,
$$
\int_0^t \bE |f(s)|^2 ds = \lambda^2\int_0^\Omega
{\om (\cosh \beta\om + 1)\over 2\sinh\beta\om} d\om
< \infty \; ,
$$
$X(t)$ is well defined almost surely and satisfies (\ref{3.34}). Moreover,
the uniformity of (\ref{4.13}) in $N$, and (\ref{xn-x}) shows that
$F(t)$ satisfies (\ref{Fest}). So we are left with
proving (\ref{xn-x}).
Let $Z_N(s):= X_N(s)-X(s)$, then it satisfies (from (\ref{4.3}) and
(\ref{Xdef})),
\bea\nonumber
Z_N(t) &=& \int_0^t a^{-1}\sin [a(t-s)]\Big[ f_N(s)-f(s)
- (M\star Z_N')(s)\\
\nonumber &&- (M_N-M)\star X_N' (s) -x (M_N-M)(s)\Big] ds \; ,
\eea
and a similar formula holds $Z_N'(t)$. Clearly $Z_N(0)=Z_N'(0)=0$.
Hence, similarly to (\ref{4.11}),
\bea
\nonumber\bE \Big( | Z_N(s)| + | Z_N'(s)|\Big)
&\leq &K \int_0^t \wt F_N(s) ds \\
\nonumber
&&+ a^{-1}t\sup_{s\leq t}\Bigg( \Big\{|x| + t\sup_{u\leq t}
\bE |X_N'(u)|\Big\} |M_N(s)- M(s)|
+ \bE |f_N(s)-f(s)|\Bigg) \; ,
\eea
with $\wt F_N (t) = \sup_{s\leq t} \bE | Z_N(s)|+
\sup_{s\leq t} \bE | Z_N'(s)|$.
We use again a Gronwall argument to obtain (\ref{xn-x}), based upon
the control of $\sup_{u\leq t}\bE |X_N'(u)|$
from Lemma \ref{apriorilemma}
and the facts that
$|M_N(s)-M(s)|\to 0$ (see (\ref{Mconv})) and $\bE |f_N(s)-f(s)|\to 0$
uniformly for $s\leq t$ as $N\to \infty$.
In order to check $\bE |f_N(s)-f(s)|\to 0$, we observe that,
$$
r_k = N^{1/2}r\Big(\Big[ {k-1\over N}, {k\over N}\Big]\Big)=
N^{1/2} \int
{\bf 1}\Big(\om \in \big[ {k-1\over N}, {k\over N}\big]\Big)
r(d\om) \; ,
$$
to obtain,
\bea
\label{fN-f}
\bE |f(s)-f_N(s)|^2 = \lambda^2\int_0^\Omega
\Bigg[ A_\beta(\om)
- \sum_{k=1}^{N\Omega} A_\beta(\om_k) \cdot
{\bf 1}\Big(\om \in \big[ {k-1\over N}, {k\over N}\big]\Big)
\Bigg]^2 d\om \; ,
\eea
which goes to zero as $N\to\infty$, uniformly in $s\leq t$.
For uniformly spaced frequencies, $\om_k = {k\over N}$, (\ref{fN-f})
is straightforward.
% as the second function in the square
%bracket above is just a Riemann approximation of the first.
For frequencies satisfying only the uniform density condition (\ref{3.2})
with $c=1$, first one has to verify that
$$
\lim_{N\to\infty} {1\over N}
\; \# \Big\{ k \; : \; \big|\om_k -\frac{k}{N} \big|
\ge \eta \Big\} = 0
$$
for any $\eta>0$,
and then using the continuity of the function $A_\beta(\om)$
to conclude the result. \qed
\bigskip
Let us remark that for the present paper there is no need to use
stochastic integrals. A reader who is unfamiliar with this concept,
can keep the finite sums $\sum_{k=1}^{N\Omega}$ instead of $\int_0^\Omega d\om$,
$f_N(t)$ instead of $f(t)$, and keep on thinking of $\bE$ as expectation $\bE_N$
with respect to the finite dimensional
measure $d\mu_N$. We shall compute various expectations involving $f(t)$.
The results are given
as an ordinary $\int_0^\Omega (\ldots) d\omega$ integral.
However, one can keep the finite
dimensional approximations $f_N(t)$, and perform the expectations
with respect to $d\mu_N$. In this case the expectations involve a finite
sum over the frequencies, like $\sum_{k=1}^{N\Omega}(\ldots)$.
It is sufficient to take the $N\to\infty$ limit only in
this sum, which is a Riemann sum for
the integral $\int_0^\Omega (\ldots) d\omega$ using (\ref{3.2}) with $c=1$.
%In this way one can avoid the concept of stochastic integrals and
% still arrive at
%the same result.
However, for notational simplicity we will use the continuous
formalism. Note that the thermodynamic limit $N\to\infty$ is always taken
before any other limits.
\bigskip
The conclusion of Section \ref{S3} is the,
\begin{lemma}\label{L3.1}
Assume (\ref{3.2}) with $c=1$ and assume (\ref{3.22}).
Let $ w^{N,\eps}_A(t)$ be defined as (\ref{3.20}),
while $ w^{N,\eps}(t)$ is the solution of (\ref{3.23})
with initial datum (\ref{3.21}).
Then, in the thermodynamic limit, we have for all $\phi(x,v) \in
C^\infty_c({\Bbb R}^2)$
locally uniformly for $t \in {\Bbb R}$,
\bea
\label{3.36}
\disp \lim_{N\to\infty}
\int_{\zr^2} w^{N,\eps}_A(t,x,v) \overline{\phi}(x,v) dx \; dv
= \disp \int_{\zr^2} w^{\eps}_A(t,x,v) \overline{\phi}(x,v)
dx \; dv\; ,
\eea
where $ w^{\eps}_A$ is defined by,
\bea
\label{3.37}
&&
\disp \int\limits_{\zr^2} \!w^{\eps}_A(t,x,v) \overline{\phi}(x,v) dx
dv=\\
\nonumber
&& \; \; \; \; \; \; \; \; \; \; \; \;
=
\bE \!\disp
\int\limits_{\zr^6} \!\hat w_0(\xi, \eta)
\overline{\hat \phi(\theta, \sigma)} e^{i(x\xi + v\eta)}
e^{-i(X(t)\theta + X'(t)\sigma)}
d\xi \; d\eta \; dx \; dv \; d\theta \;
d\sigma \; ,
\eea
and
$X$ satisfies (\ref{3.34}) .
\end{lemma}
For the proof one only has to observe that the dominated convergence
theorem applies and use Lemma \ref{stoch}
and (\ref{3.24}) (recalling that $X$ is actually $X_N$ in that formula). \qed
\ \\
{\bf Remark.} As an alternative proof which avoids any reference to probabilistic
concepts, we can easily compute the right-hand-side of (\ref{3.24}) directly
by performing a finite dimensional Gaussian integration with respect to
$d\mu_N$ (again, $X(t)$ is actually $X_N(t)$ in (\ref{3.24})).
In this case all the integrals $\int_0^{N\Omega} (\ldots )d \om$ are
finite sums and the $N\to\infty$ limit is taken only after having performed
the $d\mu_N$ integration. We easily find that
the right-hand-side of (\ref{3.24})
is equal to,
\bea
\label{3.38}
&&
\disp
\int_{\zr^2} \hat{w_0}\Big( A(t) \th + A'(t)\sigma \; ,
\; B(t) \th + B'(t) \sigma \Big)
\; \overline{\hat \phi (\th,\sigma)}
\\
&&
\nonumber
\times
\exp\Big[
\disp
-\int_{0}^{\Om}
{
[A_\om(t) \th + A'_\om(t) \sigma ]^2 \over 2 \l_\om } \; d\omega
-\int_{0}^{\Om}
{
[B_\om(t) \th + B'_\om(t) \sigma ]^2 \over 2 \mu_\om }
d\om
\Big] \;d\theta \; d\sigma \; ,
\eea
where $\l_\om=[2 \om (\cosh(\b \om) - 1)]/[\sinh(\b \om)]$,
$\mu_\om=[2\sinh(\b \om)]/[ \om (\cosh(\b \om) + 1)]$,
and,
\bea\nonumber
\Psi(t) & =&\l^2 \int_{0}^{\Om} \int_{0}^t
\om \sin(\om [t-s]) \sin(s) \; ds \; d\om \; , \\ \nonumber
A(t)&=&\cos(t)+(\Psi \star A)(t) \; , \\ \nonumber
B(t)&=&\sin(t)+(\Psi \star B)(t) \; , \\ \nonumber
A_\om(t)&=&-\int_{0}^t \l \om \cos(\om s)
\sin(t-s) \; ds + (\Psi \star A_\om)(t) \; , \\ \nonumber
B_\om(t)&=&-\int_{0}^t \l \sin(\om s)
\sin(t-s) \; ds + (\Psi \star B_\om)(t) \; .
%\end{array}
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nsection{The Fokker-Planck equation from the original
Caldeira-Leggett model}\label{S4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Evolution without friction} \label{S4.1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the spirit of \cite{CL1},
we would like to exhibit a scaling
where the solution of (\ref{3.34})
is close to the solution $\wt X(t)$ of the equation without
friction term below. The scaling parameters are $\eps=(\b, \Om, \l)$.
The frictionless equation (compare with (\ref{3.34})) is,
\bea
\label{4.1}
\wt X''(t) + a^2\wt X(t) = f(t) \; , \qquad\mbox{ with, }
\qquad X(0)= x \; , \quad X'(0)=v \; ,
\eea
recalling that $a^2=a_\eps^2= 1-\lambda^2\Omega \in [\frac{1}{4}, 1]$.
We need a continuity result ensuring that $X(t)$ and $\wt X(t)$ are
close. If $Y(t) = X(t)-\wt X(t)$, then,
\bea
\label{4.14}
Y''(t) + a^2Y(t) = - (M\star X')(t) - xM(t) \; ,
\eea
with initial conditions $Y(0)=Y'(0)=0$. Given the bound (\ref{Fest})
on $X(t)$ and (\ref{4.7})
it is trivial to see that,
\bea
\label{4.15}
\bE\Big( |Y(t)|+ |Y'(t)|\Big) \leq
Kte^{Kt}\Bigg( |x|+|v|+
K|x| + \sup_{s\leq t} \Big\{ se^{-Ks}\Big\}
\Big[\lambda^2\Omega \Big(\beta^{-1} +
\Omega\Big)\Big]^{1/2}\Bigg) \; ,
\eea
where $K= C\lambda^2(1 +{1\over |\Omega -a|})$ (see (\ref{Kdef})).
So in particular the solution of (\ref{3.34}) tends to the solution of
(\ref{4.1})
in a very strong norm
if the right-hand-side of (\ref{4.15}) goes to zero.
This happens for example for such
limiting regimes of $\eps = (\beta, \Omega, \lambda)$
that $\lambda\to 0$ and $\Omega\to\infty$ in
such a way that $a^2 =1-\lambda^2\Omega\in [\frac{1}{4}, 1]$
and $\lambda^2\beta^{-1/2}\to0$.
Hence,
as soon as one can ensure a small right-hand-side in (\ref{4.15}),
we can replace $X$ by $\wt X $
in (\ref{3.36})-(\ref{3.37}) by the Lebesgue theorem,
since the $x, v, \theta, \sigma$ integrations
range over a bounded domain ($\phi$ is compactly supported) and
we assumed $\wh w_0(\xi, \eta)\in L^1$ (see (\ref{3.22})).
This proves
\begin{lemma}\label{replace}
Let $\wt w_A^\eps$ be defined as,
\bea
\label{tildewdef}
\disp \int\limits_{\zr^2} \wt w^{\eps}_A(t,x,v) \overline{\phi}(x,v) dx dv=
\bE \!\disp
\int\limits_{\zr^6} \!\hat w_0(\xi, \eta)
\overline{\hat \phi(\theta, \sigma)} e^{i(x\xi + v\eta)}
e^{-i(\wt X(t)\theta + \wt X'(t)\sigma)}
d\xi \; d\eta \; dx \; dv \; d\theta \; d\sigma \; ,
\eea
analogously to (\ref{3.37}). Then,
\bea
\label{replaceeq}
\lim_\eps \int\limits_{\zr^2}
\wt w^{\eps}_A(t,x,v) \overline{\phi}(x,v) dx dv
=\lim_\eps \int\limits_{\zr^2}
w^{\eps}_A(t,x,v) \overline{\phi}(x,v) dx dv \; ,
\eea
for any limit of the parameters $\eps = (\beta, \Omega, \lambda)$
for which the right hand side of
(\ref{4.15}) goes to zero. \qed
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Computing the dynamics of the test-particle when the memory
vanishes} \label{S4.2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we compute $w^\eps(t,x,v)$ when $X$ is actually replaced
by $\wt X $, the solution of (\ref{4.1}), in (\ref{3.37}).
We have,
\bea
\label{4.16}
\wt X(t) &=& x\cos at + va^{-1}\sin at + \int_0^t a^{-1} \sin a(t-s)
f(s) ds \; ,\\
\nonumber
\wt X'(t) &=& -xa\sin at + v\cos at
+ \int_0^t \cos [a(t-s)] f(s) ds \; .
\eea
Hence
\bea\nonumber
\int\limits_{\zr^2} \wt w^\eps_A(t, x, v)\overline{\phi(x, v)} dx \; dv
&= & \bE \int\limits_{\zr^6} \wh w_0(\xi, \eta)
\overline{\wh \phi(\theta, \sigma)} e^{i(x\xi + v\eta)}
e^{-i(\wt X(t)\theta + \wt X'(t)\sigma)}\; d\xi \;d\eta\; dx\; dv\;
d\theta\; d\sigma\\
\label{4.17}
% & =& \bE \int_{\zr^6} \wh w_0(\xi, \eta)
% \overline{\wh \phi(\theta, \sigma)} e^{-i(x\xi + v\eta)}
%\eea
%$$
% \times e^{-i\theta\Big(
% x\cos at + va^{-1}\sin a t + \int_0^t \sin a(t-s)
% f(s) ds\Big)
% - i\sigma\Big( -x a\sin at + v\cos a t
% + \int_0^t a\cos a(t-s) f(s) ds\Big)}
%\;d\xi \;d\eta\; dx\; dv\; d\theta\; d\sigma
%$$
%$$
&= &\bE \int_{\zr^2} \wh w_0\Big( \xi_{\theta,\sigma}(t),
\eta_{\theta, \sigma}(t)\Big) \overline{\wh\phi(\theta, \sigma)}
e^{-i\int_0^t\eta_{\theta, \sigma}(t-s) f(s)ds} \; d\theta \;
d\sigma \; ,
\eea
with,
\bea
\label{4.18}
\eta_{\theta, \sigma}(t) := \theta a^{-1}\sin at
+\sigma\cos at \; , \qquad
\xi_{\theta, \sigma}(t) : = \theta \cos at - \sigma a\sin at
\; ,
\eea
which are, by the way, harmonic oscillator trajectories,
\bea
\label{4.20}
{d\over dt} \eta_{\theta, \sigma} (t)
= \xi_{\theta, \sigma}(t) \; , \; \; \; \;
{d\over dt} \xi_{\theta, \sigma}(t)
= -a^2\eta_{\theta, \sigma}(t) \; .
\eea
After performing the expectation in (\ref{4.17}), we arrive at
\begin{lemma} \label{L4.1}
With the notations above, we have for any $t\ge0$,
\bea
\label{4.21}
\int_{\zr^2} \wt w^\eps_A(t, x, v)\overline{\phi(x, v)} \;dx \;dv =
\int_{\zr^2} \wh w_0\Big( \xi_{\theta,\sigma}(t),
\eta_{\theta, \sigma}(t)\Big) \overline{\wh\phi(\theta, \sigma)}
e^{-{1\over 2}Q(t)} \;d\theta \; d\sigma\; ,
\eea
with
\bea
\label{4.22}
Q(t) := Q(t; \theta, \sigma; \beta, a)=
\lambda^2 \int_0^\Omega A_\beta^2(\om) H(t, \om)d \om
\; ,
\eea
\bea
\label{Hdef}
H(t, \om): = H(t,\om ; \theta, \sigma; a)
= \Big| \int_0^t \eta_{\theta, \sigma} (s) e^{-i\om s} ds
\Big|^2 \; .
\eea
The functions $\xi_{\theta,\sigma}$,
$\eta_{\theta, \sigma}$ are defined by (\ref{4.18}).
The function $H(t, \om)$ satisfies the following estimate
\bea
\label{Hest}
H(t,\om)\leq 2\gamma^2\Bigg\{
\Big| {e^{it(a-\om)}-1\over a-\om}\Big|^2
+ {4\over (a+\om)^2}\Bigg\}
\eea
with $\gamma^2 : = \theta^2+a^2\sigma^2$. Assuming $\Omega>1$ we also have
\bea
\label{Qest}
Q(t) = I\lambda^2t \gamma^2
{\cosh \b a + 1\over 2a\sinh \b a} +
\lambda^2 \gamma^2 B(t)
\eea
with $I:=\frac{\pi}{2}$ and with a function $B$ satisfying $B(0)=0$ and
\bea
\label{Best}
|B(t)|\leq C\big[ 1+\beta^{-1}\big]\big[ 1 + (\log t)_+\big]
\big[ 1 + \log \Omega\big]
\eea
with a universal constant $C$.
Also, we have the estimate:
\bea
\label{4.23a}
Q(t) = \bE \Big(f\star \eta_{\theta, \sigma}\Big)^2(t) = \bE \Big(
\theta\wt X(t) + \sigma\wt X'(t)\Big)^2 + {\cal O}\Big[ (|x|+|v|)
(|\theta| + |\sigma|)\Big]
\; . \qquad
\eea
\end{lemma}
\ \\
{\bf Remark.}
Notice that $Q(t)$ grows quadratically in $t$
for small $t$
(since $H$ does so). This means
that the test-particle as described by the
Wigner distribution $w^\eps_A$ has a ballistic
behaviour when the memory effects
disappear (quadratic growth of the mean squared displacement
$\bE \wt X^2(t)$).
In the rest of this paper we show that, under several specific
scaling limits, one can indeed replace $w_A^\eps$ with $\wt w_A^\eps$
(see Lemma \ref{replace}) {\it and}
recover a linear growth for $Q(t)$, i.e.
a diffusive behaviour for the test-particle.
In particular, this is where the time asymmetric condition
$t\ge0$ is used.
\ \\
{\bf Proof.} We only have to show the estimates (\ref{Hest}) and
(\ref{Best}). These are straightforward calculations.
We use the following notation,
\bea
\label{4.34}
a\sigma + i\theta = \gamma e^{i\phi} \; .
\eea
(i.e. $\theta = \gamma\sin\phi$, $a\sigma = \gamma
\cos \phi$ and $\gamma^2 = \theta^2 +a^2\sigma^2$).
Hence, from (\ref{4.18}),
\bea
\label{4.35}
\eta_{\theta, \sigma}(t) = {\gamma\over 2a}\Big(
e^{i(\phi-at)}+ e^{-i(\phi-at)}\Big) \; ,
\eea
and
\bea
\label{4.36}
H(t, \omega)
% &=& {\gamma^2\over 4a^2}\Big|e^{i\phi}\int_0^t e^{-is(a+\om)}ds
% + e^{-i\phi} \int_0^t e^{is(a-\om)}ds\Big|^2\\
%\nonumber
&=& {\gamma^2\over 4a^2}\Big|e^{2i\phi} {e^{-it(a+\om)}-1\over a+\om}
- {e^{it(a-\om)}-1\over a-\om}\Big|^2 \; ,
\eea
which proves (\ref{Hest}).
To prove (\ref{Qest})-(\ref{Best}), for any $\Omega>1$ we obtain, by
extracting the worst singularity
\bea
\label{4.37}
Q(t) &=& \lambda^2 \int_0^\Omega {\om (\cosh \beta \om +1)\over
2\sinh \beta \om} H(t, \omega) d\om\\
\nonumber
&=& \l^2 {\gamma^2 \over 4a^2} \wt B(t) + \lambda^2 {\gamma^2\over 4a^2}
\int_0^\Omega
{\om (\cosh \beta \om +1)\over 2\sinh \beta \om} \Big|
{e^{it(a-\om)}-1\over a-\om}\Big|^2d\om \; ,
\eea
with,
\bea
\label{4.38}
\wt B (t):= \int_{0}^\Om {\om (\cosh \beta \om +1)\over
2\sinh \beta \om} \Big\{
\Big| {e^{-it(a+\om)}-1\over a+\om}\Big|^2 - 2 \mbox{Re}
\Big(e^{2i\phi} {e^{-it(a+\om)}-1\over a+\om}{e^{it(a-\om)}
-1\over a-\om} \Big) \Big\}
d\om
\; ,
\eea
and $\wt B(0)=0$.
With the substitution $\om'=t(a-\om)$ in (\ref{4.38}), one easily
computes
\bea
\label{4.39}
|\wt B(t)|\leq C \big[ 1 + \beta^{-1}\big] \big[ 1 + (\log t)_+\big]
\big[ 1 + \log \Omega\big] \; .
\eea
The second integral in (\ref{4.37}) is
proportional to $t$ for large $t$ since
$\Omega >1$. Obviously it becomes
uniformly bounded if $\Omega < a \leq 1$ (a trivial behaviour),
and this is the very reason why we assumed $\Omega >1$ in this section.
Then the main contribution
comes from $\omega\sim a$, and by the same change of variables as above,
the result is,
\bea
\label{4.40}
Q(t) = \l^2 \gamma^2 B(t) + I\lambda^2 t \gamma^2
{\cosh a\beta +1 \over 2a \sinh a\beta}
\eea
with $I:=\frac{\pi}{2}$,
and $\wt B(t)$ is replaced by some $B(t)$ which also
satisfies (\ref{4.39})
and $B(0)=0$.
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Caldeira-Leggett limits: obtaining the Fokker-Planck
equation} \label{S4.3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we rigorously perform
the scaling limit introduced in \cite{CL1}. We prove the following,
\begin{theorem}\label{T4.1}
Let $w^\eps_A$ be the Wigner distribution of the test-particle after
the thermodynamic limit, as given by Lemma \ref{L3.1}.
We recall that $\eps$ stands for $(\b,\Om,\l)$.
Let $\l = \l_0 \b^{1/2}$
with some fixed $\l_0$, as in \cite{CL1}.
{\bf a) [Nonzero frequency shift.]}
Assume that $a^2=1-\lambda^2\Omega =1- \lambda_0^2\beta\Omega
\in [\frac{1}{4}, 1]$ is fixed. Then for
any $t\ge0$ the following weak limit exists
\bea
\label{4.24freq}
W(t,x,v)=\lim_{\Om \rgt \infty, \b\rgt 0\atop \b\Omega = (1-a^2)
\lambda_0^{-2}}
w_A^\eps(t,x,v) \; .
\eea
The limit holds in the topology of $C^0([0,\infty)_t
;{\cal D}'_{x,v})$.
Moreover, $W$ satisfies the Fokker-Planck equation,
\bea
\label{4.25freq}
\d_t W + v \d_x W - a^2 x \d_v W - {\lambda_0^2\pi\over 2}\D_v W = 0 \; ,
\eea
with initial datum $W(t=0) = w_0$ satisfying (\ref{3.22})
{\bf b) [No frequency shift.]}
For any $t\ge0$ the following weak limit exists,
\bea
\label{4.24}
W(t,x,v)=\lim_{\Om \rgt \infty} \lim_{\b \rgt 0} w_A^\eps(t,x,v) \; .
\eea
[the order of limits cannot be interchanged], and
$W$ satisfies the Fokker-Planck equation,
\bea
\label{4.25}
\d_t W + v \d_x W - x \d_v W - {\lambda_0^2\pi\over 2}\D_v W = 0 \; ,
\eea
with initial datum $W(t=0) = w_0$ satisfying (\ref{3.22})
\end{theorem}
{\bf Proof.}
For the proof of part a)
first notice that Lemma \ref{replace} applies since the
right hand side of
(\ref{4.15}) goes to zero
under the prescribed limits. Hence
$X$ can be replaced by $\wt X$ and
we can therefore rely on Lemma \ref{L4.1} above.
On the other hand,
since we assumed $\lambda = \lambda_0\beta^{1/2}$,
we readily observe,
\bea
\label{WN}
\lim \!{}^* \; Q(t)
\!\! = \lambda^2_0
\lim \!{}^*
\int_0^\Omega \beta A^2_\beta(\om)
H(t,\om) d\om
= \lambda^2_0 \int_0^\infty \Big| \int_0^t
\eta_{\theta, \sigma}^2(s)
e^{-i\om s} ds \Big|^2 d\om
\; ,
\eea
where $\lim^*$ stands for the simultaneous limit $\beta\to0$, $\Om\to\infty$
such that $a^2=1-\lambda_0^2\beta\Om \in [\frac{1}{4}, 1]$ is fixed.
Here we used that $\beta A_\beta(\om)^2\to 1$ in our limit
if $\om\leq \Om^{1/2}$
and that $H(t, \om)\in L^1(d\om)$, see (\ref{Hest}).
The contribution $\om\ge \Om^{1/2}$ to the
integral vanishes in the limit by the estimate (\ref{Hest})
and the trivial bound ${z\cosh z + 1\over \sinh z}\leq 2(1+ z)$.
Hence from the unitarity of the Fourier transform
\bea
\label{4.28}
\int_0^\infty
\Big|\int_0^t g(s)e^{-i\om s} ds\Big|^2 d\om
= \pi \int_0^t |g(s)|^2 ds \; ,
\eea
which is valid for any real function $g$, we obtain
\bea
\label{4.27}
\lim\!{}^* \; Q(t)
&=& \lambda^2_0 \pi\int_0^t \eta_{\theta, \sigma}^2(s) ds\; .
\eea
Here $t\ge 0$ is used, and this step
is the origin of irreversibility.
The end of the calculation is trivial. {F}rom Lemma \ref{L4.1} together
with (\ref{4.27}) we have,
\bea
\label{4.29}
\lim\!{}^* \;
\int_{\zr^2} w_A^\eps(t, x, v)\overline{\phi(x, v)} \;dx \;dv
& =& \int_{\zr^2} \!\wh w_0\Big( \xi_{\theta,\sigma}(t),
\eta_{\theta, \sigma}(t)\Big) \\
\nonumber
&&\times\overline{\wh\phi(\theta, \sigma)}
e^{-I\lambda^2_0
\int_0^t \eta_{\theta,\sigma}^2(s)ds}\; d\theta \; d\sigma \; ,
\eea
where $\eta$ and $\xi$ are defined in (\ref{4.18}) and $I=\frac{\pi}{2}$.
We can define,
\bea
\label{4.30}
W(t, x, v): =
\lim\!{}^* \;
w_A^\eps(t, x, v) \; ,
\eea
as a weak limit given by (\ref{4.29}). Then differentiating (\ref{4.29})
gives (using (\ref{4.18})),
\bea
\label{4.31}
\int_{\zr^2}\partial_t W(t, x, v)\overline{\phi(x, v)} dx \; dv
&=&\int_{\zr^2}\partial_t \wh W(t, \theta, \sigma )
\overline{\wh\phi(\theta, \sigma)} d\theta \; d\sigma
\eea
$$
= \int_{\zr^2} \Bigg[
- a^2\eta_{\theta, \sigma}(t)\partial_\xi + \xi_{\theta,\sigma}(t)
\partial_\eta - I\lambda_0^2\eta_{\theta, \sigma}^2(t)\Bigg]
\wh w_0\Big( \xi_{\theta,\sigma}(t),
\eta_{\theta, \sigma}(t)\Big) \overline{\wh\phi(\theta, \sigma)}
e^{-I\lambda^2_0\int_0^t \eta_{\theta,\sigma}^2(s)ds} d\theta\;
d\sigma \; .
$$
Letting $t=0$, we have,
\bea
\label{4.32}
\partial_t\Big|_{t=0} \wh W(t, \theta, \sigma ) = \Big[
-a^2 \sigma \partial_\theta + \theta\partial_\sigma - I
\lambda_0^2\sigma^2
\Big]
\wh W(t, \theta, \sigma )\Big|_{t=0} \; ,
\eea
which is exactly the Fokker-Planck equation (\ref{4.25}) after
Fourier transforming,
\bea
\label{4.33}
\partial_t\Big|_{t=0} W(t, x, v)
= \Big[ a^2 x\d_v -v\d_x + I\lambda_0^2
\Delta_v\Big] W(t, x, v)\Big|_{t=0}\; .
\eea
Considering $t=0$ is not a restriction, since the proof
works for any $L^1$ initial condition.
\bigskip
The proof of part b) is completely analogous. We again notice that
under the prescribed limits the
right hand side of
(\ref{4.15}) goes to zero, hence Lemma \ref{replace} applies.
Here
$\eta_{\theta, \sigma}$ and $\xi_{\theta, \sigma}$ depend on
the limiting parameters, since $a^2=1-\lambda^2\Om = 1-\lambda^2_0\beta\Om$.
But $\lim_{\beta\to0} a =1$, hence
\bea\label{alim}
\lim_{\beta\to0}\eta_{\theta, \sigma}(s)
= \theta \sin s + \sigma\cos s \; , \qquad
\lim_{\beta\to0}\xi_{\theta, \sigma}(s)
= \theta \cos s - \sigma\sin s
\eea
uniformly for $s\in [0, t]$. Therefore
\bea\label{Qlimit}
\lim_{\Om\to\infty}\lim_{\beta\to0}
Q(t) &= &\lambda_0^2 \lim_{\Om\to\infty} \int_0^\Om
\Big| \int_0^t
\big[ \theta \sin s + \sigma\cos s \big] e^{-i\om s} ds \Big|^2
d\om \\
\nonumber
&=& \lambda_0^2 \int_0^\infty
\Big| \int_0^t
\big[ \theta \sin s + \sigma\cos s \big] e^{-i\om s} ds \Big|^2
d\om \\
\nonumber
& = & \pi
\lambda_0^2 \int_0^t\big[ \theta \sin s + \sigma\cos s \big]^2
ds \; .
\eea
Again, the last step is robust in a sense that it does not use the particular
form of the function $\big[ \theta \sin s + \sigma\cos s \big]$,
instead it uses (\ref{4.28}).
But it is rigid in a sense that $\Omega =\infty$ is essential to get
diffusive (linear) behaviour for the mean square displacement
(\ref{4.23a}).
To conclude,
we follow the calculation (\ref{4.29})-(\ref{4.33}). In addition
to the limit (\ref{Qlimit}), we have to replace
$\xi_{\theta, \sigma}(s), \eta_{\theta, \sigma}(s)$
by their limiting values (\ref{alim}) in the
argument of $\wh w_0$ to arrive at the analogue of (\ref{4.29}).
Dominated convergence theorem applies
if we assume, additionally, that $\wh w_0$ is continuous
and bounded. However $\wh w_0\in L^1$, hence it can be
approximated by such functions in $L^1$-norm.
Using that the flow $(\theta, \sigma)\mapsto \Big( \xi_{\theta, \sigma}
(s), \eta_{\theta, \sigma}(s)\Big)$ is measure preserving
and that $\wh \phi$ is bounded,
one can easily see that the approximation error can be made
arbitrarily small.
%We use that the flow $(\theta, \sigma)\mapsto \Big( \xi_{\theta, \sigma}
%(s), \eta_{\theta, \sigma}(s)\Big)$ is measure preserving.
%This allows to change variables
%\bea\label{changevar}
% \int_{\zr^2} \!\wh w_0\Big( \xi_{\theta,\sigma}(t),
% \eta_{\theta, \sigma}(t)\Big) \overline{\wh\phi(\theta, \sigma)}
% e^{-\frac{1}{2}Q(t)}
% \; d\theta \; d\sigma
% \qquad\qquad\qquad\qquad\qquad\qquad \\ \nonumber
% \qquad\qquad\qquad\qquad\qquad\qquad
% = \int_{\zr^2} \!\wh w_0(\theta, \sigma)
% \overline{\wh\phi\Big(\xi_{\theta,\sigma}^*(t),
% \eta_{\theta, \sigma}^*(t) \Big)}
% e^{-\frac{1}{2}Q^*(t)}
% \; d\theta \; d\sigma \; ,
%\eea
%where $\eta^*(t):= \eta(-t)$, $\xi^*(t): = \xi(-t)$
%are the backward trajectories and
%$$
% Q^*(t): = \lambda^2 \int_0^\Om A_\beta^2(\om)
% \Big| \int_0^t \eta_{\theta, \sigma}^*(s) ds\Big|^2 d\om\; .
%$$
%Clearly (see (\ref{Qlimit}))
%$$
% \lim_{\Om\to\infty}\lim_{\beta\to0}
% Q^*(t) = \pi
% \lambda_0^2 \int_0^t\big[ \theta \sin (-s) + \sigma\cos (-s) \big]^2
% ds \; .
%$$
%We can perform the limit $\beta\to 0$ ($a\to 1$) then $\Om\to\infty$
%on the right hand side of (\ref{changevar}) since dominated convergence
%theorem applies ($\wh\phi \in L^\infty\cap C^0$
%and $\wh w_0 \in L^1$). I.e. we can substitute
%the limiting
%value of $Q^*(t)$ and to replace the $a$-dependent $\xi_{\theta,\sigma}^*,
%\eta_{\theta,\sigma}^*$ by their $a=1$ counterparts.
%Then we can switch back to forward trajectories to get the
%same formula as the right hand side of (\ref{4.29}) but with
%trajectories defined with $a=1$.
%Notice that this change of variables procedure was necessary
%only because we did not assume continuity on $\wh w_0$,
%hence the trajectories can be replaced by their limiting values
%only in the argument of $\wh\phi$.
The rest of the calculation is identical to the proof of part a)
and we obtain
(\ref{4.25}).
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nsection{Scaling limit at high temperature:
the frictionless heat equation}
\label{S4.4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We propose a different
%(\ref{Qlimit}))
way to get diffusion from the Hamiltonian
(\ref{1.1}).
%The underlying physical mechanism
%is more universal in this context (see \cite{CTDRG}):
%the test-particle - with frequency $a$ - resonates
%with those oscillators which have frequency close to $a$ (see below),
%and the diffusion is a long time cumulative effect of "resonance kicks".
As we mentioned, obtaining diffusion for the test-particle
means that we have
to extract linear dependence in time
for $Q(t)$.
%(it also means that we have to ensure
%a small right-hand-side in (\ref{4.15}), or, in other words,
%a vanishing memory effect, in order to use $\wt w_A^\eps$ instead of
%$w_A^\eps$.
In this section, linear growth is obtained from time rescaling and
from the special form of linear combinations of
$\sin s$ and $\cos s$ in Lemma \ref{L4.1}.
It relies on a resonance effect which comes from a
singularity near $\omega\sim a$.
The system $\wt X''(t) + a^2\wt X(t)$
(see (\ref{4.1}))
picks up those modes
from the forcing term $f(t)$ in (\ref{3.32}) for which
the frequency $\om$ is close to its eigenfrequency.
So, in this section we assume $\Omega > 1$ but
finite, contrary to the previous section.
% Moreover, since the
%present analysis relies on a long time effect - see below -,
%the reader should think
%of $t$ as large in the subsequent computations.
This effect
is more robust (see the remark after (\ref{Qlimit}))
in the sense that one
{\it could} leave the hyperbolic functions $\beta A_\beta^2$
in (\ref{WN})
without ensuring a limit where it goes to 1.
In other terms,
we do not
need the high temperature limit $\beta\to0$ to obtain diffusion,
unlike in Section \ref{S4.3}, where
this limit made the $d\om$ measure
uniform and we recovered a white noise forcing term.
%Now in view of Lemma \ref{L4.1} above and the
%subsequent remark, the computations (\ref{4.40})
%above seem to characterize the large-time
%dynamics of the test-particle like a diffusive
%motion (linear behaviour for $Q$) for any given $\Om>1$ and
%without assuming $\b \rgt 0$.
Nevertheless, Lemma \ref{L4.1} needs the right-hand-side of (\ref{4.15})
to go to zero in order to be applicable (one needs
the friction to vanish), and this cannot be achieved
keeping $\beta$ fixed (Section \ref{compmod}), hence we again
set $\l=\l_0 \b^{1/2}$, $\beta\to0$.
% from the estimate (\ref{4.15})
%one needs $\lambda^2 t\to 0$.
%But then $Q$ disappears as well (see (\ref{Qest})), unless $\beta\to0$.
%This is the reason why we again prescribe $\l=\l_0 \b^{1/2}$,
%as in \cite{CL1} and as in Section \ref{S4}.
\subsection {Large space/time convergence of the Wigner distribution}
%We introduce the macroscopic scale we shall
%be interested in.
Let $\a$ be a small parameter.
We describe the behaviour of the test-particle,
as given by its Wigner distribution $w_A^\eps$
on time scales
of order $1/\a^{2}$. We consider the diffusive scaling, i.e.
the space coordinate scales as $1/\a$. Since the test-particle is
a fast harmonic oscillator, and energies are transferred back and forth
between space and velocity,
we also have to consider
velocities of order $1/\a$.
Hence we introduce the following scaling,
\bea
\label{scaling}
t = T\a^{-2}, \qquad x = X\a^{-1}, \qquad v= V\a^{-1} \; ,
\eea
where the capital letters are unscaled quantities (macroscopic
variables).
The rescaled reduced Wigner transform is defined as,
\bea
\label{Walpha}
W^{\eps,\alpha}_T(X, V):=
w_A^\eps (T\alpha^{-2}, X\alpha^{-1}, V\alpha^{-1})\; ,
\eea
where $w_A^\eps$ is defined in Lemma \ref{L3.1} (after the
thermodynamic limit). Its Fourier
transform is,
\bea
\label{Walphahat}
\wh W^{\eps,\alpha}_T
( \Theta , \Sigma) =
\a^{2}\wh w^\eps_A(T\a^{-2}, \Theta\a, \Sigma\a) \; ,
\eea
where we use $\Theta = \theta\a^{-1}$ and $\Sigma = \sigma \a^{-1}$
rescaled dual variables.
The initial condition is,
\bea
W^{\eps,\alpha}_{T=0}( X, V) = W_0(X, V)\; , \qquad
\wh W^{\eps,\alpha}_{T=0}( \Theta , \Sigma) = \wh W_0
(\Theta , \Sigma) \; ,
\eea
and we assume that,
\bea
\label{W0inL1}
\wh W_0(\Theta , \Sigma) \in L^1({\Bbb R}_\Theta
\times {\Bbb R}_\Sigma ) \; .
\eea
%In microscopic variables, the initial data $w_0(x, v)=w^\eps_A(t=0, x, v)$
%satisfies,
%\bea
%\label{W0alpha}
% W_0 ( X, V) =
% w_0 ( X\alpha^{-1}, V\alpha^{-1})\; , \qquad
% \wh W_0(\Theta, \Sigma) =\a^2 \wh w_0 (\Theta \a, \Sigma\a) \; .
%\eea
The macroscopic testfunction $\Phi(X, V)$ is a smooth function with compact
support, the microscopic testfunction is defined as,
\bea
\label{phi}
\phi(x, v) = \Phi (x\a, v\a) = \Phi (X, V)\; ,
\eea
and in Fourier variables,
$ \wh \phi(\theta, \sigma) = \a^{-2}\wh \Phi (\theta\a^{-1}$,
$ \sigma\a^{-1}) = \a^{-2}\wh \Phi (\Theta, \Sigma)$.
We are now in position to state the theorem of this section,
\begin{theorem}\label{T4.2}
Define the large time/space scale Wigner distribution
$W^{\eps,\a}_T( X, V)$ as in (\ref{Walpha}).
Assume (\ref{W0inL1}) for the initial data.
Assume that $\l=\l_0 \b^{1/2}$ with a fixed $\lambda_0>0$
and fix the frequency cutoff $\Om>1$. Hence the limits of
the parameters $\eps = (\beta, \Omega, \l)$ are reduced to
$\beta\to0$.
Then:
{\bf a)} The following high-temperature limit
exists in the weak sense for any $T\ge 0$:
\bea
\label{4.51.0}
W^\a_T( X, V ) : = \lim_{\beta\to0} W^{\eps,\a}_T
(X, V ) \; .
\eea
{\bf b)} Define the following time average of $W^\a$ over one cycle
of the harmonic oscillator (\ref{4.18}),
%-- in macroscopic time $T=t\a^{2}$ --,
\bea
\label{Wsh}
W^{\#,\a}_T( X, V): =
{1\over 2\pi\a^{2}}\int_T^{T+2\pi\a^{2}}
W^\a_S(X, V)dS \; .
\eea
Then the weak limit,
\bea
\label{W+}
W^+_T (X, V) : = \lim_{\a\to0}
W^{\#,\a}_T( X, V) \; ,
\eea
exists for each $T\ge 0$ and it
satisfies the heat equation in phase space,
\bea
\label{heateq}
\partial_T W^+_T = {\pi\lambda_0^2 \over 4}
(\Delta_X + \Delta_V) W^+_T \; ,
\eea
with initial condition $W^+_{T=0} ( X, V)$ given by
\bea
\label{4.63.0}
\wh W^+_0 (X, V) = {1\over 2\pi}
\int_0^{2\pi} \wh W_0\Big( X
\sin s + V \cos s, \; X\cos s - V \sin s\Big)ds \; .
\eea
{\bf c)}
Define the radial average,
\bea
\label{radialdef}
W^{*, \a}_T( X, V): = {1\over 2\pi}\int_0^{2\pi}
W^\a_T( R\cos s , R\sin s)ds\;
\eea
with $R:=\sqrt{X^2 + V^2}$, and clearly $ W^{*, \a}_T$ depends on $R$
only.
Again, the weak limit,
\bea
\label{4.56.1}
W^{\dagger}_T ( X, V) : = \lim_{\a\to0}
W^{*, \a}_T( X, V) \; ,
\eea
exists and the radially symmetric function
$W^{\dagger}_T$ satisfies the heat equation (\ref{heateq})
with initial condition,
$$
W^\dagger_{T=0}(X, V) : = {1\over 2\pi}\int_0^{2\pi}
W_0 (R\cos s, R\sin s)ds \; .
$$
\end{theorem}
{\bf Remark.} Here we identified the equation in a weak sense
in the space and velocity variables, but in a strong sense in
the time variable and some averaging ((\ref{Wsh}) or (\ref{radialdef}))
was needed to
ensure the existence of the limit.
If we want to consider the limit in a weak sense in time as well,
then there is no need for averaging.
Based upon part b), one can easily prove
that $W_T^+(X, V)$ can also be identified as the weak limit in
space, velocity and time, i.e. we have the following
\begin{corollary}\label{weakcor} Under the above conditions the
weak limit
$$
W_T^+(X, V) : = \lim_{\alpha\to0}
\lim_{\beta\to0} W^{\eps, \alpha}_T(X, V)
$$
exists in the topology of ${\cal D}'\big( \; [0,\infty)_T\times
\zr_X \times \zr_V \; \big)$,
it coincides with (\ref{W+}) and satisfies (\ref{heateq}). \qed
\end{corollary}
\ \\
{\bf Proof of Theorem \ref{T4.2}.} Using the rescaling and the definition of
$w^\eps_A$ (\ref{3.37}), we have,
\bea
\label{rescale}\nonumber
\langle W^{\eps, \alpha}_T, \Phi \rangle & = &
\int_{\zr^2} W^{\eps, \a}_T( X, V)
\overline{ \Phi (X, V)} dX \; dV
= \a^2 \int_{\zr^2} w^\eps_A (T\a^{-2}, x, v)
\overline{\phi(x, v)} dx\; dv \\
& = & \a^{2} \;
\bE \int_{\zr^6} \wh w_0 (\xi, \eta)
\overline{\wh \phi(\theta, \sigma)}
e^{i(x\xi +v\eta)} e^{-i(\theta X(t) +\sigma X'(t))}
d\xi \; d\eta \; dx \; dv \; d\theta \; d\sigma \\
\nonumber & =&
\bE \int_{\zr^6} \wh W_0 (\xi\a^{-1}, \eta\a^{-1})
\overline{\wh \Phi(\Theta, \Sigma)}
e^{i(x\xi +v\eta)} e^{-i\a(\Theta X(t) +\Sigma X'(t))}
d\xi \; d\eta \; dx \; dv \; d\Theta \; d\Sigma \; ,
\eea
where $t=T\a^{-2}$.
\medskip
\noindent
{\it First Step: the limit $\b \rgt 0$.}
\medskip
Due to the choice $\l=\l_0 \b^{1/2}$, we can replace $X(t)$
by $\wt X(t)$ in the $\beta\to0$ limit. For, the
right hand side of (\ref{4.15}) goes to zero as $\b\to 0$,
hence Lemma \ref{replace} applies.
% (here $\a$ is fixed and
%we also used (\ref{W0inL1})).
Hence,
\bea
\label{betalimit}
\nonumber
&&\lim_{\b\to0}\langle W^{\eps, \alpha}_T, \Phi \rangle=\\
\nonumber
&= &\lim_{\b\to0} \bE \int_{\zr^6} \wh W_0 (\xi\a^{-1}, \eta\a^{-1})
\overline{\wh \Phi(\Theta, \Sigma)}
e^{i(x\xi +v\eta)} e^{-i\a(\Theta \wt X(t) +\Sigma \wt X'(t))}
d\xi \; d\eta \; dx \; dv \; d\Theta \; d\Sigma \\
& = &
\lim_{\b\to0} \bE \int_{\zr^2} \wh W_0 \Big(\xi_{\Theta, \Sigma}
(T\a^{-2}),
\eta_{\Theta, \Sigma} (T\a^{-2})\Big)
\overline{\wh \Phi(\Theta, \Sigma)} e^{-{1\over 2} Q(T\a^{-2})}
\; d\Theta \; d\Sigma \; ,
\eea
where in the second step
we also used Lemma \ref{L4.1} and the fact that
$\a^{-1}\xi_{\a\Theta, \a\Sigma} = \xi_{\Theta, \Sigma}$ and
$\a^{-1}\eta_{\a\Theta, \a\Sigma} = \eta_{\Theta, \Sigma}$
(see (\ref{4.18})).
Recall that both $Q(t)$ and the trajectories $\xi_{\Theta, \Sigma},
\eta_{\Theta, \Sigma}$ depend on $\beta$, since $a^2=1-\lambda^2\Om
= 1-\lambda^2_0\beta\Omega$ appears in their definition (see (\ref{4.18})).
Similarly to the argument
at the end of the proof of part b) of Theorem \ref{T4.1},
using that $\wh W_0\in L^1 (d\Theta \; d\Sigma)$,
$\wh\Phi\in L^\infty\cap C^0$, $Q\ge 0$,
we see that the limit can be taken inside the integral and
the trajectories $\xi_{\Theta, \Sigma},
\eta_{\Theta, \Sigma}$ can be replaced by their
limiting values (as $a\to 1$)
\bea
\label{limhar}
\eta_{\Theta, \Sigma}^* (s) :
=\theta \sin t + \sigma\cos t \qquad
\xi_{\Theta, \Sigma}^* (s) :
=\theta \cos t - \sigma\sin t \; .
\eea
We also use (see (\ref{Qest})) that
\bea
\label{4.48}
\lim_{\beta\to0} Q(t) =
I\lambda_0^2 t \gamma^2 + \lambda_0^2 \gamma^2 B_0(t) \; .
\eea
with $B_0(t)$ satisfying $B_0(0)=0$ and
\bea\label{4.49}
|B_0(t)|\leq C\big[ 1 + (\log t)_+\big]\big[ 1 + \log\Omega\big]
\eea
(see (\ref{Best})).
We also recall that $\gamma^2 =
\theta^2 + \sigma^2 = \a^2 (\Theta^2 + \Sigma^2) = :\a^2 \Gamma^2$.
Hence,
\bea
\label{betalimit1}
\lim_{\b\to0}\langle W^{\eps, \alpha}_T, \Phi \rangle
&= & \int_{\zr^2} \wh W_0 \Big( \xi_{\Theta, \Sigma}^*(T\a^{-2}),
\eta_{\Theta, \Sigma}^*(T\a^{-2})\Big) \times\\
\nonumber
&&\times \overline{\wh\Phi (\Theta, \Sigma) }
\exp{ \Big\{ -{1\over 2}\Big[I\lambda_0^2 T\a^{-2}
+\l_0^2 B_0(T\a^{-2})\Big] \a^2 (\Theta^2 + \Sigma^2) \Big\}}
d\Theta \; d\Sigma \; .
\eea
This relation defines the Fourier transform,
\bea
\label{4.51}
\wh W^\a_T( \Theta, \Sigma ) : = \lim_{\beta\to0} \wh W^{\eps,\a}_T
( \Theta, \Sigma ) \; ,
\eea
as a weak limit, and its inverse Fourier transform,
$$
W^\a_T( X, V ) : = \lim_{\beta\to0} W^{\eps,\a}_T
( X, V ) \; .
$$
We can compute its time derivative in Fourier space,
\bea
\label{4.52}
\langle \partial_T \wh W^\a_T, \wh\Phi \rangle
&=& \int \a^{-2} \Bigg[
-\eta_{\Theta, \Sigma}^*(T\a^{-2})
\partial_\xi + \xi_{\Theta, \Sigma}^*(T\a^{-2})
\partial_\eta - \\
\nonumber &&
-{\a^2\over 2} \Big[I\lambda_0^2 +
\l_0^2 B_0'(T\a^{-2})\Big]
(\Theta^2 + \Sigma^2)\Bigg]
\wh W_0 \Big( \xi_{\Theta, \Sigma}^*(T\a^{-2}),
\eta_{\Theta, \Sigma}^*(T\a^{-2})\Big) \\
\nonumber &&
\times \overline{\wh\Phi (\Theta, \Sigma)} \exp{\Big\{ -
{1\over 2}\Big[I\lambda_0^2 T\a^{-2} +
\l_0^2 B_0(T\a^{-2})\Big]
\a^2 (\Theta^2 + \Sigma^2)\Big\} } d\Theta \; d\Sigma \; .
\eea
As usual, we can let $T=0$ to obtain,
\bea
\label{4.53}
\partial_T\Big|_{T=0}\wh W^\a_T( \Theta, \Sigma)
= \a^{-2} \Bigg[ -\Sigma
\partial_\Theta + \Theta
\partial_\Sigma - {\a^2\over 2} \Big[I\lambda_0^2 +
\l_0^2 B_0'(0)\Big]
(\Theta^2 + \Sigma^2)\Bigg] \wh W_0( \Theta, \Sigma) \; .
\eea
\medskip
\noindent
{\it Second Step: the macroscopic limit $\a \rgt 0$.}
\medskip
Now the difficulty in (\ref{4.53}) is that the convective term is too big
compared to the last diffusive term since
the motion takes place on two different time scales. There
is the fast (microscopic) time scale of the harmonic
oscillator described by $ \a^{-2} [ -\Sigma
\partial_\Theta + \Theta\partial_\Sigma]$. Then there is a
slow, macroscopic diffusive scale.
%We mention that it is clear from (\ref{4.53})
%that the scaling (\ref{scaling})
%is the natural one in this context.
We present two ways to average out the fast motion.
\bigskip\noindent
{\it Part b) of Theorem \ref{T4.2}: Averaging over a cycle.}
\medskip
Here we define $W^{\#,\a}$ according to (\ref{Wsh}).
Now for any $T$ fixed the formula,
\bea
\label{4.55}
&&\lim_{\a\to0}\langle \wh W^{\#,\a}_T, \wh \Phi \rangle
= \lim_{\a\to0} \int \wh W^{\#,\a}_T( \Theta, \Sigma)
\overline{\wh \Phi(\Theta, \Sigma)} d\Theta d \Sigma \\
\nonumber
&=& \lim_{\a\to0}\int \Bigg[ {1\over 2\pi\a^2}
\int_T^{T+2\pi\a^2} \wh W_0 \Big( \xi_{\Theta, \Sigma}^*
(S\a^{-2} ), \eta_{\Theta, \Sigma}^*
(S\a^{-2} ) \Big) e^{-I_1\lambda_0^2 S (\Theta^2 + \Sigma^2)}
dS\Bigg]
\overline{\wh \Phi(\Theta, \Sigma)} d\Theta d \Sigma \; ,
\eea
defines a function,
\bea
\label{4.56}
\wh W^+_T (\Theta, \Sigma) : = \lim_{\a\to0}
\wh W^{\#,\a}_T( \Theta, \Sigma) \; ,
\eea
weakly,
as we show below. Here $I_1:= {I\over 2} = {\pi\over 4}$ for
brevity.
Note that in (\ref{4.55}) we neglected the term involving $B_0$
in the exponential (see (\ref{betalimit1})) since the estimate (\ref{4.49})
readily implies $\a^2 B_0(T \a^{-2}) \rgt 0$. The exponential
factor in (\ref{betalimit1}) converges to that in (\ref{4.55})
uniformly for all $S\leq T$. Using $\wh \Phi\in L^1$,
we can apply the dominated convergence
theorem along with approximating $\wh W_0$ by bounded
functions, similarly to the argument at the end of the proof of
Theorem \ref{T4.1}.
%after a change of variables similarly to (\ref{changevar}).
%Again, this is just a technical step to avoid the assumption
%that $\wh W_0$ is bounded.
We have to show that the limit on the right-hand-side of (\ref{4.55}) exists,
\bea
\label{4.57}\nonumber
\langle \wh W^{\#,\a}_T,
\wh \Phi \rangle &=& \int_{\zr^2} \Bigg[ {1\over 2\pi\a^{2}}
\int_T^{T+2\pi\a^2}\wh W_0 \Big( \xi_{\Theta, \Sigma}^*
(S\a^{-2} ), \eta_{\Theta, \Sigma}^*
(S\a^{-2} ) \Big) e^{-I_1\lambda_0^2 T (\Theta^2 + \Sigma^2)}
dS\\
&&+ {1\over 2\pi\a^{2}}
\int_T^{T+2\pi\a^2}\wh W_0 \Big( \xi_{\Theta, \Sigma}^*
(S\a^{-2} ), \eta_{\Theta, \Sigma}^*
(S\a^{-2} ) \Big) \\
\nonumber
&&\times\Big[e^{-I_1\lambda_0^2 S (\Theta^2 + \Sigma^2)}
- e^{-I_1\lambda_0^2 T (\Theta^2 + \Sigma^2)}\Big]
dS\Bigg]
\overline{\wh \Phi(\Theta, \Sigma)} d\Theta d \Sigma \; .
\eea
The first term in (\ref{4.57}) is independent of $\a$, as it is just the
integral of $\wh W_0(\xi^*(s), \eta^*(s))$ over one full cycle of the
harmonic oscillator (\ref{limhar}),
\bea
\label{4.58}
{1\over 2\pi\a^{2}}
\int_T^{T+2\pi\a^2}\wh W_0 \Big( \xi_{\Theta, \Sigma}^*
(S\a^{-2} ), \eta_{\Theta, \Sigma}^*
(S\a^{-2} ) \Big) dS = {1\over 2\pi}\int_0^{2\pi}
\wh W_0 \Big( \xi_{\Theta, \Sigma}^*
(s ), \eta_{\Theta, \Sigma}^*(s)\Big)ds \; .
\eea
The second term in (\ref{4.57}) vanishes in the limit $\a \rgt 0$ since,
\bea
\label{4.59}
\Big|e^{-I_1\lambda_0^2 S (\Theta^2 + \Sigma^2)}
- e^{-I_1\lambda_0^2 T (\Theta^2 + \Sigma^2)}\Big|
\leq 2\pi I_1\lambda_0 \a^{2} (\Theta^2 + \Sigma^2)
e^{-I_1\lambda_0^2 T (\Theta^2 + \Sigma^2)}
\eea
(use that $|S-T|\leq 2\pi\a^2$),
which kills the factor $\a^{-2}$ in (\ref{4.57})
and then the length of
the integration interval goes to zero. Dominated convergence
theorem again has to be applied after an approximation.
This shows that the limit in (\ref{4.56}) makes sense
and,
\bea
\label{4.60}
\nonumber
\langle W^+_T, \Phi\rangle &=&
\langle \wh W^+_T, \wh\Phi\rangle\\
&&= \int_{\zr^2} \Big[ {1\over 2\pi}\int_0^{2\pi}
\wh W_0 \Big( \xi_{\Theta, \Sigma}^*
(s ), \eta_{\Theta, \Sigma}^*(s)\Big)ds\Big]
e^{-I_1\lambda_0^2 T (\Theta^2 + \Sigma^2)}
\overline{\wh \Phi(\Theta, \Sigma)} d\Theta d \Sigma
\; .
\eea
The time derivative is,
\bea
\label{4.61}
\nonumber
\nonumber
&&\langle \partial_T W^+_T, \Phi\rangle=\\
\nonumber
&=& -I_1\lambda_0^2 \int_{\zr^2} (\Theta^2 + \Sigma^2)\Big[
{1\over 2\pi}\int_0^{2\pi}
\wh W_0 \Big( \xi_{\Theta, \Sigma}^*
(s ), \eta_{\Theta, \Sigma}^*(s)\Big)ds\Big]
e^{-I_1\lambda_0^2 T (\Theta^2 + \Sigma^2)}
\overline{\wh \Phi(\Theta, \Sigma)} d\Theta d \Sigma \\
&=& -I_1\lambda_0^2 \Big\langle \wh W^+_T, (\Theta^2 + \Sigma^2)
\wh \Phi \Big\rangle
= -I_1\lambda_0^2 \Big\langle W^+_T, -(\Delta_X + \Delta_V)
\Phi\Big\rangle \; ,
\eea
which completes the proof of (\ref{heateq}). The initial condition
(\ref{4.63.0})
is easily obtained from (\ref{4.60}) by setting $T=0$ and
taking inverse Fourier transform.
\bigskip
\noindent
{\it Part c) of Theorem \ref{T4.2}: Radial average}
\medskip
The other possibility to eliminate the fast modes is to use the
radial function $W^{*,\a}_T$ defined in (\ref{radialdef}).
Now the formula,
\bea
\label{4.55b}
\lim_{\a\to0}\langle \wh W^{*,\a}_T ,
\wh\Phi \rangle
= \lim_{\a\to0} \int \wh W^{*,\a}_T(\Theta, \Sigma )
\overline{\wh \Phi(\Theta, \Sigma)} \; d\Theta \; d\Sigma
\eea
$$
= \lim_{\a\to0}\int \Bigg[
{1\over 2\pi}\int_0^{2\pi} \wh W_0 \Big( \xi_{\Gamma\cos s,
\Gamma\sin s}^*
(T\a^{-2} ), \eta_{\Gamma\cos s, \Gamma\sin s}^*
(T\a^{-2} ) \Big) ds \Bigg]e^{-I_1\lambda_0^2 T(\Theta^2+\Sigma^2)}
\overline{\wh \Phi(\Theta, \Sigma)} \; d\Theta \; d\Sigma \; ,
$$
(with $\Gamma: = \sqrt{\Theta^2+\Sigma^2}$)
defines a radial function,
\bea
\label{4.56b}
\wh W^{\dagger}_T(\Theta, \Sigma) : = \lim_{\a\to0}
\wh W^{*, \a}_T(\Theta, \Sigma) \; ,
\eea
(depending only on $\Theta^2+\Sigma^2$)
as a weak limit, as we show below. Note that in (\ref{4.55b}) we
again neglected the term involving $B_0$
in the exponential for the same reason as in (\ref{4.55}).
We have to show that the limit on the right-hand-side of (\ref{4.55b}) exists.
But,
$$
\xi_{\Gamma\cos s, \Gamma\sin s}^* (T\a^{-2} )
% = \Gamma\cos s \cos (T\a^{-2}) - \Gamma\sin s \sin (T\a^{-2})
= \Gamma \cos (s+ T\a^{-2}) \; , \qquad
\eta_{\Gamma\cos s, \Gamma\sin s}^* (T\a^{-2} )
% = \Gamma\cos s \sin (T\a^{-2}) + \Gamma\sin s \cos (T\a^{-2})
= \Gamma \sin (s+ T\a^{-2}) \; ,
$$
hence,
$$
{1\over 2\pi}\int_0^{2\pi} \wh W_0
\Big( \xi_{\Gamma\cos s, \Gamma\sin s}^*
(T\a^{-2} ), \eta_{\Gamma\cos s, \Gamma\sin s}^*
(T\a^{-2} ) \Big) ds
$$
$$
= {1\over 2\pi}\int_0^{2\pi}
\wh W_0 (\Gamma\cos s, \Gamma\sin s)ds =: \wh W_0^{\dagger}(\Theta,
\Sigma) \; ,
$$
independently of $\a$,
which is the "radialized" initial condition in Fourier space.
So it is clear that the limit on the right-hand-side of (\ref{4.55b}) exists,
$$
\lim_{\a\to0}\langle \wh W^{*,\a}_T, \wh\Phi \rangle
= \int \wh W_0^\dagger(\Theta, \Sigma) e^{-I_1\lambda_0^2 T(\Theta^2
+\Sigma^2)}
\overline{\wh \Phi(\Theta, \Sigma)} \; d\Theta \; d\Sigma
= :\langle \wh W^\dagger_T , \wh \Phi
\rangle \; ,
$$
and clearly $ W^{\dagger}_T $ also satisfies the heat equation
(\ref{heateq}).
This ends the proof of Theorem \ref{T4.2}.
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nsection{Heat equation with friction at finite temperature}
\label{Ssmooth}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we choose a scaling where the Markovian
part of the friction term does not vanish,
i.e. we can keep $\beta$ fixed and still get finite
diffusion.
Again we look at large time $t=T\delta^{-1}$
but now we do not scale the space variable.
To eliminate the fast mode, we again
integrate the angle. The result is a radial
Fokker-Planck equation with friction.
While the test-particle performs many cycles, it slowly diffuses out,
and this diffusion is slowed down by a friction. The diffusion
comes from resonance.
In this scaling limit
the solution of (\ref{3.34})
is close to the solution $\wt X(t)$ of an equation without a
time delayed (non-Markovian) friction term, but a Markovian friction term
will be present.
Let us choose,
\bea\label{lambdadelta}
\lambda : = \lambda_0 \delta^{1/2} \; ,
\eea
with some $\lambda_0\leq \sqrt{3}/2$ fixed.
We compare the solution of (\ref{3.34}) to that of
\bea
\label{Xtildenew}
\wt X''(t) + I\lambda^2 \wt X'(t)+ a^2\wt X(t) = f(t) \; ; \qquad
\wt X(0)=x\; , \quad \wt X'(0)=v \; ,
\eea
with $ a^2 : = 1-\l^2\Omega =1-\lambda_0^2\delta^{-1}\Omega$,
and,
\bea
\label{I}
I = \int_0^\infty {\sin \Omega s\over s} ds = {\pi \over 2} \; .
\eea
We choose the scaling such that $a \in [{1\over 2}, 1]$,
hence we always assume that
$\Omega \leq \delta^{-1} $,
but to exploit resonance, we also assume $\Omega >2$.
The new term $\lambda^2 I \wt X'(t)$ for the approximate characteristic
is due to the fact that
$M(t) \sim \lambda^2 I \delta_0(t)$ as $\Om\to0$, where
$\delta_0$ denotes the Dirac delta measure. This term is the main
part of the full friction $(M\star X')$ in (\ref{3.34}). Notice
that it is small compared with the pure harmonic oscillator
terms, $\wt X'' + a^2 \wt X$, but it is not negligible, since
we will consider long times $t\sim \l^{-2}$.
% On this
%time scale this small memory effect will give an effective
%friction (slow mode of the motion),
% while the big pure harmonic oscillator terms will
%average out (fast mode of the motion).
\subsection {A priori bounds and continuity results}
As in Section \ref{S4.1} we need a priori estimates for $X$,
i.e. for,
$$
F(t): = \sup_{s\leq t}\bE |X(s)|+\sup_{s\leq t}
\bE |X'(s)| \; ,
$$
and estimates on the difference between $\wt X(t)$ and $X(t)$.
The estimate (\ref{Fest}) in Lemma \ref{stoch}
(which originates in
(\ref{4.13}) in Lemma \ref{apriorilemma}), however,
is not precise enough for large times.
%The estimate on the forcing term (\ref{4.9}) is too robust,
%it does not reflect the fact that only frequencies close to
%the eigenfrequency $\om\sim a$ are used effectively.
The following
estimate is a more precise version of Lemma \ref{stoch}.
\begin{lemma}\label{apricontlemma}
Let $t=T\de^{-1}$, $\l=\l_0\de^{1/2}$ with fixed $\l_0\leq \sqrt{3}/2$
and $T\ge0$
and we assume that $2\leq |\log \delta|^7\leq \Omega \leq \delta^{-1}$
We also fix $\beta >0$, hence the limit of
scaling parameters $\eps = (\beta, \Omega,
\l)$ is reduced to $\de\to0$, $\Om\to\infty$ with the
side condition that $\Om\in \big[ |\log\delta|^7, \delta^{-1}\big]$.
Let $X$ be the solution
to (\ref{3.34}), then,
\bea\label{aprieq}
F(T\de^{-1})
\leq C (\beta, \lambda_0, T)\Big(1+|x| + |v|
\Big) \; ,
\eea
where $C$ is monotone increasing in $T$.
Moreover, if $\wt X$ is the solution to (\ref{Xtildenew}), then
the difference $Y(t)=: X(t)-\wt X(t)$ satisfies,
\bea\label{conteq}
\lim_{\de\to0} \Big(
\sup_{s\leq T\de^{-1}}\bE |Y(s)|+\sup_{s\leq T\de^{-1}}\bE |Y'(s)|
\Big)=0 \; .
\eea
In particular,
\bea\label{replaceeqnew}
\lim_{\de\to 0} \int\limits_{\zr^2}
\wt w^{\eps}_A(s,x,v) \overline{\phi}(x,v) dx dv
= \lim_{\de\to 0} \int\limits_{\zr^2}
w^{\eps}_A(s,x,v) \overline{\phi}(x,v) dx dv \; ,
\eea
uniformly for all $s\leq T\de^{-1}$,
where $ \wt w^{\eps}_A(t,x,v)$ is the Wigner transform
corresponding to $\wt X$, defined exactly
as (\ref{tildewdef}), but $\wt X(t)$ now being the solution to
(\ref{Xtildenew}).
\end{lemma}
\noindent
{\bf Proof.}
We follow essentially the proof of Lemma \ref{apriorilemma}.
The characteristics (\ref{3.34}) fulfill
\bea\label{charact}
X(t) &=& x\cos at + va^{-1}\sin at +
\int_0^t a^{-1}\sin a(t-s)\Big[ f(s)-
(M\star X')(s) - xM(s)\Big] ds \; ,\\
\nonumber
X'(t) &=& -xa \sin a t + v\cos at+
\int_0^t \cos a(t-s)\Big[ f(s) -
(M\star X')(s) - xM(s)\Big] ds \; .
\eea
Similarly to the proof of (\ref{4.8}) one obtains
\bea\label{memoryex}
\bE\Big| \int_0^t a^{-1}\sin a(t-s) \Big[
(M\star X')(s) + xM(s)\Big] ds\Big|
\leq K \Big[\int_0^t F(s)ds + |x|\Big] \; ,
\eea
recalling the value of $K$ (\ref{Kdef}), and the cosine term in $X'(t)$
is similar.
\\
Now we estimate the random forcing term. First we use
\bea\label{schwarz}
\bE \Big| \int_0^t f(s)\,\, a^{-1}\sin a(t-s)\; ds\Big| \leq \Bigg(
\bE \Big| \int_0^t f(s)\,\,
a^{-1}\sin a(t-s)\; ds\Big|^2\Bigg)^{1/2} \; ,
\eea
then notice that
$a^{-1}\sin a(t-s) = \eta_{\theta, \sigma}(t-s)$ with $\theta =1$, $\sigma =0$
(see (\ref{4.18})). Hence (cf. (\ref{Hdef}))
\bea
\bE \Big| \int_0^t f(s)a^{-1}\sin a(t-s)\; ds\Big|^2
\leq\lambda^2 \int_0^\Omega A_\beta^2(\om) H(t, \om; 1, 0; a) \;
\eea
which is just $Q(t)= Q(t; 1, 0; \beta, a)$, see (\ref{4.22}).
Hence from (\ref{Qest}), (\ref{Best}) we get
\bea\label{forceex}
\bE \Big| \int_0^t f(s)\,\, a^{-1}\sin a(t-s)\; ds\Big|^2
\leq C_1^2(\beta, \lambda_0, T)
\eea
using the relations among the parameters; $t=T\delta^{-1}$,
$\lambda= \lambda_0\delta^{1/2}$ and $\Omega\leq \delta^{-1}$.
Similar estimate is valid for the cosine term.
The estimates (\ref{memoryex}), (\ref{schwarz})
and (\ref{forceex}) lead to the a priori
bound,
\bea
\label{45.11}
F(t) \leq |x|+|v|
+ K \Big[\int_0^t F(s) ds+ |x|\Big]+
C_1 (\beta, \lambda_0, T) \; ,
\eea
and by the standard Gronwall argument we obtain,
\bea\label{festfin}
F(t) &\leq&
% (const)
% \Big(|x|(1+K) + |v| + C_1 (\beta, \lambda_0, T) \Big)
% e^{2Kt} \\
%\nonumber
% &\leq &
C_2 (\beta, \lambda_0, T)\Big(1+ |x| + |v|
\Big) \; .
\eea
%where we also used the value of $K=K(\lambda, \Omega)\leq (const)\lambda^2$
%and that $\lambda^2t=\lambda_0^2T$.
By monotonicity of $C_2$ in $T$, we get
the a priori bound (\ref{aprieq}) on $X(t)$ and $X'(t)$.
\medskip
{F}rom the equation (\ref{3.34}) we also get a similar bound for $X''(t)$.
We estimate
\bea
\nonumber
\bE |X''(t)| &\leq& a^2 \bE |X(t)| + \Big(
\bE |f(t)|^2\Big)^{1/2} +
|x| |M(t)| + \int_0 ^t |M(s)|\;
\bE| X'(t-s)| ds
\; .
\eea
For the forcing term we use
$$
\bE |f(t)|^2 = \lambda^2\int_0^\Om {\om (\cosh\beta \om +1)\over
2\sinh \beta \om}\; d\om \leq C_3(\beta)\lambda^2\Om^2
$$
(see (\ref{4.10})) and that
\bea\label{Mest}
|M(s)| = \lambda^2\Big| {\sin \Omega s\over s}\Big|
\leq {2\Om\lambda^2\over 1 + \Omega s} \; .
\eea
These estimates, along with $t=T\de^{-1}$, $\lambda = \lambda_0\de^{1/2}$
and $\Om\leq \de^{-1}$, give that
\bea\label{secondder}
\sup_{s\leq T\de^{-1}} \bE |X''(s)|\leq
C_4 (\beta, \lambda_0, T)\Big(|x| + |v| +
\Om^{1/2} \Big) \; ,
\eea
using the a priori bounds (\ref{Fest}), and $C_4$
is monotone in $T$.
\bigskip
For the continuity result, notice that
$Y(t): = X(t)-\wt X(t)$ satisfies the equation,
\bea
\label{45.14}
Y''(t) + I\lambda^2 Y'(t)+
a^2 Y(t) = I\lambda^2 X'(t) - (M\star X')(t) - xM(t) \; ,
\eea
with initial conditions $Y(0)=Y'(0)=0$.
Using (\ref{I}) we obtain,
\bea
\Big| I\lambda^2 X'(s) - (M\star X')(s) \Big|
&\leq& \lambda^2 \Big| \int_0^s
{\sin \Om u\over u}\Big( X'(s)-
X'(s-u)\Big)\; du\Big| \\
\nonumber
& & + \; \lambda^2 \; |X'(s)| \; \Big|
\int_{s}^\infty {\sin \Om u\over u} \; du \Big| \; .
\eea
The second term is estimated by $(const)\l^2 |X'(s)|$
with a universal constant if $s\leq 1$ and by $(const)\l^2 (\Om s)^{-1}
|X'(s)| \leq (const)\l^2 \Om^{-1} |X'(s)|$
if $s\ge 1$.
In the first term we split the integration domain.
For $u\ge \Om^{-2/3}$ we use integration by parts, (\ref{Fest})
and (\ref{secondder})
$$
\lambda^2 \; \bE \; \Big| \int_{\Om^{-2/3}}^s
{d\over du } \Big( {\cos\Om u\over \Om}\Big)
u^{-1}\Big( X'(s)-
X'(s-u)\Big)\; du\Big| \leq C_5(\beta, \lambda_0, T)
\delta |\log \delta| \Om^{-1/3}\Big(1+|x| + |v| \Big)
$$
for all $s\leq T\de^{-1}$.
For the domain $0\leq u \leq \Om^{-2/3}$, we use Taylor expansion:
$|X'(s)-X'(s-u)|\leq |u|\sup_{\sigma\leq s}
|X''(\sigma)|$ and the bound (\ref{secondder}).
We obtain finally, using (\ref{Fest}),
\bea\label{memoryest}
\bE \; \Big| I\lambda^2X'(s) - (M\star X')(s)\Big|
&\leq & C_6( \beta, \l_0, T, x, v) \de|\log\de| \Om^{-1/6} \; ,
\eea
if $1\leq s \leq T\de^{-1}$ and
\bea\label{tleq1}
\bE \; \Big| I\lambda^2 X'(s) - (M\star X')(s)\Big|
\leq \pi \l_0^2\de F(t) \leq C_7(\beta, \l_0, x, v)\de
\Big(1 + |\log\de|\Om^{-1/6}\Big)\; ,
\eea
if $s< 1$.
We now introduce the two fundamental solutions $\varphi$ and $\psi$
of $Y'' + I \lambda^2 Y' + a^2 Y=0$ with $\varphi(0)=0$, $\varphi'(0)=1$
and $\psi(0)=1, \psi'(0)=0$.
They are explicitly given as,
\bea
\label{varphi}
\varphi(t) = b^{-1}e^{-I \lambda^2 t/2}
\sin bt \; , \qquad \qquad
\psi(t) = e^{-I\lambda^2 t/2}
\cos b t
+ {I \lambda^2\over 2} \varphi(t) \; ,
\eea
with $ b: = (a^2-I^2\lambda^4/ 4)^{1/2}$.
Note that they are
bounded functions for small enough $\de$. Hence, by (\ref{Mest}),
(\ref{memoryest}) and
(\ref{tleq1}),
\bea
\label{yestimate}
\bE \; |Y(t)|& =& \bE\; \Big|\int_0^t \varphi (t-s)
\Big( I\lambda^2X'(s) - (M\star X')(s) - xM(s)\Big)
\; ds\Big| \\
\nonumber
&\leq& \Bigg( C_8( \beta, \l_0, T, x, v) |\log\de|\Om^{-1/6}
+ C_7(\beta, \l_0, x, v)\de + 2\l^2 |x|\Big[1+ (\log \Om t)_+\Big]
\Bigg)\|\phi\|_\infty \\
\nonumber
&\leq & C_9( \beta, \l_0, T, x, v) \Om^{-1/6}|\log\de| \; .
\eea
The constants $C_8$ and $C_9$
can be chosen monotone in $T$, so the same estimate is valid
for $\sup_{s\leq T\de^{-1}} \bE \; |Y(s)|$.
The argument for $Y'$ is similar, which proves
(\ref{conteq}). \qed
\subsection {Transport equation before scaling limits}
Armed with (\ref{replaceeqnew}), it
is enough to compute $\wt w^{\eps}_A(t,x,v)$.
The calculation is the same
as in Section \ref{S4.2} except for
the different fundamental solutions $\varphi$ and $\psi$ given in
(\ref{varphi}).
We redefine,
\bea
\label{etaundtau}
\eta_{\theta, \sigma} &:=& \theta \varphi (t) +\sigma\varphi'(
t)
\; ,\\
\nonumber
\xi_{\theta, \sigma} &: =& \theta\psi( t) +\sigma \psi'(t)
\; ,
\eea
and in complete analogy to Lemma \ref{L4.1} we state the,
\begin{lemma}\label{L4.3}
We have for $t\ge0$,
\bea
\label{45.21}
\int_{\zr^2} \wt w^\eps_A(t, x, v)\overline{\phi(x, v)} \;dx \; dv =
\int_{\zr^2} \wh w_0\Big( \xi_{\theta,\sigma}(t),
\eta_{\theta, \sigma}(t)\Big) \overline{\wh\phi(\theta, \sigma)}
e^{-{1\over2}Q(t)} \;d\theta \; d\sigma \;,
\eea
with
\bea
\label{45.22}
Q(t) : = \lambda^2 \int_0^\Om A_\beta^2(\om)
H(t, \om)d \om
\; ,
\eea
and $H$ is given again as
$H(t, \om) = \Big| \int_0^t \eta_{\theta, \sigma} (s)e^{-is\om}ds\Big|^2$,
but with the new $\eta_{\theta, \sigma}$ defined in
(\ref{etaundtau}). We also have exactly the same
estimate as (\ref{4.23a}), but with the redefined quantities. \qed
\end{lemma}
\subsection {Obtaining diffusion from scaling limit}
In this section, and
with similar arguments as in Section \ref{S4.4}, we again obtain
linear dependence in time of $Q(t)$ for large $t$.
Indeed,
we first write,
\bea
\varphi(t) = {1\over 2i b}
\Big( e^{tu} - e^{t\bar u}\Big) \; , \qquad
\mbox{ with, } \quad
u: = -{I\lambda^2\over2} + ib \; .
\eea
With these notations, we have,
\bea
\eta_{\theta, \sigma} (t) = {1\over 2i b}
\Bigg( \theta \Big( e^{tu} - e^{t\bar u}\Big)
+ \sigma \Big( ue^{tu} -\bar u e^{t\bar u}\Big)\Bigg) \; ,
\eea
hence,
\bea
H(t, \om) &=&
{1\over 4b^2} \Bigg|
(\theta + \sigma u){e^{t(u-i\om)}-1\over u-i\om}
- (\theta +\sigma \bar u){e^{t(\bar u-i\om)}-1\over \bar u-i\om}
\Bigg|^2 \; .
\eea
We now take the scaling $t= T\delta^{-1}$ for a fixed $T$ and
$\de\to0$.
The terms with denominator
$\bar u - i\om = -I\lambda^2/2 - i( \sqrt{a^2-I^2\lambda^4/4} + \om)$
have no singularity (they are bounded)
so the first term of $H$ is the main term.
Extracting the main term, we can write (cf. (\ref{4.37})),
$$
H(t, \om) =(\theta^2 +a^2\sigma^2)\Bigg[
{1\over 4a^2}
\Big| {e^{t(u-i\om)}-1\over u-i\om} \Big|^2 +
U(t, \om) \Bigg]\; .
$$
Using $u = ai + O(\delta)$, $3/4\leq a^2 \leq 1$, $b^2= a^2 + O(\de^2)$
we obtain
for small enough $\de$ that,
$$
\int_0^\infty \big| U(T\de^{-1}, \om)\big|
d\om \leq C_{10}(\beta, \l_0, T)
|\log\de| \; .
$$
With some elementary calculations this implies,
\bea
Q(T\de^{-1})
& =&\lambda^2(\theta^2 +a^2\sigma^2)\Bigg[
\frac{1}{4a^2}
\int_0^\Om
A_\beta^2(\om)
\Bigg| {e^{T\de^{-1}(u-i\om)}-1\over u-i\om} \Bigg|^2 d\om +
B_1(T\de^{-1})
\Bigg]\\
%\nonumber
% &=&\lambda^2
% (\theta^2 +a^2\sigma^2)\Bigg[\frac{1}{4a^2}
% \int_{a+\sqrt{\de}}^{a-\sqrt{\de}}
% A_\beta^2(\om)
% \Bigg| {e^{T\de^{-1}(u-i\om)}-1\over u-i\om} \Bigg|^2 d\om
% + B_2(T\de^{-1})\Bigg] \\
\nonumber
&=& \lambda^2
(\theta^2 +a^2\sigma^2)\Bigg[\frac{A_\beta^2(a)}{4a^2}
\int_{a+\sqrt{\de}}^{a-\sqrt{\de}}
\Bigg| {e^{T\de^{-1}(u-i\om)}-1\over u-i\om} \Bigg|^2 d\om
+ B_3(T\de^{-1})\Bigg] \; ,
\eea
where the functions $B_j$ ($j=1, 2, 3$) satisfy
$|B_j(T\de^{-1})|\leq C_{11} (\beta, \l_0, T) \de^{-1/2}$.
We used that the function $\om\mapsto A_\beta^2(\om)$
is bounded with a bounded derivative around $\om\sim a$,
and that the function $z\mapsto (e^{tz}-1)/z$
is uniformly bounded by $t$ in the vicinity of the imaginary axis.
Since the derivative of $z\mapsto |(e^{tz}-1)/z|^2$ is bounded by $t^2$,
one can replace $u$ by $ai$ in the last integral at the expense of
an error $2\sqrt{\de}|u-ia|t^2 = O(\de^{-1/2})$. Finally
one can evaluate,
$$
\int_{a+\sqrt{\de}}^{a-\sqrt{\de}}
\Bigg| {e^{T\de^{-1}(a-\om)i}-1\over a-\om} \Bigg|^2 d\om
= 2\pi T\de^{-1} + O(\de^{-1/2}) \;
$$
At this step $T\ge 0$ is used.
In summary, we obtained,
\bea\label{Qsum}
Q(T\de^{-1}) =(\theta^2 + a^2\sigma^2)\Big( \l_0^2 T
\; {\pi(\cosh (\beta a)+ 1)\over 4a\sinh \beta a}
+ B_4 (T\de^{-1}) \Big) \; .
\eea
The error satisfies
$ \big| B_4 (T\de^{-1})\big|\leq C_{12} (\beta, \l_0, T) \de^{1/2} $,
hence,
\bea
\label{Qlim}
\lim_{\de\to0}Q(T\de^{-1}) = c_\beta \l_0^2 \gamma^2 T \; ,
\eea
with $\gamma: = \theta^2 + \ca^2\sigma^2$ and
\bea\label{cbeta}
c_\beta := {\pi(\cosh (\beta \ca)+ 1)\over 4\ca\sinh \beta \ca} \; ,
\eea
assuming that
\bea
\label{ca}
\ca : = \lim_{\de\to0,\Om\to\infty}a \;
= \lim_{\de\to0,\Om\to\infty}\big( 1-\l_0\Om \delta^{-1}\big)
\eea
exists, and $\ca\in [\frac{1}{4}, 1]$.
\medskip
%The effect pointed out here is obviously again
%a pure resonance effect; the system $\wt X''(t) +I\l^2 \wt X'(t)+
% a^2 \wt X(t)$ (see (\ref{Xtildenew}))
%picks up those modes
%from the forcing term $f(t)$ in (\ref{3.32}) for which
%the frequency $\om$ is $\sim a$, i.e. close to its eigenfrequency.
Since we will keep $\beta$ fixed
and choose $\l= \l_0\de^{1/2}$ with a fixed $\l_0$,
$\de$ and $\Om$ are left as a scaling parameters from the triple
$\eps = (\beta, \Om, \l)$.
Like in Section \ref{S4.4} (cf.(\ref{Walpha})) we introduce,
\bea
\label{45.44b}
W^\eps_T( x, v) : = w_A^\eps(T\de^{-1}, x, v)\; ,
\eea
and notice that only the time is rescaled. We will assume that
$\Om\to\infty$ along with $\de\to0$ in such a way that
the limit (\ref{ca}) exists and
$\Om\in \big[ |\log\delta|^7, \delta^{-1}\big]$. Clearly either
$\Om\sim \de^{-1}$, in which case $\ca<1$, or $\Om\ll \de^{-1}$,
when $\ca =1$. In the latter case, however, we need $\Om \ge |\log\de|^7$.
\subsection{Derivation of the limiting equation}
We need the notion of "radial" function with respect to the elliptical
phase space trajectories of the oscillator $Y'' + \ca^2 Y$. As usual,
the dual variables to the phase space coordinates $(x, v)$ are $(\theta,
\sigma)$. With $\ca>0$ fixed, let
$$
\gamma = \gamma (\theta, \sigma) : = \sqrt{ \theta^2+ \ca^2\sigma^2}
\; , \qquad
r = r (x, v) : = \sqrt{x^2 + \ca^{-2} v^2} \; ,
$$
which will be considered either variables or functions, depending
on the context.
If a function $u(x,v)$ depends only on $x^2+ \ca^{-2} v^2$, then it
can be written as $u(x, v) = u^*(r)$ with some function $u^*$ defined
on $\zr_+$. Then the {\it two dimensional} Fourier transform
$\wh u(\theta, \sigma)= \int \exp{\big[ -i (\theta x + \sigma v)\big]}
u(x,v ) dx dv$ is a function of $ \theta^2+ \ca^2\sigma^2$ only,
hence it can be written as $\wh u(\theta, \sigma) = \wt u^*(\gamma)$.
Here $\wt u^*$ can be thought of as the "elliptical-radial" Fourier
transform of $u^*$, but in order to avoid confusion, we will always
perform Fourier transforms on $\zr^2$, i.e. between $u(x, v)\leftrightarrow
\wh u(\theta, \sigma)$, even if these functions are "radial".
For any function $u(x, v)$ we can form the "radial" average
of its Fourier transform $\wh u(\theta, \sigma)$ by defining
$$
\wh u^\# (\theta, \sigma)
:= {1\over 2\pi} \int_0^{2\pi} \wh u
\big( \gamma \cos s, \ca^{-1}\gamma \sin s\big) ds
\qquad\qquad \Bigg( \; = {1\over 2\pi\gamma} \int_{\tilde\theta^2
+ \ca^2 \tilde\sigma^2=\gamma^2} \wh u
(\tilde\theta, \tilde\sigma ) d\tilde\theta d\tilde\sigma \;
\Bigg) \; ,
$$
which is a function of $\gamma$, hence it can be written as
$$
\wh u^\# (\theta, \sigma) = \wt u^{\#, *}(\gamma) \; .
$$
In this notation $\#$ refers to "radial" averaging, and $*$ indicates
that we consider the radial part of the function. Tilde indicates that
it comes from the two dimensional Fourier transform $\wh u$ of
the original function $u$.
\begin{theorem}\label{T4.3}
Define the large time scale Wigner function
$W^\eps_T( x, v)$ as in (\ref{45.44b}).
Assume that $\l=\l_0 \de^{1/2}$, $\l_0\leq \sqrt{3}/2$ and fix $\beta>0$,
$\ca\in [\frac{1}{2}, 1]$.
The initial condition $W^\eps_0(x, v) = w_0(x, v)$
satisfies $\wh w_0(\theta, \sigma)\in L^1(\zr_\theta\times \zr_\sigma)$.
Consider the "radial" average of $\wh W^\eps_T$,
\bea
\wt W^{\#,\eps}_T( \gamma):
= {1\over 2\pi}\int_0^{2\pi} \wh W^\eps_T( \gamma\cos s,
\ca^{-1}\gamma\sin s )
ds \; .
\eea
Then for any $T\ge0$ the limit,
\bea\label{longlim}
\wh W^+_T( \theta, \sigma ): = \lim_{ \de\to0, \Om\to\infty
\atop {1-\l_0^2 \Om\de \to \ca \atop \Om\ge |\log\de|^7}}
\wh W^{\#,\eps}_T( \theta, \sigma ) \; ,
\eea
exists in a weak sense and it is a function
of $\gamma = (\theta^2 + \ca^2\sigma^2)^{1/2}$ only.
Hence, its inverse Fourier transform $W^+_T ( x, v)$
is a function of $r = (x^2 + \ca^{-2}v^2)^{1/2}$ only
and it can be written as $ W^{+, *}_T(r): =W^+_T ( x, v)$.
This function satisfies the "radial" Fokker-Planck equation,
\bea
\label{45.62.0}
\partial_T W^{+,*}_T =
\frac{\pi\lambda_0^2}{4} \; \partial_r (rW^{+,*}_T)
+ \frac{c_\beta\lambda_0^2}{2} \;
\Delta_r W^{+,*}_T \; ,
\eea
($c_\beta$ is given in (\ref{cbeta}))
with initial condition $W^{+,*}_0(r): = W^+_{T=0}( x, v)$
whose Fourier transform $\wh W^+_0(\theta, \sigma)$
is given by,
\bea
\label{Winit}
\wh W^+_0(\theta, \sigma) : = \wh w_0^\#(\theta, \sigma)
= {1\over 2\pi}\int_0^{2\pi} \wh w_0\big( \gamma \cos s,
\ca^{-1}\gamma \sin s\big) ds \; .
\eea
\end{theorem}
\ \\
{\bf Remark 1.} The weak limit
$ \lim\!{}^{**} \wh W^\eps_T ( \theta, \sigma ) $
(without averaging over the angular variables)
does not exist (here $\lim\!{}^{**}$ stands for the same
limit as in (\ref{longlim})). However, time averaging can again replace
angular averaging (see Remark and Corollary \ref{weakcor}),
i.e. our method easily proves that $\lim\!{}^{**} W^\eps_T(x, v)$
exists in a weak sense in all variables $(x, v, T)$, i.e. in the
topology of ${\cal D}'\big( \; \zr_x\times \zr_v\times [0,\infty)_T \; \big)$,
and it
satisfies (\ref{45.62.0}) weakly in space, velocity and time.
\ \\
{\bf Remark 2.} Since the diffusion coefficient
${1\over 2}\l_0^2 c_\beta$
in (\ref{45.62.0}) behaves as $\beta^{-1}$ for small
$\beta$ (high temperature), we see that Einstein's relation is
satisfied at high temperatures. At small temperatures
the diffusion does not disappear ($\lim_{\beta\to\infty}c_\beta >0$),
which is due to the ground state quantum fluctuations of the heat bath.
\ \\
{\bf Proof.} The proof is similar to the proof of Theorem
\ref{T4.2}, hence we skip certain steps. Let $\phi(x, v) \in C_0^\infty(
\zr\times\zr)$.
Similarly to (\ref{rescale})
we obtain from (\ref{3.37}),
\bea
\label{45.47b}
\langle W_T^\eps, \phi \rangle & = &
\int \wh w^\eps_A(T\de^{-1}, \theta, \sigma)
\overline{ \wh \phi (\theta, \sigma)} d\theta \; d\sigma\\
\nonumber
&=& \bE \int \wh w_0 (\xi, \eta) \overline{\wh \phi(\theta, \sigma)}
e^{i(x\xi +v\eta)} e^{-i(\theta X(t) +\sigma X'(t))}
d\xi \; d\eta \; dx \; dv \; d\theta \; d\sigma \; .
\eea
Thanks to (\ref{replaceeqnew}), in the limit $\de\to0$ we can
replace $X$ by $\wt X$
and to take the limiting value (\ref{Qlim}) of $Q$ in the formulae
(we again have to approximate $\wh w_0$ by bounded functions first).
% change variables back and forth, see (\ref{changevar}),
%to push the trajectories into the argument of $\wh\phi$ to apply
%dominated convergence).
We obtain (cf. (\ref{betalimit})),
\bea
\label{45.50b}
\nonumber
\lim\!{}^{**} \langle W_T^\eps, \phi \rangle
&=& \lim\!{}^{**} \bE \int \wh w_0 (\xi, \eta)
\overline{\wh \phi(\theta, \sigma)}
e^{i(x\xi +v\eta)} e^{-i(\theta \wt X(T\de^{-1}) +\sigma \wt
X'(T\de^{-1}))}
d\xi \; d\eta \; dx \; dv \; d\theta \; d\sigma\\
&=&
\lim\!{}^{**}\int \wh
w_0\Big( \xi_{\theta, \sigma}(T\de^{-1} ),
\eta_{\theta, \sigma}(T\de^{-1})\Big)
\overline{\wh \phi(\theta, \sigma)} e^{- {1\over 2}Q(T\de^{-1})}
d\theta \; d\sigma
%\nonumber
% &=& \int \lim\!{}^{**}
% \wh w_0\Big( \check\xi_{\theta, \sigma}(T\de^{-1}),
% \check\eta_{\theta, \sigma}(T\de^{-1})\Big)
% \overline{\wh \phi(\theta, \sigma)} e^{-{1\over2}
% c_\beta\lambda_0^2 \gamma^2T}
% d\theta \; d\sigma \; ,
\eea
where $\lim\!{}^{**}$ stands for the limit in (\ref{longlim}).
Recall that the functions $\xi_{\theta, \sigma}$
and $\eta_{\theta, \sigma}$ now depend on the limiting parameters,
since $\varphi$ and $\psi$ do, and they are oscillating,
which again
prevents the existence of the weak limit
in the last line of (\ref{45.50b}) without
averaging.
Time averaging is analogous to part b) of Theorem \ref{T4.2},
and it gives the weak limit in space, velocity and time. We skip
the details of the proof of the statement of Remark 1.
Performing a radial avegaring (with respect to the limiting
ellipses given by the level curves of $r=r(x, v)$ or $\gamma=\gamma(\theta,
\eta)$) is the same
as using "radial" testfunctions $\phi$ which
depend only on $r$;
i.e. $\wh \phi(\theta, \sigma)$
depends only on $\gamma$ hence it can be written as
$\wh \phi(\theta, \sigma)
=\wt\phi^*(\gamma)$.
In this case
$$
\langle \wh W^{ \#, \eps}_T, \wh\phi\rangle =
\langle \wh W^{\eps}_T, \wh\phi\rangle \; .
$$
{F}rom the explicit formulas (\ref{varphi}), (\ref{etaundtau})
it is straightforward to check
that
\bea\label{trajlim}
\lim\!^{**}\sup_{s\leq T\de^{-1}}
\Bigg| \Big(\big[\xi_{\theta, \sigma}(s)\big]^2
+ \ca^2\big[\eta_{\theta, \sigma}(s)\big]^2\Big)
- e^{-I\l_0^2 s\delta}\Big(\big[\check\xi_{\theta, \sigma}(s)\big]^2
+ \ca^2\big[\check\eta_{\theta, \sigma}(s)\big]^2\Big)\Bigg| =0 \; ,
\eea
where $\check\xi$ and $\check\eta$ are the solutions to $Y'' + \ca^2 Y=0$,
i.e.
$$
\check \xi_{\theta, \sigma}(s) : =\theta \cos(\ca s)
- \sigma \ca \sin (\ca s) \; , \qquad
\check \eta_{\theta, \sigma}(s) : =
\theta \ca^{-1}\sin (\ca s) +\sigma \cos (\ca s) \; .
$$
Since the flow $(\theta, \sigma)\mapsto \Big( \xi_{\theta, \sigma}(s),
\eta_{\theta, \sigma}(s) \Big)$ is measure preserving,
one can change variables
\bea\label{changevar}
\int_{\zr^2} \!\wh w_0\Big( \xi_{\theta,\sigma}(t),
\eta_{\theta, \sigma}(t)\Big) \overline{\wh\phi(\theta, \sigma)}
e^{-\frac{1}{2}Q(t)}
\; d\theta \; d\sigma
\qquad\qquad\qquad\qquad\qquad\qquad \\ \nonumber
\qquad\qquad\qquad\qquad\qquad\qquad
= \int_{\zr^2} \!\wh w_0(\theta, \sigma)
\overline{\wh\phi\Big(\xi_{\theta,\sigma}^*(t),
\eta_{\theta, \sigma}^*(t) \Big)}
e^{-\frac{1}{2}Q^*(t)}
\; d\theta \; d\sigma \; ,
\eea
where $\eta^*(t):= \eta(-t)$, $\xi^*(t): = \xi(-t)$
are the backward trajectories.
In this way we pushed the trajectories into the argument of $\wh\phi$,
where only their $\xi^2 + \ca^2 \eta^2$ combination matters,
and we can apply (\ref{trajlim})
to replace $\xi, \eta$ by $\check\xi, \check\eta$, finally we
can change variables backwards, now along these new trajectories.
Hence together with (\ref{Qlim}) and
with $c_\beta' : = c_\beta/2$ for simplicity, we have
\bea\nonumber
\lim\!{}^{**} \langle \wh W^{ \#, \eps}_T, \wh\phi\rangle &=&
\lim\!{}^{**} \langle \wh W^{\eps}_T, \wh\phi\rangle \\
\nonumber
&=&
\lim\!{}^{**} \int_{\zr^2}
\wh w_0\Big( e^{-I\l_0^2T/2} \check \xi_{\theta, \sigma}(T\de^{-1})
\, , e^{-I\l_0^2T/2} \check \eta_{\theta, \sigma}(T\de^{-1})\Big)
\overline{\wt \phi^*(\gamma)}
e^{-c_\beta'\lambda_0^2 T \gamma^2}
d\theta \, d\sigma \; ,
\eea
if we can show that this latter limit exists.
But the right hand side above is in fact independent of the limiting
parameters $\de, \Om$, since
we can first
integrate on ellipses $\theta^2 + \ca^2\sigma^2= (const)$,
similarly to the same calculation in
the proof of part c), Theorem \ref{T4.2}.
Hence,
\bea
\int_{\zr^2}
\wh w_0\Big( e^{-I\l_0^2T/2} \check \xi_{\theta, \sigma}(T\de^{-1})
\, , e^{-I\l_0^2T/2} \check \eta_{\theta, \sigma}(T\de^{-1})\Big)
\overline{\wt \phi^*(\gamma)}
e^{-c_\beta'\lambda_0^2 T \gamma^2}
d\theta \, d\sigma
\eea
$$
= \int_{\zr^2} \wt W^{+,*}_0\Big(\gamma e^{-I\l_0^2T/2} \Big)
\overline{\wt \phi^*(\gamma)}
e^{-c_\beta'\lambda_0^2 T \gamma^2}
d\theta \, d\sigma \; ,
$$
where we recall the definition of $ \wt W^{+}_0$ (\ref{Winit}), which
depends only on $\gamma^2=\theta^2 + \ca^2\sigma^2$, and we let
$\wt W^{+,*}_0 (\gamma) : = \wt W^{+}_0(\theta, \sigma)$.
Therefore, the relation,
$$
\lim\!{}^{**} \langle \wh W^{\#, \eps}_T, \wh\phi\rangle
= \int_{\zr^2} \wt W^{+,*}_0\Big(\gamma e^{-I\l_0^2T/2} \Big)
\overline{\wt \phi^*(\gamma)}
e^{-c_\beta'\lambda_0^2 T \gamma^2}
d\theta \, d\sigma \;
$$
defines the weak limit,
$$
\wh W_T^+(\theta, \sigma) : = \lim\!{}^{**}
\wh W^{\#, \eps}_T(\theta, \sigma)
$$
and it is a function depending only on $\theta^2+\ca^2\sigma^2$, i.e.
it can be written as $ \wt W_T^{+,*}(\gamma):=\wh W_T^+(\theta, \sigma)$.
Also, we readily obtain the equation satisfied by
$ \wt W_T^{+,*}(\gamma)$ by computing,
\bea
\Big\langle \partial_T\Big|_{T=0} \wh W^+_T , \wh\phi\Big\rangle
&=& \partial_T\Big|_{T=0}
\int_{\zr^2} \wt W^{+,*}_0\Big(\gamma e^{-I\l_0^2T/2} \Big)
\overline{\wt \phi^*(\gamma)}
e^{-c_\beta'\lambda_0^2 T \gamma^2}
d\theta \, d\sigma\\
\nonumber
&=& \int_{\zr^2} \Big[ -{I\lambda_0^2\over2}\gamma\partial_\gamma
- c_\beta'\lambda_0^2 \gamma^2\Big]
\wt W_0^{+,*} (\gamma ) \overline{\wt \phi^*(\gamma)}
d\theta \, d\sigma \; ,
\eea
from which
(\ref{45.62.0}) follows, recalling that $I={\pi\over 2}$
and the value of $c_\beta' = c_\beta/2$ from (\ref{cbeta}). \qed
%\section{Concluding remarks}
%The two problems described in sections 4.3 and 4.4
%really describe two different situations.
%
%%The first is a situation where short time is
%considered: the base motion $\wt X''(t)+ \wt X(t) $
%in equation (\ref{4.1}) performs finitely
%many cycles. At the same time the forcing frequencies are
%driven to infinity. Essentially one forces it by a Brownian
%motion (more precisely by white noise).%%
%The second situation is different. Here the focus is on long time.
%The frequency does not have to go up to infinity, only to reach
%the resonant frequency (this is the assumption $\Om>1$). The
%full range of frequencies needed for Brownian motion is obtained
%from the scaling limit. Also, the frequency distribution can be
%arbitrary, hence a distribution function $\phi(\om)$ is allowed
%in the original Hamiltonian (see (\ref{3.3})), and then only its value at
%the resonant frequency $\om=1$ is picked up.
%The particle performs essentially infinitely
%many oscillations in its own harmonic oscillator $\wt X''(t)+ \wt X(t)$.
%But it resonates with the frequencies
%close to its eigenfrequency, so that the total energy gained via
%these resonances is proportional to $t$ (this is clear from (\ref{4.40})),
%and the displacement is proportional to
%$\sqrt{t}$. In addition to its fast harmonic oscillator motion,
%its mean displacement describes a diffusion in phase space.
%The second situation is a more sophisticated way to get diffusion,
%and it is slightly closer to reality as mentioned at the beginning
%of this section 5. The first one assumes
%frequencies up to infinity and proves diffusion on a short
%time scale, while the second one allows Ultra-Violet cutoff
%and the diffusion emerges only after a long time.
%But still we have the limit $\beta\to0$.%
%In the last problem in section 6
%we can obtain diffusion without letting the temperature
%%tend to infinity and again this is due to a resonance effect.
%But still the linear coupling assumption remains questionable, as we
%explained in section 2.
\bigskip
\noindent
{\bf Acknowledgements.} The authors are indebted to H. Spohn for
discussions. F.C. and L.E. were partially supported by
the Erwin Schr\"odinger Institute in Vienna (Austria) during their visit,
and they thank this institution
for its hospitality.
This work was supported by the TMR-Network ``Asymptotic Methods in
Kinetic Theory'' number ERB FMBX CT97 0157 (F.C., F.F. and P.A.M.) and
by NSF-grant DMS-9970323 (L.E.).
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\end{thebibliography}
\noindent
{\it E-mail addresses:} {\verb castella@maths.univ-rennes1.fr,
lerdos@math.gatech.edu, \\
Florian.Frommlet@esi.ac.at, marko@aurora.tuwien.ac.at}
\end{document}