%fmt=latex %\documentclass[amstex]{article} \documentclass[12pt,a4,amstex]{article} %\usepackage{Newbib} \usepackage{amssymb} \setlength{\textwidth}{17cm} \setlength{\textheight}{22cm} \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt} \setlength{\topmargin}{-2cm} \renewcommand{\baselinestretch}{1.2} \begin{document} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{definition}{Definition}[section] \newtheorem{corollary}{Corollary}[section] \def\l{\lambda} \renewcommand\theequation{\thesection.\arabic{equation}} \newcommand{\qed}{\nopagebreak\nolinebreak\hfill\rule{3mm}{3mm}} \newcommand\nsection[1]{\section{#1}\setcounter{equation}{0}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\beas}{\begin{eqnarray*}} \newcommand{\eeas}{\end{eqnarray*}} \newcommand{\noi}{\noindent} \newcommand{\disp}{\displaystyle} \newcommand{\zn}{{\mathrm {I\!N}}} \newcommand{\zr}{{\mathrm {I\!R}}} \def\a{\alpha} \def\e{\varepsilon} \def\s{\sigma} \def\l{\lambda} \def\o{\otimes} \def\om{\omega} \def\Om{\Omega} \def\b{\beta} \def\g{\gamma} \def\de{\delta} \def\p{\phi} \def\th{\theta} \def\eps{\varepsilon} \def\s{\sigma} \def\V{\hat V} \def\d{\partial} \def\D{\Delta} \def\rgt{\rightarrow} \def\ca{\check{a}} \newcommand\bN{{\Bbb N}} \newcommand\bR{{\Bbb R}} \newcommand\bZ{{\Bbb Z}} \newcommand\bT{{\Bbb T}} \newcommand\bE{{\bold E}} \newcommand\bk{{\bold k}} \def\tomega{{\tilde\omega}} \def\wt{\widetilde} \def\wh{\widehat} \thispagestyle{empty} \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}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \nsection{Preliminary results} \label{S3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Reducing the number of physical parameters} \label{S3.1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %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 %\ \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The Wigner formalism} \label{S3.2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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. 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