Content-Type: multipart/mixed; boundary="-------------0504130819417" This is a multi-part message in MIME format. ---------------0504130819417 Content-Type: text/plain; name="05-135.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-135.comments" 29 pages ---------------0504130819417 Content-Type: text/plain; name="05-135.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-135.keywords" canonical ensemble in quantum theory; probability measures on Hilbert space; Gaussian measures; density matrices ---------------0504130819417 Content-Type: application/x-tex; name="thermo1.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="thermo1.tex" \documentclass[12pt]{article} \usepackage{graphicx,amsfonts,amsmath,amssymb,mathrsfs} \title{On the Distribution of the Wave Function for Systems in Thermal Equilibrium} \author{ Sheldon Goldstein\footnote{Departments of Mathematics and Physics, Hill Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: oldstein@math.rutgers.edu}, Joel L. Lebowitz\footnote{Departments of Mathematics and Physics, Hill Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: lebowitz@math.rutgers.edu},\\ Roderich Tumulka\footnote{Dipartimento di Fisica dell'Universit\`a di Genova and INFN sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy. E-mail: tumulka@mathematik.uni-muenchen.de}, and Nino Zangh\`\i\footnote{Dipartimento di Fisica dell'Universit\`a di Genova and INFN sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy. E-mail: zanghi@ge.infn.it} } \date{April 4, 2005} \addtolength{\textwidth}{2.0cm} \addtolength{\hoffset}{-1.0cm} \addtolength{\textheight}{3.0cm} \addtolength{\voffset}{-1.5cm} \newcommand{\Hilbert}{\mathscr{H}} \newcommand{\conf}{\mathcal{Q}} \newcommand{\tr}{\mathrm{tr}} \renewcommand{\Re}{\mathrm{Re}} \renewcommand{\Im}{\mathrm{Im}} \newcommand{\EEE}{\mathbb{E}} \newcommand{\PPP}{\mathbb{P}} \newcommand{\RRR}{\mathbb{R}} \newcommand{\CCC}{\mathbb{C}} \newcommand{\QQQ}{\mathbb{Q}} \newcommand{\NNN}{\mathbb{N}} \newcommand{\prob}{\mathrm{Prob}} \newcommand{\sphere}{\mathscr{S}} % unit sphere \renewcommand{\sp}[2]{\langle #1|#2 \rangle} \newcommand{\spec}{\mathrm{spec}} % spectrum \newcommand{\support}{\mathrm{support}} \newcommand{\R}{\mathscr{R}} \newcommand{\G}[1]{G(#1)} \newcommand{\GA}[1]{GA(#1)} \newcommand{\GAP}[1]{GAP(#1)} \newcommand{\EIG}[1]{EIG(#1)} \newcommand{\red}{\mathrm{red}} \newcommand{\proj}{P} \newcommand{\B}{\mathscr{B}} \newcommand{\M}{\mathcal{M}} % set of prob measures \newcommand{\D}{\mathcal{D}} % set of density matrices \newcommand{\onb}{$\{ |q_2\rangle\}$} \newtheorem{thm}{Theorem} \newtheorem{lemma}{Lemma} \newenvironment{proof}[1]{\medskip\noindent\textit{Proof#1.}} {$\hfill\square$\medskip} \renewcommand{\baselinestretch}{1.15} %\renewcommand{\baselinestretch}{1.25} \begin{document} \maketitle \begin{abstract} A density matrix that is not pure can arise, via averaging, from many different distributions of the wave function. This raises the question, which distribution of the wave function, if any, should be regarded as corresponding to systems in thermal equilibrium as represented, for example, by the density matrix $\rho_\beta = (1/Z) \exp(- \beta H)$ of the canonical ensemble. To answer this question, we construct, for any given density matrix $\rho$, a measure on the unit sphere in Hilbert space, denoted $\GAP{\rho}$, using the Gaussian measure on Hilbert space with covariance $\rho$. We argue that $\GAP{\rho_\beta}$ corresponds to the canonical ensemble. \medskip \noindent Key words: canonical ensemble in quantum theory; probability measures on Hilbert space; Gaussian measures; density matrices. \end{abstract} %\tableofcontents \section{Introduction} In classical mechanics, ensembles, such as the microcanonical and canonical ensembles, are represented by probability distributions on the phase space. In quantum mechanics, ensembles are usually represented by density matrices. These density matrices could be thought of as arising from probability distributions on the (normalized) wave functions associated with the thermodynamical ensembles, so that members of the ensemble are represented by a random state vector. There are, however, as is well known, many probability distributions which give rise to the same density matrix, and thus to the same predictions for experimental outcomes \cite[sec.~IV.3]{Neumann}. Moreover, as emphasized by Landau and Lifshitz \cite[sec. I.5]{LL59}, the energy levels for macroscopic systems are so closely spaced (exponentially small in the number of particles in the system) that ``the concept of stationary states [energy eigenfunctions] becomes in a certain sense unrealistic'' because of the difficulty of preparing a system with such a sharp energy and keeping it isolated. Landau and Lifshitz are therefore wary of, and warn against, regarding the density matrix for such a system as arising solely from our lack of knowledge about the wave function of the system. We shall argue, however, that despite these caveats such distributions can be both useful and physically meaningful. In particular we describe here a novel probability distribution, to be associated with any thermal ensemble such as the canonical ensemble. One motivation for considering probability distributions on wave functions is to exploit the analogy between classical and quantum statistical mechanics \cite{schr,schrbook,Bloch,joel2,joel3,joel1}. This analogy suggests that some relevant classical reasonings can be transferred to quantum mechanics by formally replacing the classical phase space by the unit sphere $\sphere (\Hilbert)$ of the quantum system's Hilbert space $\Hilbert$. In particular, with a natural measure $\mu(d \psi)$ on $\sphere(\Hilbert)$ one could attempt an analysis of the second law of thermodynamics in quantum mechanics along the lines of Boltzmann's analysis of the second law in classical mechanics, involving an argument to the effect that the behavior described in the second law (such as entropy increase) occurs for typical states of an isolated macroscopic system, i.e.\ for the overwhelming majority of points on $\sphere(\Hilbert)$ with respect to $\mu(d\psi)$. Probability distributions on wave functions of a composite system, with Hilbert space $\Hilbert$, have also been used to establish the typical properties of the reduced density matrix of a subsystem arising from the wave function of the composite. For example, Page \cite{Pag93} considers the uniform distribution on $\sphere(\Hilbert)$ for a finite-dimensional Hilbert space $\Hilbert$, in terms of which he shows that the von Neumann entropy of the reduced density matrix is typically nearly maximal under appropriate conditions on the dimensions of the relevant Hilbert spaces. Given a probability distribution $\mu$ on the unit sphere $\sphere(\Hilbert)$ of the Hilbert space $\Hilbert$ there is always an associated density matrix $\rho_\mu$ \cite{Neumann}: it is the density matrix of the mixture, or the statistical ensemble of systems, defined by the distribution $\mu$, given by \begin{equation}\label{rhomupsi} \rho_\mu = \int\limits_{\sphere(\Hilbert)} \!\!\! \mu(d\psi) \, |\psi\rangle \langle\psi| \,. \end{equation} For any projection operator $P$, $\tr\,(\rho_\mu P)$ is the probability of obtaining in an experiment a result corresponding to $P$ for a system with a $\mu$-distributed wave function. It is evident from \eqref{rhomupsi} that $\rho_\mu$ is the second moment, or covariance matrix, of $\mu$, provided $\mu$ has mean 0 (which may, and will, be assumed without loss of generality since $\psi$ and $-\psi$ are equivalent physically). While a probability measure $\mu$ on $\sphere(\Hilbert)$ determines a unique density matrix $\rho$ on $\Hilbert$ via \eqref{rhomupsi}, the converse is not true: the association $\mu \mapsto \rho_\mu$ given by \eqref{rhomupsi} is many-to-one.\footnote{For example, in a $k$-dimensional Hilbert space the uniform probability distribution $u=u_{\sphere(\Hilbert)}$ over the unit sphere has density matrix $\rho_u = \tfrac{1}{k}I$ with $I$ the identity operator on $\Hilbert$; at the same time, for every orthonormal basis of $\Hilbert$ the uniform distribution over the basis (which is a measure concentrated on just $k$ points) has the same density matrix, $\rho = \tfrac{1}{k} I$. An exceptional case is the density matrix corresponding to a pure state, $\rho = |\psi \rangle \langle \psi|$, as the measure $\mu$ with this density matrix is almost unique: it must be concentrated on the ray through $\psi$, and thus the only non-uniqueness corresponds to the distribution of the phase.} There is furthermore no unique ``physically correct'' choice of $\mu$ for a given $\rho$ since for any $\mu$ corresponding to $\rho$ one could, in principle, prepare an ensemble of systems with wave functions distributed according to this $\mu$. However, the situation is different for systems in thermal equilibrium, and it is not unreasonable to ask: which measure on $\sphere(\Hilbert)$ corresponds to a thermodynamic ensemble? Let us start with the \emph{microcanonical} ensemble, corresponding to the energy interval $[E, E+\delta]$, where $\delta$ is small on the macroscopic scale but large enough for the interval to contain many eigenvalues. To this there is associated the spectral subspace $\Hilbert_{E,\delta}$, the span of the eigenstates $|n\rangle$ of the Hamiltonian $H$ corresponding to eigenvalues $E_n$ between $E$ and $E + \delta$. Since $\Hilbert_{E,\delta}$ is finite dimensional, one can form the \emph{microcanonical density matrix} \begin{equation}\label{microcanrho} \rho_{E,\delta} = (\dim \Hilbert_{E,\delta})^{-1} P_{\Hilbert_{E,\delta}} \end{equation} with $P_{\Hilbert_{E,\delta}} = 1_{[E,E+\delta]}(H)$ the projection to $\Hilbert_{E,\delta}$. This density matrix is diagonal in the energy representation and gives equal weight to all energy eigenstates in the interval $[E, E+\delta]$. But what is the corresponding \emph{microcanonical measure}? The most plausible answer, given long ago by E.~Schr\"odinger \cite{schr,schrbook} and F.~Bloch \cite{Bloch}, is the (normalized) uniform measure $u_{E,\delta} = u_{\sphere (\Hilbert_{E,\delta})}$ on the unit sphere in this subspace. $\rho_{E,\delta}$ is associated with $u_{E,\delta}$ via \eqref{rhomupsi}. Note that a wave function $\Psi$ chosen at random from this distribution is almost certainly a nontrivial superposition of the eigenstates $|n \rangle$ with random coefficients $\sp{n}{\Psi}$ that are identically distributed, but not independent. The measure $u_{E,\delta}$ is clearly stationary, i.e., invariant under the unitary time evolution generated by $H$, and it is as spread out as it could be over the set $\sphere(\Hilbert_{E,\delta})$ of allowed wave functions. This measure provides us with a notion of a ``typical wave function'' from $\Hilbert_{E,\delta}$ which is very different from the one arising from the measure $\mu_{E,\delta}$ that, when $H$ is nondegenerate, gives equal probability $(\dim \Hilbert_{E,\delta})^{-1}$ to every eigenstate $|n \rangle$ with eigenvalue $E_n \in [E,E+\delta]$. The measure $\mu_{E,\delta}$, which is concentrated on these eigenstates, is, however, less robust to small perturbations in $H$ than is the smoother measure $u_{E,\delta}$. Our proposal for the canonical ensemble is in the spirit of the uniform microcanonical measure $u_{E,\delta}$ and reduces to it in the appropriate cases. It is based on a mathematically natural family of probability measures $\mu$ on $\sphere(\Hilbert)$. For every density matrix $\rho$ on $\Hilbert$, there is a unique member $\mu$ of this family, satisfying \eqref{rhomupsi} for $\rho_\mu = \rho$, namely the \emph{Gaussian adjusted projected measure} $\GAP{\rho}$, constructed roughly as follows: Eq.~\eqref{rhomupsi} (i.e., the fact that $\rho_\mu$ is the covariance of $\mu$) suggests that we start by considering the Gaussian measure $\G{\rho}$ with covariance $\rho$ (and mean 0), which could, in finitely many dimensions, be expressed by $\G{\rho}(d\psi) \propto \exp (- \sp {\psi} {\rho^{-1}| \psi}) \, d\psi$ (where $d\psi$ is the obvious Lebesgue measure on $\Hilbert$). This is not adequate, however, since the measure that we seek must live on the sphere $\sphere (\Hilbert)$ whereas $\G{\rho}$ is spread out over all of $\Hilbert$. We thus adjust and then project $\G{\rho}$ to $\sphere (\Hilbert)$, in the manner described in Section~\ref{sec:defmeasure}, in order to obtain the measure $\GAP{\rho}$, having the prescribed covariance $\rho$ as well as other desirable properties. It is our contention that \textit{a quantum system in thermal equilibrium at inverse temperature $\beta$ should be described by a random state vector whose distribution is given by the measure $\GAP{\rho_\beta}$ associated with the density matrix for the canonical ensemble,} \begin{equation}\label{rhobetaH} \rho_\beta = \rho_{\Hilbert,H,\beta} = \frac{1}{Z} \exp(-\beta H) \text{ with } Z:=\tr\, \exp(-\beta H). \end{equation} In order to convey the significance of $\GAP{\rho}$ as well as the plausibility of our proposal that $\GAP{\rho_\beta}$ describes thermal equilibrium, we recall that a system described by a canonical ensemble is usually regarded as a subsystem of a larger system. It is therefore important to consider the notion of the distribution of the wave function of a subsystem. Consider a composite system in a pure state $\psi \in \Hilbert_1 \otimes \Hilbert_2$, and ask what might be meant by the wave function of the subsystem with Hilbert space $\Hilbert_1$. For this we propose the following. Let $\{|q_2\rangle\}$ be a (generalized) orthonormal basis of $\Hilbert_2$ (playing the role, say, of the eigenbasis of the position representation). We define the \emph{conditional wave function} \cite{DGZ} of system 1 to be the random vector $\Psi_1\in \Hilbert_1$ given by \begin{equation}\label{Psi1def} \Psi_1 = \mathcal{N} \: \sp{Q_2} {\psi} \end{equation} where the (partial) scalar product is taken in $\Hilbert_2$, $\mathcal{N}= \mathcal{N} (\psi,Q_2) = \bigl\| \sp{Q_2}{\psi} \bigr\|^{-1}$ is the normalizing factor, and $|Q_2\rangle$ is a random element of the basis $\{|q_2\rangle\}$, chosen with the quantum distribution \begin{equation}\label{marg} \PPP(Q_2 = q_2) = \bigl\| \sp{q_2} {\psi} \bigr\|^2. \end{equation} Note that $\Psi_1$ becomes doubly random when we start with a random wave function in $\Hilbert_1 \otimes \Hilbert_2$ instead of a fixed one. The distribution of $\Psi_1$ corresponding to (\ref {Psi1def}) and (\ref{marg}) is given by the probability measure on $\sphere (\Hilbert_1)$, \begin{equation}\label{mu1} \mu_1(d\psi_1)= \PPP(\Psi_1 \in d\psi_1) = \sum_{q_2} \bigl\| \sp{q_2} {\psi} \bigr\|^2 \, \delta\bigl( \psi_1 - \mathcal{N} (\psi, q_2) \, \sp{q_2} {\psi} \bigr) \, d\psi_1 \,, \end{equation} where $\delta(\psi - \phi) \, d\psi$ denotes the ``delta'' measure concentrated at $\phi$. While the density matrix $\rho_{\mu_1}$ associated with ${\mu_1}$ always equals the reduced density matrix $\rho_1^\red$ of system 1, given by \begin{equation}\label{rhoreddef} \rho_1^\red = \tr_2 |\psi \rangle \langle \psi | = \sum_{q_2} \sp{q_2}{\psi} \sp{\psi}{q_2} \,, \end{equation} the measure $\mu_1$ itself usually depends on the choice of the basis $\{| q_2 \rangle\}$. It turns out, nevertheless, as we point out in Section~\ref{sec:typicality}, that $\mu_1(d\psi_1)$ is a universal function of $\rho_1^\red$ in the special case that system 2 is large and $\psi$ is typical (with respect to the uniform distribution on all wave functions with the same reduced density matrix), namely $\GAP{\rho_1^\red}$. Thus $\GAP{\rho}$ has a distinguished, universal status among all probability measures on $\sphere(\Hilbert)$ with density matrix $\rho$. To support our claim that $\GAP{\rho_\beta}$ is the right measure for $\rho_\beta$, we shall regard, as is usually done, the system described by $\rho_\beta$ as coupled to a (very large) heat bath. The interaction between the heat bath and the system is assumed to be (in some suitable sense) negligible. We will argue that if the wave function $\psi$ of the combined ``system plus bath'' has microcanonical distribution $u_{E,\delta}$, then the distribution of the conditional wave function of the (small) system is approximately $\GAP{\rho_{\beta}}$; see Section~\ref{sec:hb1}. Indeed, a stronger statement is true. As we argue in Section~\ref{sec:hb2}, even for a typical \emph{fixed} microcanonical wave function $\psi$ of the composite, i.e., one typical for $u_{E,\delta}$, the conditional wave function of the system, defined in \eqref{Psi1def}, is then approximately $\GAP{\rho_\beta}$-distributed, for a typical basis $\{|q_2\rangle\}$. This is related to the fact that for a typical microcanonical wave function $\psi$ of the composite the reduced density matrix for the system is approximately $\rho_\beta$ \cite{thermo4, schr, schrbook}. Note that the analogous statement in classical mechanics would be wrong: for a fixed phase point $\xi$ of the composite, be it typical or atypical, the phase point of the system could never be random, but rather would merely be the part of $\xi$ belonging to the system. The remainder of this paper is organized as follows. In Section~\ref{sec:defmeasure} we define the measure $\GAP{\rho}$ and obtain several ways of writing it. In Section~\ref{sec:prop} we describe relevant mathematical properties of these measures. In Section~\ref{sec:hb1} we argue that $\GAP{\rho_\beta}$ represents the canonical ensemble. In Section~\ref{sec:Typicality} we outline the proof that $\GAP{\rho}$ is the distribution of the conditional wave function for \emph{most} wave functions in $\Hilbert_1 \otimes \Hilbert_2$ with reduced density matrix $\rho$ if system 2 is large, and show that $\GAP{\rho_\beta}$ is the typical distribution of the conditional wave function arising from a fixed microcanonical wave function of a heat bath. In Section~\ref{sec:rem} we discuss other measures that have been or might be considered as the thermal equilibrium distribution of the wave function. Finally, in Section~\ref{sec:two} we compute explicitly the distribution of the coefficients of a $\GAP{\rho_\beta}$-distributed state vector in the simplest possible example, the two-level system. \section{Definition of $\GAP{\rho}$} \label{sec:defmeasure} In this section, we define, for any given density matrix $\rho$, the Gaussian adjusted projected measure $\GAP{\rho}$ on $\sphere (\Hilbert)$. This definition makes use of two auxiliary measures, $\G{\rho}$ and $\GA{\rho}$, defined as follows. $\G{\rho}$ is the Gaussian measure on $\Hilbert$ with covariance matrix $\rho$ (and mean 0). More explicitly, let $\{|n\rangle\}$ be an orthonormal basis of eigenvectors of $\rho$ and $p_n$ the corresponding eigenvalues, \begin{equation} \rho = \sum_n p_n \, |n \rangle \langle n|. \end{equation} Such a basis exists because $\rho$ has finite trace. Let $Z_n$ be a sequence of independent complex-valued random variables having a (rotationally symmetric) Gaussian distribution in $\CCC$ with mean $0$ and variance \begin{equation}\label{variance} \EEE |Z_n|^2 = p_n \end{equation} (where $\EEE$ means expectation), i.e., $\Re \, Z_n$ and $\Im \, Z_n$ are independent real Gaussian variables with mean zero and variance $p_n/2$. We define $\G{\rho}$ to be the distribution of the random vector \begin{equation}\label{psiGdef} \Psi^G := \sum_n Z_n |n\rangle \,. \end{equation} Note that $\Psi^G$ is not normalized, i.e., it does not lie in $\sphere (\Hilbert)$. In order that $\Psi^G$ lie in $\Hilbert$ at all, we need that the sequence $Z_n$ be square-summable, $\sum_n |Z_n|^2 < \infty$. That this is almost surely the case follows from the fact that $\EEE \sum_n |Z_n|^2$ is finite. In fact, \begin{equation}\label{Esumvariances} \EEE \sum_n |Z_n|^2 = \sum_n \EEE |Z_n|^2 = \sum_n p_n = \tr\, \rho = 1. \end{equation} More generally, we observe that for any measure $\mu$ on $\Hilbert$ with (mean 0 and) covariance given by the trace class operator $C$, \[ \int_{\Hilbert} \mu(d\psi) \, |\psi \rangle \langle \psi| = C\,, \] we have that, for a random vector $\Psi$ with distribution $\mu$, $\EEE \| \Psi\|^2 = \tr\, C$. It also follows that $\Psi^G$ almost surely lies in the positive spectral subspace of $\rho$, the closed subspace spanned by those $|n\rangle$ with $p_n \neq 0$, or, equivalently, the orthogonal complement of the kernel of $\rho$; we shall call this subspace $\support(\rho)$. Note further that, since $\G{\rho}$ is the Gaussian measure with covariance $\rho$, it does not depend (in the case of degenerate $\rho$) on the choice of the basis $\{|n\rangle \}$ among the eigenbases of $\rho$, but only on $\rho$. We define the adjusted Gaussian measure $\GA{\rho}$ on $\Hilbert$ by \begin{equation}\label{muNdef} \GA{\rho} (d\psi) = \|\psi\|^2 \; \G{\rho} (d\psi). \end{equation} Since $\EEE \|\Psi^G\|^2 =1$ by \eqref{Esumvariances}, $\GA{\rho}$ is a probability measure. Let $\Psi^{GA}$ be a $\GA{\rho}$-distributed random vector. We define $\GAP{\rho}$ to be the distribution of \begin{equation}\label{mudef} \Psi^{GAP} := \frac{\Psi^{GA}}{\|\Psi^{GA}\|} = \proj(\Psi^{GA}) \end{equation} with $\proj$ the projection to the unit sphere (i.e., the normalization of a vector), \begin{equation}\label{projdef} \proj: \Hilbert \setminus \{0\} \to \sphere(\Hilbert)\,, \quad \proj(\psi) = \|\psi\|^{-1} \psi. \end{equation} Putting \eqref{mudef} differently, for a subset $B \subseteq \sphere(\Hilbert)$, \begin{equation}\label{muB} \GAP{\rho}(B) = \GA{\rho} (\RRR^+ B) = \int\limits_{\RRR^+ B} \G{\rho}(d\psi) \; \|\psi\|^2 \end{equation} where $\RRR^+B$ denotes the cone through $B$. More succinctly, \begin{equation}\label{GAPprojGA} \GAP{\rho} = \proj_* \bigl( \GA{\rho} \bigr) = \GA{\rho} \circ \proj^{-1}\,. \end{equation} where $\proj_*$ denotes the action of $\proj$ on measures. More generally, one can define for any measure $\mu$ on $\Hilbert$ the ``adjust-and-project'' procedure: let $A(\mu)$ be the adjusted measure $A(\mu)(d\psi) = \|\psi\|^2 \, \mu(d\psi)$; then the adjusted-and-projected measure is $\proj_* \bigl( A(\mu) \bigr) = A(\mu) \circ \proj^{-1}$, thus defining a mapping $\proj_* \circ A$ from the measures on $\Hilbert$ with $\int \mu(d\psi) \, \|\psi\|^2 = 1$ to the probability measures on $\sphere(\Hilbert)$. We then have that $\GAP{\rho} = \proj_* \bigl( A(\G{\rho}) \bigr)$. We remark that $\Psi^{GAP}$, too, lies in $\support(\rho)$ almost surely, and that $\proj(\Psi^G)$ does \emph{not} have distribution $\GAP{\rho}$---nor covariance $\rho$ (see Sect.~\ref{sec:densitymatrix}). We can be more explicit in the case that $\rho$ has finite rank $k = \dim \, \support(\rho)$, e.g.\ for finite-dimensional $\Hilbert$: then there exists a Lebesgue volume measure $\lambda$ on $\support(\rho) = \CCC^k$, and we can specify the densities of $\G{\rho}$ and $\GA{\rho}$, \begin{subequations}\label{muNdensity} \begin{align} \frac{d\G{\rho}}{d\lambda}(\psi) &= \frac{1}{\pi^k \, \det \rho_+} \exp(-\langle \psi |\rho^{-1}_+| \psi \rangle),\\ \frac{d\GA{\rho}}{d\lambda}(\psi) &= \frac{ \|\psi\|^2}{\pi^k \, \det \rho_+} \exp(-\langle \psi |\rho^{-1}_+| \psi \rangle), \end{align} \end{subequations} with $\rho_+$ the restriction of $\rho$ to $\support(\rho)$. Similarly, we can express $\GAP{\rho}$ relative to the $(2k-1)$--dimensional surface measure $u$ on $\sphere(\support(\rho))$, \begin{subequations}\label{mudensity} \begin{align} \frac{d\GAP{\rho}}{du}(\psi) &= \frac{1}{\pi^k \, \det \rho_+} \int\limits_0^\infty dr \, r^{2k-1} \, r^2 \exp(-r^2 \langle \psi | \rho^{-1}_+ | \psi \rangle) = \\ &= \frac{k!}{2\pi^{k} \, \det \rho_+} \, \langle \psi | \rho^{-1}_+ | \psi \rangle^{-k-1} \,.\label{mupowerlaw} \end{align} \end{subequations} We note that \begin{equation}\label{microGAP} \GAP{\rho_{E,\delta}} = u_{E,\delta}\,, \end{equation} where $\rho_{E\delta}$ is the microcanonical density matrix given in \eqref{microcanrho} and $u_{E\delta}$ is the microcanonical measure. \section{Properties of $\GAP{\rho}$} \label{sec:prop} In this section we prove the following properties of $\GAP{\rho}$: \bigskip %\begin{thm}\label{thma} \noindent{\bf Property 1}\textit{ The density matrix associated with $\GAP{\rho}$ in the sense of \eqref{rhomupsi} is $\rho$, i.e., $\rho_{\GAP{\rho}} = \rho$. %\end{thm} } \bigskip %\begin{thm}\label{thmb} \noindent{\bf Property 2}\textit{ The association $\rho \mapsto \GAP{\rho}$ is \emph{covariant}: For any unitary operator $U$ on $\Hilbert$, \begin{equation}\label{covariance} U_* \GAP{\rho} = \GAP{U\rho U^*} \end{equation} where $U^* = U^{-1}$ is the adjoint of $U$ and $U_*$ is the action of $U$ on measures, $U_* \mu = \mu \circ U^{-1}$. In particular, $\GAP{\rho}$ is stationary under any unitary evolution that preserves $\rho$. %\end{thm} } %Another way of expressing this fact is to say that the following %diagram commutes: %\begin{equation} %\begin{array}{rcl} % \rho & \xrightarrow{\quad U \quad} & U\rho U^* \\ % {}^{GAP} \bigg\downarrow && \bigg\downarrow {}^{GAP} \\ % \GAP{\rho} & \xrightarrow{\quad U_* \quad} & \GAP{U\rho U^*} %\end{array} %\end{equation} \bigskip %\begin{thm}\label{thmc} \noindent{\bf Property 3}\textit{ If $\Psi \in \Hilbert_1 \otimes \Hilbert_2$ has distribution $\GAP{\rho_1 \otimes \rho_2}$ then, for any basis $\{|q_2\rangle\}$ of $\Hilbert_2$, the conditional wave function $\Psi_1$ has distribution $\GAP{\rho_1}$. (``GAP of a product density matrix has GAP marginal.'') %\end{thm} } \medskip We will refer to the property expressed in Property~3 by saying that the family of GAP measures is \emph{hereditary}. We note that when $\Psi \in \Hilbert_1 \otimes \Hilbert_2$ has distribution $\GAP{\rho}$ and $\rho$ is not a tensor product, the distribution of $\Psi_1$ need not be $\GAP{\rho_1^\red}$ (as we will show after the proof of Property~3). Before establishing these properties let us formulate what they say about our candidate $\GAP{\rho_\beta}$ for the canonical distribution. As a consequence of Property~1, the density matrix arising from $\mu = \GAP{\rho_\beta}$ in the sense of \eqref{rhomupsi} is the density matrix $\rho_{\beta}$. As a consequence of Property~2, $\GAP{\rho_\beta}$ is stationary, i.e., invariant under the unitary time evolution generated by $H$. As a consequence of Property~3, if $\Psi \in \Hilbert = \Hilbert_1 \otimes \Hilbert_2$ has distribution $\GAP{\rho_{\Hilbert, H, \beta}}$ and systems 1 and 2 are decoupled, $H = H_1 \otimes I_2 + I_1 \otimes H_2$, where $I_i$ is the identity on $\Hilbert_i$, then the conditional wave function $\Psi_1$ of system $1$ has a distribution (in $\Hilbert_1$) of the same kind with the same inverse temperature $\beta$, namely $\GAP{\rho_{\Hilbert_1, H_1, \beta}}$. This fits well with our claim that $\GAP{\rho_\beta}$ is the thermal equilibrium distribution since one would expect that if a system is in thermal equilibrium at inverse temperature $\beta$ then so are its subsystems. The following lemma, proven in Section~\ref{gmgm}, is convenient for showing that a random wave function is GAP-distributed: \begin{lemma}\label{GaussGAP} Let $\Omega$ be a measurable space, $\mu$ a probability measure on $\Omega$, and $\Psi: \Omega \to \Hilbert$ a Hilbert-space-valued function. If $\Psi(\omega)$ is $\G{\rho}$-distributed with respect to $\mu(d\omega)$, then $\Psi(\omega)/\|\Psi(\omega)\|$ is $\GAP{\rho}$-distributed with respect to $\|\Psi(\omega)\|^2 \mu(d\omega)$. \end{lemma} \subsection{The Density Matrix} \label{sec:densitymatrix} In this subsection we establish Property~1. We then add a remark on the covariance matrix. \begin{proof}{ of Property~1} {}From \eqref{rhomupsi} we find that \begin{align}\nonumber \rho_{\GAP{\rho}} &= \int\limits_{\sphere(\Hilbert)} \!\! \GAP{\rho} (d\psi) \, |\psi\rangle \langle\psi|= \EEE \biggl( |\Psi^{GAP}\rangle \langle\Psi^{GAP}| \biggr) = \\\nonumber &\stackrel{\eqref{mudef}}{=} \EEE \biggl( \|\Psi^{GA}\|^{-2} \, |\Psi^{GA}\rangle \langle\Psi^{GA}| \biggr) = \int\limits_\Hilbert \GA{\rho} (d\psi) \, \|\psi\|^{-2} \, |\psi \rangle \langle \psi | = \\\nonumber &\stackrel{\eqref{muNdef}}{=} \int\limits_\Hilbert \G{\rho} (d\psi) \, |\psi\rangle \langle\psi| = \rho \end{align} because $\int_\Hilbert \G{\rho} (d\psi) \, |\psi\rangle \langle \psi|$ is the covariance matrix of $\G{\rho}$, which is $\rho$. (A number above an equal sign refers to the equation used to obtain the equality.) \end{proof} \noindent {\bf Remark on the covariance matrix.} The equation $\rho_{\GAP{\rho}} = \rho$ can be understood as expressing that $\GAP{\rho}$ and $\G{\rho}$ have the same covariance. For a probability measure $\mu$ on $\Hilbert$ with mean 0 that need not be concentrated on $\sphere(\Hilbert)$, the covariance matrix $C_\mu$ is given by \begin{equation}\label{Cdef} C_\mu = \int\limits_\Hilbert \mu(d\phi) \, |\phi \rangle \langle \phi|. \end{equation} Suppose we want to obtain from $\mu$ a probability measure on $\sphere(\Hilbert)$ having the same covariance. The projection $\proj_*\mu$ of $\mu$ to $\sphere(\Hilbert)$, defined by $\proj_*\mu(B) = \mu (\RRR^+ B)$ for $B \subseteq \sphere(\Hilbert)$, is not what we want, as it has covariance \[ C_{\proj_* \mu} = \int\limits_{\sphere(\Hilbert)} \!\! \proj_*\mu(d\psi) \, |\psi \rangle \langle \psi| = \int\limits_\Hilbert \mu(d\phi) \, \|\phi\|^{-2} \, |\phi \rangle \langle \phi| \neq C_\mu. \] However, $\proj_* \bigl( A(\mu) \bigr)$ does the job: it has the same covariance. As a consequence, a naturally distinguished measure on $\sphere(\Hilbert)$ with given covariance is the Gaussian adjusted projected measure, the GAP measure, with the given covariance. \subsection{$\GAP{\rho}$ is Covariant} We establish Property~2 and then discuss in more general terms under which conditions a measure on $\sphere(\Hilbert)$ is stationary. \begin{proof}{ of Property~2} Under a unitary transformation $U$, a Gaussian measure with covariance matrix $C$ transforms into one with covariance matrix $UCU^*$. Since $\|U\psi\|^2 = \|\psi\|^2$, $\GA{C}$ transforms into $\GA{UCU^*}$; that is, $U\Psi^{GA}_C$ and $\Psi^{GA}_{UCU^*}$ are equal in distribution, and since $\|U\Psi^{GA}_C\| = \|\Psi^{GA}_C\|$, we have that $U\Psi_C^{GAP}$ and $\Psi_{UCU^*}^{GAP}$ are equal in distribution. In other words, $\GAP{C}$ transforms into $\GAP{UCU^*}$, which is what we claimed in \eqref{covariance}. \end{proof} \subsubsection{Stationarity} \label{sec:stationarity} In this subsection we discuss a criterion for stationarity under the evolution generated by $H= \sum_n E_n \, |n \rangle \langle n|$. Consider the following property of a sequence of complex random variables $Z_n$: \begin{equation}\label{independ} \begin{split} &\text{The phases $Z_n/|Z_n|$, when they exist, are independent of the moduli } |Z_n|\\ &\text{and of each other, and are uniformly distributed on }S^1 =\{ e^{i \theta}: \theta \in \RRR \}. \end{split} \end{equation} (The phase $Z_n/|Z_n|$ exists when $Z_n \neq 0$.) Condition \eqref{independ} implies that the distribution of the random vector $\Psi = \sum_n Z_n |n\rangle$ is stationary, since $Z_n(t) = \exp(-iE_n t/\hbar) Z_n(0)$. Note also that \eqref{independ} implies that the distribution has mean 0. We show that the $Z_n = \sp{n}{\Psi^{GAP}}$ have property \eqref{independ}. To begin with, the $Z_n = \sp{n}{\Psi^G}$ obviously have this property since they are independent Gaussian variables. Since the density of $\GA{\rho}$ relative to $\G{\rho}$ is a function of the moduli alone, also the $Z_n = \sp{n}{\Psi^{GA}}$ satisfy \eqref{independ}. Finally, since the $|\sp{n}{\Psi^{GAP}}|$ are functions of the $|\sp{n}{\Psi^{GA}}|$ while the phases of the $\sp{n}{\Psi^{GAP}}$ equal the phases of the $\sp{n}{\Psi^{GA}}$, also the $Z_n = \sp{n}{\Psi^{GAP}}$ satisfy \eqref{independ}. We would like to add that \eqref{independ} is not merely a sufficient, but also almost a necessary condition (and \emph{morally} a necessary condition) for stationarity. Since for any $\Psi$, the moduli $|Z_n| = |\sp{n}{\Psi}|$ are constants of the motion, the evolution of $\Psi$ takes place in the (possibly infinite-dimensional) torus \begin{equation}\label{torus} \Big\{ \sum\limits_{n} |Z_n| e^{i \theta_n} \, |n\rangle : 0 \le \theta_n < 2\pi \Big\} \cong \prod\limits_{n: Z_n \neq 0} S^1, \end{equation} contained in $\sphere(\Hilbert)$. Independent uniform phases correspond to the uniform measure $\lambda$ on $\prod_n S^1$. $\lambda$ is the only stationary measure if the motion on $\prod_n S^1$ is uniquely ergodic, and this is the case whenever the spectrum $\{ E_n\}$ of $H$ is linearly independent over the rationals $\QQQ$, i.e., when every finite linear combination $\sum_n r_n E_n$ of eigenvalues with rational coefficients $r_n$, not all of which vanish, is nonzero, see \cite{ergodic,vN29}. This is true of generic Hamiltonians, so that $\lambda$ is generically the unique stationary distribution on the torus. But even when the spectrum of $H$ is linearly dependent, e.g.\ when there are degenerate eigenvalues, and thus further stationary measures on the torus exist, these further measures should not be relevant to thermal equilibrium measures, because of their instability against perturbations of $H$ \cite{HKTP74,LAG75}. The stationary measure $\lambda$ on $\prod_n S^1$ corresponds, for given moduli $|Z_n|$ or, equivalently, by setting $|Z_n| = p(E_n)^{1/2}$ for a given probability measure $p$ on the spectrum of $H$, to a stationary measure $\lambda_p$ on $\sphere(\Hilbert)$ that is concentrated on the embedded torus \eqref{torus}. The measures $\lambda_p$ are (for generic $H$) the extremal stationary measures, i.e., the extremal elements of the convex set of stationary measures, of which all other stationary measures are mixtures. \subsection{GAP Measures and Gaussian Measures}\label{gmgm} Lemma~\ref{GaussGAP} is more or less immediate from the definition of $\GAP{\rho}$. A more detailed proof looks like this: \begin{proof}{ of Lemma~\ref{GaussGAP}} By assumption the distribution $\mu \circ \Psi^{-1}$ of $\Psi$ with respect to $\mu$ is $\G{\rho}$. Thus for the distribution of $\Psi$ with respect to $\mu' (d\omega) = \|\Psi(\omega)\|^2 \mu (d\omega)$, we have $\mu' \circ \Psi^{-1} (d\psi) = \|\psi\|^2 \, \mu \circ \Psi^{-1} (d\psi) = \|\psi\|^2 \, \G{\rho} (d\psi) = \GA{\rho} (d\psi)$. Thus, $\proj(\Psi(\omega))$ has distribution $\proj_* \GA{\rho} = \GAP{\rho}$. \end{proof} \subsection{Generalized Bases} We have already remarked in the introduction that the orthonormal basis $\{|q_2\rangle\}$ of $\Hilbert_2$, used in the definition of the conditional wave function, could be a \emph{generalized} basis, such as a ``continuous'' basis, for which it is appropriate to write \[ I_2 = \int dq_2 \, |q_2 \rangle \langle q_2| \] instead of the ``discrete'' notation \[ I_2 = \sum_{q_2} |q_2 \rangle \langle q_2| \] we used in \eqref{Psi1def}--\eqref{rhoreddef}. We wish to elucidate this further. A generalized orthonormal basis $\{|q_2 \rangle: q_2 \in \conf_2\}$ indexed by the set $\conf_2$ is mathematically defined by a unitary isomorphism $\Hilbert_2 \to L^2(\conf_2,dq_2)$, where $dq_2$ denotes a measure on $\conf_2$. We can think of $\conf_2$ as the configuration space of system 2; as a typical example, system 2 may consist of $N_2$ particles in a box $\Lambda \subset \RRR^3$, so that its configuration space is $\conf_2 = \Lambda^{N_2}$ with $dq_2$ the Lebesgue measure (which can be regarded as obtained by combining $N_2$ copies of the volume measure on $\RRR^3$).\footnote{In fact, in the original definition of the conditional wave function in \cite{DGZ}, $q_2$ was supposed to be the \emph{configuration}, corresponding to the positions of the particles belonging to system 2. For our purposes here, however, the physical meaning of the $q_{2}$ is irrelevant, so that any generalized orthonormal basis of $\Hilbert_{2}$ can be used.} The formal ket $|q_2 \rangle$ then means the delta function centered at $q_2$; it is to be treated as (though strictly speaking it is not) an element of $\Hilbert_2$. The definition of the conditional wave function $\Psi_1$ then reads as follows: The vector $\psi \in \Hilbert_1 \otimes \Hilbert_2$ can be regarded, using the isomorphism $\Hilbert_2 \to L^2(\conf_2,dq_2)$, as a function $\psi : \conf_2 \to \Hilbert_1$. Eq.~\eqref{Psi1def} is to be understood as meaning \begin{equation}\label{Psi1defcont} \Psi_1 = \mathcal{N} \: \psi(Q_2) \end{equation} where \[ \mathcal{N}= \mathcal{N} (\psi,Q_2) = \bigl\|\psi(Q_2) \bigr\|^{-1} \] is the normalizing factor and $Q_2$ is a random point in $\conf_2$, chosen with the quantum distribution \begin{equation}\label{margcont} \PPP(Q_2 \in dq_2) = \bigl\| \psi(q_2) \bigr\|^2 dq_2 \,, \end{equation} which is how \eqref{marg} is to be understood in this setting. As $\psi$ is defined only up to changes on a null set in $\conf_2$, $\Psi_1$ may not be defined for a particular $Q_2$. Its distribution in $\Hilbert_1$, however, is defined unambiguously by \eqref{Psi1defcont}. In the most familiar setting with $\Hilbert_1 = L^2(\conf_1,dq_1)$, we have that $(\psi(Q_2))(q_1) = \psi(q_1,Q_2)$. In the following, we will allow generalized bases and use continuous instead of discrete notation, and set $\sp{Q_2}{\psi} = \psi(Q_2)$. \subsection{Distribution of the Wave Function of a Subsystem} \label{sec:sub} \begin{proof}{ of Property~3} The proof is divided into four steps. \textit{Step 1.} We can assume that $\Psi = \proj (\Psi^{GA})$ where $\Psi^{GA}$ is a $\GA{\rho}$-distributed random vector in $\Hilbert = \Hilbert_1 \otimes \Hilbert_2$. We then have that $\Psi_1 = \proj_1 \bigl( \sp{Q_2}{\Psi} \bigr) = \proj_1 \bigl( \sp{Q_2}{ \Psi^{GA}} \bigr)$ where $\proj_1$ is the normalization in $\Hilbert_1$, and where the distribution of $Q_2$, given $\Psi^{GA}$, is \[ \PPP(Q_2 \in dq_2|\Psi^{GA}) = \frac{\| \sp{q_2}{\Psi^{GA}} \|^2}{\| \Psi^{GA} \|^2} dq_2\,. \] $\Psi^{GA}$ and $Q_2$ have a joint distribution given by the following measure $\nu$ on $\Hilbert \times \conf_2$: \begin{equation} \nu(d\psi \times dq_2) = \| \sp{q_2}{\psi} \|^2 \, \G{\rho}(d\psi) \, dq_2 \,. \end{equation} Thus, what needs to be shown is that with respect to $\nu$, $\proj_1 (\sp{q_2}{\psi})$ is $\GAP{\rho_1}$-distributed. \textit{Step 2.} \emph{If $\Psi \in \Hilbert_1 \otimes \Hilbert_2$ is $\G{\rho_1 \otimes \rho_2}$-distributed and $q_2 \in \conf_2$ is fixed, then the random vector $f(q_2) \, \sp{q_2}{\Psi} \in \Hilbert_1$ with $f(q_2) = \langle q_2 | \rho_2 | q_2 \rangle^{-1/2}$ is $\G{\rho_1}$-distributed.} This follows, more or less, from the fact that a subset of a set of jointly Gaussian random variables is also jointly Gaussian, together with the observation that the covariance of $\sp{q_2}{\Psi}$ is \[ \int_\Hilbert \G{\rho_1 \otimes \rho_1} (d\psi) \, \sp{q_2}{\psi}\sp{\psi}{q_2} = \sp{q_2} {\rho_1 \otimes \rho_2| q_2} = \rho_1 \, \sp{q_2}{\rho_2|q_2}\,. \] More explicitly, pick an orthonormal basis $\{|n_i \rangle \}$ of $\Hilbert_i$ consisting of eigenvectors of $\rho_i$ with eigenvalues $p^{(i)}_{n_i}$, and note that the vectors $| n_1,n_2 \rangle:= |n_1 \rangle \otimes |n_2 \rangle$ form an orthonormal basis of $\Hilbert = \Hilbert_1 \otimes \Hilbert_2$ consisting of eigenvectors of $\rho_1 \otimes \rho_2$ with eigenvalues $p_{n_1,n_2} = p^{(1)}_{n_1} p^{(2)}_{n_2}$. Since the random variables $Z_{n_1,n_2} := \sp{ n_1,n_2 }{ \Psi}$ are independent Gaussian random variables with mean zero and variances $\EEE|Z_{n_1},{n_2}|^2 = p_{n_1,n_2}$, so are their linear combinations \[ Z_{(1)n_1} := \sp{ n_1 }{ f(q_2) \, \Psi(q_2) } = f(q_2) \sum_{n_2} \sp{ q_2 }{ n_2} \, Z_{n_1,n_2} \] with variances (because variances add when adding independent Gaussian random variables) \[ \EEE |Z_{(1)n_1}|^2 = f^2(q_2) \sum_{n_2} \bigl| \sp{q_2}{ n_2} \bigr|^2 \EEE |Z_{n_1,n_2}|^2 = p^{(1)}_{n_1} \frac{ \sum_{n_2} |\sp{q_2}{n_2}|^2 \, p^{(2)}_{n_2}} {\sp{q_2}{ \rho_2 | q_2}} = p^{(1)}_{n_1}. \] Thus $f(q_2) \, \sp{q_2}{\Psi}$ is $\G{\rho_1}$-distributed, which completes step 2. \textit{Step 3.} \emph{If $\Psi \in \Hilbert_1 \otimes \Hilbert_2$ is $\G{\rho_1 \otimes \rho_2}$-distributed and $Q_2 \in \conf_2$ is random with any distribution, then the random vector $f(Q_2) \, \sp{Q_2}{\Psi}$ is $\G{\rho_1}$-distributed.} This is a trivial consequence of step 2. \textit{Step 4.} Apply Lemma~\ref{GaussGAP} as follows. Let $\Omega = \Hilbert \times \conf_2$, $\Psi(\omega) = \Psi(\psi, q_2) = f(q_2) \, \sp{q_2}{\psi}$, and $\mu(d\psi \times dq_2) = \G{\rho}(d\psi) \, \sp{q_2}{\rho_2|q_2} \, dq_2$ (which means that $q_2$ and $\psi$ are independent). By step 3, the hypothesis of Lemma~\ref{GaussGAP} (for $\rho = \rho_1$) is satisfied, and thus $\proj_1 (\Psi) = \proj_1 (\sp{q_2}{\psi})$ is $\GAP{\rho_1}$-distributed with respect to \[ \|\Psi(\omega)\|^2 \mu(d\omega) = {f^2(q_2)} \|\sp{q_2}{\psi}\|^2 \, \G{\rho}(d\psi) \, \sp{q_2}{\rho_2|q_2} \, dq_2 = \nu(d\omega)\,, \] where we have used that $f^2(q_2)=\langle q_2|\rho_2|q_2 \rangle^{-1}$. But this is, according to step 1, what we needed to show. \end{proof} %Another way of expressing Property~3 is to say that the %following diagram commutes for every density matrix $\rho_2$ in %$\Hilbert_2$: %\begin{equation} %\begin{array}{rcl} % \rho_1 & \xrightarrow{\quad \otimes\rho_2 \quad} & \rho_1\otimes\rho_2 \\ % {}^{GAP} \bigg\downarrow && \bigg\downarrow {}^{GAP} \\ % \GAP{\rho_1} & \xleftarrow{\quad (\mu_1)_* \quad} & % \GAP{\rho_1 \otimes \rho_2} %\end{array} %\end{equation} %Here, $\mu_1$ is the mapping that associates with every $\psi \in %\Hilbert_1 \otimes \Hilbert_2$ the distribution \eqref{mu1} of the %conditional wave function, and $(\mu_1)_*$ the corresponding mixture %for random $\psi$. To verify the statement after Property~3, consider the density matrix $\rho = |\Phi \rangle \langle \Phi|$ for a pure state $\Phi$ of the form $\Phi = \sum_n \sqrt{p_n} \, \psi_n \otimes \phi_n$, where $\{\psi_n\}$ and $\{\phi_n\}$ are respectively orthonormal bases for $\Hilbert_1$ and $\Hilbert_2$ and the $p_n$ are nonnegative with $\sum_n p_n =1$. Then a $\GAP{\rho}$-distributed random vector $\Psi$ coincides with $\Phi$ up to a random phase, and so $\rho_1^\red = \sum_n p_n \, |\psi_n \rangle \langle \psi_n|$. Choosing for $\{|q_2 \rangle\}$ the basis $\{\phi_n\}$, the distribution of $\Psi_1$ is not $\GAP{\rho_1^\red}$ but rather is concentrated on the eigenvectors of $\rho_1^\red$. When the $p_n$ are pairwise-distinct this measure is the measure $\EIG{\rho_1^\red}$ we define in Section~\ref{sec:nu}. \section{Microcanonical Distribution for a Large System Implies the Distribution $\GAP{\rho_\beta}$ for a Subsystem} %{GAP Measure Arising {}From a Heat Bath} \label{sec:hb1} In this section we use Property~3, i.e., the fact that GAP measures are hereditary, to show that $\GAP{\rho_\beta}$ is the distribution of the conditional wave function of a system coupled to a heat bath when the wave function of the composite is distributed microcanonically, i.e., according to $u_{E,\delta}$. Consider a system with Hilbert space $\Hilbert_1$ coupled to a heat bath with Hilbert space $\Hilbert_2$. Suppose the composite system has a random wave function $\Psi \in \Hilbert = \Hilbert_1 \otimes \Hilbert_2$ whose distribution is microcanonical, $u_{E,\delta}$. Assume further that the coupling is negligibly small, so that we can write for the Hamiltonian \begin{equation}\label{nocoupling} H = H_1 \otimes I_2 + I_1 \otimes H_2 \,, \end{equation} and that the heat bath is large (so that the energy levels of $H_2$ are very close). By equivalence of ensembles \cite{martinlof}, we should have that $\rho_{E,\delta} \approx \rho_\beta$ for suitable $\beta = \beta(E)$. Then, since $\GAP{\rho}$ depends continuously on $\rho$, we have that $u_{E,\delta} = \GAP{\rho_{E,\delta}} \approx \GAP{\rho_\beta}$. Thus we should have that the distribution of the conditional wave function $\Psi_1$ of the system is approximately the same as would be obtained when $\Psi$ is $\GAP{\rho_\beta}$-distributed. But since, by \eqref{nocoupling}, the canonical density matrix is then of the form \begin{equation} \rho_\beta = \rho_{\Hilbert,H,\beta} = \rho_{\Hilbert_1, H_1, \beta} \otimes \rho_{\Hilbert_2, H_2, \beta}\,, \end{equation} we have by Property~3 that $\Psi_1$ is approximately $\GAP{\rho_{\Hilbert_1, H_1, \beta}}$-distributed, which is what we wanted to show. \section{Typicality of GAP Measures} \label{sec:Typicality} The previous section concerns the distribution of the conditional wave function $\Psi_1$ arising from the microcanonical distribution of the wave function of the composite. It concerns, in other words, a \emph{random} wave function of the composite. The result there is the analogue, on the level of measures on Hilbert space, of the well known result that if a microcanonical density matrix \eqref{microcanrho} is assumed for the composite, the reduced density matrix $\rho_1^\red$ of the system, defined as the partial trace $\tr_2 \, \rho_{E,\delta}$, is canonical if the heat bath is large \cite{LL59}. As indicated in the introduction, a stronger statement about the canonical density matrix is in fact true, namely that for a \emph{fixed} (nonrandom) wave function $\psi$ of the composite which is typical with respect to $u_{E,\delta}$, $\rho_1^\red \approx \rho_{\Hilbert_1, H_1, \beta}$ when the heat bath is large (see \cite{thermo4, schr, schrbook}; for a rigorous study of special cases of a similar question, see \cite{Tasaki1}).\footnote{It is a consequence of the results in \cite{Pag93} that when $\dim\Hilbert_2\to\infty$, the reduced density matrix becomes proportional to the identity on $\Hilbert_1$ for typical wave functions relative to the uniform distribution on $\sphere(\Hilbert)$ (corresponding to $u_{E,\delta}$ for $E=0$ and $H=0$).} This stronger statement will be used in Section~\ref{sec:hb2} to show that a similar statement holds for the distribution of $\Psi_1$ as well, namely that it is approximately $\GAP{\rho_{\Hilbert_1, H_1, \beta}}$-distributed for a typical fixed $\psi \in \Hilbert_{E,\delta}$ and basis \onb\ of $\Hilbert_2$. But we must first consider the distribution of $\Psi_1$ for a typical $\psi\in \Hilbert$. \subsection{Typicality of GAP Measures for a Subsystem of a Large System} \label{sec:typicality} In this section we argue that for a typical wave function of a big system the conditional wave function of a small subsystem is approximately GAP-distributed, first giving a precise formulation of this result and then sketching its proof. We give the detailed proof in \cite{thermo6}. \subsubsection{Statement of the Result} Let $\Hilbert = \Hilbert_1 \otimes \Hilbert_2$, where $\Hilbert_1$ and $\Hilbert_2$ have respective dimensions $k$ and $m$, with $k < m < \infty$. For any given density matrix $\rho_1$ on $\Hilbert_1$, consider the set \begin{equation}\label{Rdef} \R (\rho_1) = \bigl\{\psi \in \sphere(\Hilbert): \rho_1^\red (\psi) = \rho_1 \bigr\} \,, \end{equation} where $\rho_1^\red(\psi) = \tr_2 |\psi \rangle \langle \psi|$ is the reduced density matrix for the wave function $\psi$. There is a natural notion of (normalized) uniform measure $u_{\rho_1}$ on $\R(\rho_1)$; we give its precise definition in Section~\ref{sec:typoutline}. We claim that for fixed $k$ and large $m$, the distribution $\mu_1^{\psi}$ of the conditional wave function $\Psi_1$ of system 1, defined by \eqref{Psi1def} and \eqref{marg} for a basis $\{ |q_2\rangle\}$ of $\Hilbert_2$, is close to $\GAP{\rho_1}$ for the overwhelming majority, relative to $u_{\rho_1}$, of vectors $\psi \in \Hilbert$ with reduced density matrix $\rho_1$. More precisely: \bigskip \noindent\textit{For every $\varepsilon > 0$ and every bounded continuous function $f: \sphere(\Hilbert_1) \to \RRR$, \begin{equation}\label{GAPtyp} u_{\rho_1} \Bigl\{\psi \in \R(\rho_1): \bigl| \mu_1^{\psi}(f) - \GAP{\rho_1}(f) \bigr| < \varepsilon \Bigr\} \to 1 \quad \text{as } \, m \to \infty \,, \end{equation} regardless of how the basis $\{ |q_2\rangle\}$ is chosen.} \bigskip \noindent Here we use the notation \begin{equation} \mu(f) := \int\limits_{\sphere(\Hilbert)} \mu(d\psi)\, f(\psi) \,. \end{equation} % where $\mu_1^{\psi}$ is the distribution of the conditional wave % function, i.e., of $\Psi_1 = \proj_1 (\sp{Q_2}{\psi}) \in % \sphere(\Hilbert_1)$ with respect to the quantum distribution of % $Q_2$, $\PPP(Q_2 = q_2) = \|\sp{q_2}{\psi}\|^2$. } \subsubsection{Measure on $\Hilbert$ Versus Density Matrix} It is important to resist the temptation to translate $u_{\rho_1}$ into a density matrix in $\Hilbert$. As mentioned in the introduction, to every probability measure $\mu$ on $\sphere(\Hilbert)$ there corresponds a density matrix $\rho_\mu$ in $\Hilbert$, given by \eqref{rhomupsi}, which contains all the empirically accessible information about an ensemble with distribution $\mu$. It may therefore seem a natural step to consider, instead of the measure $\mu = u_{\rho_1}$, directly its density matrix $\rho_\mu=\frac1m\rho_1\otimes I_2$, where $I_2$ is the identity on $\Hilbert_2$. But since our result concerns properties of most wave functions relative to $\mu$, it cannot be formulated in terms of the density matrix $\rho_\mu$. In particular, the corresponding statement relative to another measure $\mu' \neq \mu$ on $\sphere(\Hilbert)$ with the same density matrix $\rho_{\mu'} = \rho_\mu$ could be false. Noting that $\rho_\mu$ has a basis of eigenstates that are product vectors, we could, for example, take $\mu'$ to be a measure concentrated on these eigenstates. For any such state $\psi$, $\mu_1^{\psi}$ is a delta-measure. \subsubsection{Outline of Proof} \label{sec:typoutline} The result follows, by \eqref{marg}, Lemma~\ref{GaussGAP}, and the continuity of $\proj_* \circ A$, from the corresponding statement about the Gaussian measure $\G{\rho_1}$ on $\Hilbert_1$ with covariance $\rho_1$: \bigskip \noindent\textit{For every $\varepsilon > 0$ and every bounded continuous $f: \Hilbert_1 \to \RRR$,} \begin{equation}\label{Gtyp} u_{\rho_1} \Bigl\{\psi \in \R(\rho_1): \bigl| \bar \mu_1^{\psi}(f) - \G{\rho_1}(f) \bigr| < \varepsilon \Bigr\} \to 1 \quad \text{\textit{as} } \, m \to \infty \,, \end{equation} \textit{where $\bar \mu_1^\psi$ is the distribution of $\sqrt{m} \, \sp{Q_2}{\psi} \in \Hilbert_1$ (not normalized) with respect to the uniform distribution of $Q_2 \in \{1, \ldots, m\}$.} \bigskip We sketch the proof of \eqref{Gtyp} and give the definition of $u_{\rho_1}$. According to the Schmidt decomposition, every $\psi \in \Hilbert$ can be written in the form \begin{equation}\label{Schmidt} \psi = \sum_i c_i \, \chi_i \otimes \phi_{i} \,, \end{equation} where $\{\chi_i \}$ is an orthonormal basis of $\Hilbert_{1}$, $\{\phi_i\}$ an orthonormal system in $\Hilbert_2$, and the $c_i$ are coefficients which can be assumed real and nonnegative. {}From \eqref{Schmidt} one reads off the reduced density matrix of system 1, \begin{equation}\label{Schmidtrho} \rho_1^\red = \sum_{i} c_i^2 \, |\chi_{i} \rangle \langle \chi_{i} |\,. \end{equation} As the reduced density matrix is given, $\rho_1^\red = \rho_1$, the orthonormal basis $\{\chi_{i} \}$ and the coefficients $c_i$ are determined (when $\rho_1$ is nondegenerate) as the eigenvectors and the square-roots of the eigenvalues of $\rho_1$, and, $\R(\rho_1)$ is in a natural one-to-one correspondence with the set $ONS(\Hilbert_2,k)$ of all orthonormal systems $\{\phi_{i} \}$ in $\Hilbert_2$ of cardinality $k$. (If some of the eigenvalues of $\rho_1$ vanish, the one-to-one correspondence is with $ONS(\Hilbert_2,k')$ where $k'=\dim\support(\rho_1)$.) The Haar measure on the unitary group of $\Hilbert_2$ defines the uniform distribution on the set of orthonormal bases of $\Hilbert_2$, of which the uniform distribution on $ONS(\Hilbert_2,k)$ is a marginal, and thus defines the uniform distribution $u_{\rho_1}$ on $\R(\rho_1)$. (When $\rho_1$ is degenerate, $u_{\rho_1}$ does not depend upon how the eigenvectors $\chi_{i}$ of $\rho_1$ are chosen.) The key idea for establishing \eqref{Gtyp} from the Schmidt decomposition \eqref{Schmidt} is this: $\bar\mu_1^\psi$ is the average of $m$ delta measures with equal weights, $\bar\mu_1^\psi = m^{-1} \sum_{q_2} \delta_{\psi_1(q_2)}$, located at the points \begin{equation} \psi_1(q_2) = \sum_{i=1}^{k} c_i \, \sqrt{m} \, \sp{q_2}{\phi_{i}} \, \chi_{i} \,. \end{equation} Now regard $\psi$ as random with distribution $u_{\rho_1}$; then the $\psi_1(q_2)$ are $m$ random vectors, and $\bar\mu^\psi_1$ is their empirical distribution. If the $mk$ coefficients $\sp{q_2}{\phi_{i}}$ were \emph{independent Gaussian} (complex) random variables with (mean zero and) variance $m^{-1}$, then the $\psi_1(q_2)$ would be $m$ independent drawings of a $\G{\rho_1}$-distributed random vector; by the weak law of large numbers, their empirical distribution would usually be close to $\G{\rho_1}$; in fact, the probability that $\bigl| \bar\mu_1^{\psi}(f) - \G{\rho_1}(f) \bigr| < \varepsilon$ would converge to 1, as $m \to \infty$. However, when $\{\phi_{i}\}$ is a random orthonormal system with uniform distribution as described above, the expansion coefficients $\sp{q_2}{\phi_{i}}$ in the decomposition of the $\phi_{i}$'s \begin{equation}\label{expand} \phi_i=\sum_{q_2} \sp{q_2}{\phi_{i}}|q_2\rangle \end{equation} will not be independent---since the $\phi_{i}$'s must be orthogonal and since $\|\phi_i\|=1$. Nonetheless, replacing the coefficients $\sp{q_2}{\phi_{i}}$ in \eqref{expand} by independent Gaussian coefficients $a_i(q_2)$ as described above, we obtain a system of vectors \begin{equation} \phi_i'=\sum_{q_2} a_i(q_2)|q_2\rangle \end{equation} that, in the limit $m\to\infty$, form a uniformly distributed orthonormal system: $\|\phi_i'\|\to1$ (by the law of large numbers) and $\sp{\phi_i'}{\phi_j'}\to0$ for $i\neq j$ (since a pair of randomly chosen vectors in a high-dimensional Hilbert space will typically be almost orthogonal). This completes the proof. \subsubsection{Reformulation}\label{sec:ref} While this result {\it suggests} that $\GAP{\rho_\beta}$ is the distribution of the conditional wave function of a system coupled to a heat bath when the wave function of the composite is a typical {\it fixed} microcanonical wave function, belonging to $\Hilbert_{E,\delta}$, it does not quite {\it imply} it. The reason for this is that $\Hilbert_{E,\delta}$ has measure 0 with respect to the uniform distribution on $\Hilbert,$ even when the latter is finite-dimensional. Nonetheless, there is a simple corollary, or reformulation, of the result that will allow us to cope with microcanonical wave functions. We have indicated that for our result the choice of basis \onb\ of $\Hilbert_2$ does not matter. In fact, while $\mu_1^{\psi}$, the distribution of the conditional wave function $\Psi_1$ of system 1, depends upon both $\psi\in\Hilbert$ and the choice of basis \onb\ of $\Hilbert_2$, the distribution of $\mu_1^{\psi}$ itself, when $\psi$ is $u_{\rho_1}$-distributed, does not depend upon the choice of basis. This follows from the fact that for any unitary $U$ on $\Hilbert_2$ \begin{equation}\label{U} \sp{U^{-1}q_2}{\psi}=\sp{q_2}{I_1\otimes U\psi} \end{equation} (and the invariance of the Haar measure of the unitary group of $\Hilbert_2$ under left multiplication). It similarly follows from \eqref{U} that for fixed $\psi\in\Hilbert$, the distribution of $\mu_1^{\psi}$ arising from the uniform distribution $\nu$ of the basis \onb, in the set $ONB(\Hilbert_2)$ of all orthonormal bases of $\Hilbert_2$, is the same as the distribution of $\mu_1^{\psi}$ arising from the uniform distribution $u_{\rho_1}$ of $\psi$ with a fixed basis (and the fact that the Haar measure is invariant under $U\mapsto U^{-1}$). We thus have the following corollary: \bigskip \noindent \textit{Let $\psi\in\Hilbert$ and let $\rho_1=\tr_2 |\psi \rangle \langle \psi|$ be the corresponding reduced density matrix for system 1. Then for a typical basis \onb\ of $\Hilbert_2$, the conditional wave function $\Psi_1$ of system 1 is approximately $\GAP{\rho_1}$-distributed when $m$ is large: For every $\varepsilon > 0$ and every bounded continuous function $f: \sphere(\Hilbert_1) \to \RRR$, \begin{equation}\label{GAPtyp2} \nu \Bigl\{\{ |q_2\rangle\}\in ONB(\Hilbert_2): \bigl| \mu_1^{\psi}(f) - \GAP{\rho_1}(f) \bigr| < \varepsilon \Bigr\} \to 1 \quad \text{as } \, \dim(\Hilbert_2) \to \infty \,. \end{equation} } \subsection{Typicality of $\GAP{\rho_\beta}$ for a Subsystem of a Large System in the Microcanonical Ensemble} \label{sec:hb2} It is an immediate consequence of the result of Section~\ref{sec:ref} that for any fixed microcanonical wave function $\psi$ for a system coupled to a (large) heat bath, the conditional wave function $\Psi_1$ of the system will be approximately GAP-distributed. When this is combined with the ``canonical typicality'' described near the beginning of Section~\ref{sec:Typicality}, we obtain the following result: \bigskip \noindent \emph{Consider a system with finite-dimensional Hilbert space $\Hilbert_1$ coupled to a heat bath with finite-dimensional Hilbert space $\Hilbert_2$. Suppose that the coupling is weak, so that we can write $H = H_1 \otimes I_2+ I_1\otimes H_2$ on $\Hilbert=\Hilbert_1\otimes\Hilbert_2$, and that the heat bath is large, so that the eigenvalues of $H_2$ are close. Then for any wave function $\psi$ that is typical relative to the microcanonical measure $u_{E,\delta}$, the distribution $\mu_1^{\psi}$ of the conditional wave function $\Psi_1$, defined by \eqref{Psi1def} and \eqref{marg} for a typical basis $\{ |q_2\rangle\}$ of the heat bath, is close to $\GAP{\rho_\beta}$ for suitable $\beta=\beta(E)$, where $\rho_\beta=\rho_{\Hilbert_1, H_1, \beta}$. In other words, in the thermodynamic limit, in which the volume $V$ of the heat bath and $dim(\Hilbert_2)$ go to infinity and $E/V=e$ is constant, we have that for all $\varepsilon, \delta >0$, and for all bounded continuous functions $f: \sphere(\Hilbert_1) \to \RRR$, \begin{equation}\label{microcan} u_{E,\delta}\times\nu \Bigl\{(\psi,\{|q_2\rangle\}) \in \sphere(\Hilbert)\times ONB(\Hilbert_2) : \bigl| \mu_1^{\psi}(f) - \GAP{\rho_\beta}(f) \bigr| < \varepsilon \Bigr\} \to 1 \end{equation} where $\beta=\beta(e)$. } We note that if $\{ |q_2\rangle\}$ were an energy eigenbasis rather than a typical basis, the result would be false. \section{Remarks}\label{sec:rem} \subsection{Other Candidates for the Canonical Distribution} We review in this section other distributions that have been, or may be, considered as possible candidates for the distribution of the wave function of a system from a canonical ensemble. \subsubsection{A Distribution on the Eigenvectors}\label{sec:nu} One possibility, which goes back to von Neumann \cite[p.~329]{Neumann}, is to consider $\mu(d\psi)$ as concentrated on the eigenvectors of $\rho$; we denote this distribution $\EIG{\rho}$ after the first letters of ``eigenvector''; it is defined as follows. Suppose first that $\rho$ is nondegenerate. To select an $\EIG{\rho}$-distributed vector, pick a unit eigenvector $|n \rangle$, so that $\rho|n\rangle = p_n |n\rangle$, with probability $p_n$ and randomize its phase. This definition can be extended in a natural way to degenerate $\rho$: \begin{equation}\label{nudef} \EIG{\rho} = \sum_{p \in \spec (\rho)} p \, \dim \Hilbert_p \:\: u_{\sphere(\Hilbert_p)}, \end{equation} where $\Hilbert_p$ denotes the eigenspace of $\rho$ associated with eigenvalue $p$. The measure $\EIG{\rho}$ is concentrated on the set $\bigcup_p \Hilbert_p$ of eigenvectors of $\rho$, which for the canonical $\rho = \rho_{\Hilbert, H, \beta}$ coincides with the set of eigenvectors of $H$; it is a mixture of the microcanonical distributions $u_{\sphere(\Hilbert_p)}$ on the eigenspaces of $H$ in the same way as in classical mechanics the canonical distribution on phase space is a mixture of the microcanonical distributions. Note that $\EIG{\rho_{E,\delta}}=u_{E,\delta}$, and that in particular $\EIG{\rho_{E,\delta}}$ is not, when $H$ is nondegenerate, the uniform distribution $\mu_{E,\delta}$ on the energy eigenstates with energies in $[E,E+\delta]$, against which we have argued in the introduction. The distribution $\EIG{\rho}$ has the same properties as those of $\GAP{\rho}$ described in Properties~1--3, except when $\rho$ is degenerate: \textit{ The measures $\EIG{\rho}$ are such that (a) they have the right density matrix: $\rho_{\EIG{\rho}} = \rho$; (b) they are covariant: $U_* \EIG{\rho} = \EIG{U\rho U^*}$; (c) they are hereditary at nondegenerate $\rho$: when $\Hilbert = \Hilbert_1 \otimes \Hilbert_2$ and $\rho$ is nondegenerate and uncorrelated, $\rho = \rho_1 \otimes \rho_2$, then $\EIG{\rho}$ has marginal (i.e., distribution of the conditional wave function) $\EIG{\rho_1}$. } \begin{proof}{} (a) and (b) are obvious. For (c) let, for $i=1,2$, $|n_i\rangle$ be a basis consisting of eigenvectors of $\rho_i$ with eigenvalues $p^{(i)}_{n_i}$. Note that the tensor products $|n_1\rangle \otimes |n_2 \rangle$ are eigenvectors of $\rho$ with eigenvalues $p^{(1)}_{n_1} p^{(2)}_{n_2}$, and by nondegeneracy all eigenvectors of $\rho$ are of this form up to a phase factor. Since an $\EIG{\rho}$-distributed random vector $\Psi$ is almost surely an eigenvector of $\rho$, we have $\Psi = e^{i\Theta} |N_1\rangle |N_2 \rangle$ with random $N_1$, $N_2$, and $\Theta$. The conditional wave function $\Psi_1$ is, up to the phase, the eigenvector $|N_1 \rangle$ of $\rho_1$ occurring as the first factor in $\Psi$. The probability of obtaining $N_1 = n_1$ is $\sum_{n_2} p^{(1)}_{n_1} p^{(2)}_{n_2} = p^{(1)}_{n_1}$.\footnote{The relevant condition for (c) follows from nondegeneracy but is weaker: it is that the eigenvalues of $\rho_1$ and $\rho_2$ are multiplicatively independent, in the sense that $p^{(1)}_{n_1} p^{(2)}_{n_2} = p^{(1)}_{m_1} p^{(2)}_{m_2}$ can occur only trivially, i.e., when $p^{(1)}_{n_1}=p^{(1)}_{m_1}$ and $p^{(2)}_{n_2}=p^{(2)}_{m_2}$. In particular, the nondegeneracy of $\rho_1$ and $\rho_2$ is irrelevant.} \end{proof} In contrast, for a \emph{degenerate} $\rho = \rho_1 \otimes \rho_2$ the conditional wave function need not be $\EIG{\rho_1}$-distributed, as the following example shows. Suppose $\rho_1$ and $\rho_2$ are nondegenerate but $p^{(1)}_{n_1} p^{(2)}_{n_2} = p^{(1)}_{m_1} p^{(2)}_{m_2}$ for some $n_1\neq m_1$; then an $\EIG{\rho}$-distributed $\Psi$, whenever it happens to be an eigenvector associated with eigenvalue $p^{(1)}_{n_1} p^{(2)}_{n_2}$, is of the form $c|n_1\rangle |n_2\rangle + c' |m_1 \rangle |m_2 \rangle$, almost surely with nonvanishing coefficients $c$ and $c'$; as a consequence, the conditional wave function is a multiple of $c|n_1\rangle \langle Q_2|n_2\rangle + c' |m_1 \rangle \langle Q_2|m_2 \rangle$, which is, for typical $Q_2$ and unless $|n_2\rangle$ and $|m_2 \rangle$ have disjoint supports, a nontrivial superposition of eigenvectors $|n_1\rangle$, $|m_1\rangle$ with different eigenvalues---and thus cannot arise from the $\EIG{\rho_1}$ distribution.\footnote{A property weaker than (c) does hold for $\EIG{\rho}$ also in the case of the degeneracy of $\rho = \rho_1 \otimes \rho_2$: if the orthonormal basis $\{ |q_2\rangle\}$ used in the definition of conditional wave function consists of eigenvectors of $\rho_2$, then the distribution of the conditional wave function is $\EIG{\rho_1}$.} Note also that $\EIG{\rho}$ is discontinuous as a function of $\rho$ at every degenerate $\rho$; in other words, $\EIG{\rho_{\Hilbert, H, \beta}}$ is, like $\mu_{E,\delta}$, unstable against small perturbations of the Hamiltonian. (And, as with $\mu_{E,\delta}$, this fact, quite independently of the considerations on behalf of GAP-measures in Sections~\ref{sec:hb1} and \ref{sec:Typicality}, suggests against using $\EIG{\rho_\beta}$ as a thermal equilibrium distribution.) Moreover, $\EIG{\rho}$ is highly concentrated, generically on a one-dimensional subset of $\sphere(\Hilbert)$, and in the case of a finite-dimensional Hilbert space $\Hilbert$ fails to be absolutely continuous relative to the uniform distribution $u_{\sphere (\Hilbert)}$ on the unit sphere. For further discussion of families $\mu(\rho)$ of measures satisfying the analogues of Properties~1--3, see Section~\ref{sec:uniqueness}. \subsubsection{An Extremal Distribution} Here is another distribution on $\Hilbert$ associated with the density matrix $\rho$. Let the random vector $\Psi$ be \begin{equation}\label{anotherdef} \Psi = \sum_{p \in \spec(\rho)} \sqrt{p} \, \Psi_p, \end{equation} the $\Psi_p$ being independent random vectors with distributions $u_{\sphere(\Hilbert_p)}$. In case all eigenvalues are nondegenerate, this means the coefficients $Z_n$ of $\Psi$, $\Psi = \sum_n Z_n |n\rangle$, have independent uniform phases but fixed moduli $|Z_n| = \sqrt{p_n}$---in sharp contrast with the moduli when $\Psi$ is $\GAP{\rho}$-distributed. And in contrast to the measure $\EIG{\rho}$ considered in the previous subsection, the weights $p_n$ in the density matrix now come from the fixed size of the coefficients of $\Psi$ when it is decomposed into the eigenvectors of $\rho$, rather than from the probability with which these eigenvectors are chosen. This measure, too, is stationary under any unitary evolution that leaves $\rho$ invariant. In particular, it is stationary in the thermal case $\rho = \rho_{\Hilbert, H, \beta}$, and for generic $H$ it is an extremal stationary measure as characterized in Section~\ref{sec:stationarity}; in fact it is, in the notation of the last paragraph of Section~\ref{sec:stationarity}, $\lambda_p$ with $p(E_n) = (1/Z) \exp(-\beta E_n)$. This measure, too, is highly concentrated: For a Hilbert space $\Hilbert$ of finite dimension $k$, it is supported by a submanifold of real dimension $2k-m$ where $m$ is the number of distinct eigenvalues of $H$, hence generically it is supported by a submanifold of just half the dimension of $\Hilbert$. \subsubsection{The Distribution of Guerra and Loffredo} In \cite{GL81}, Guerra and Loffredo consider the canonical density matrix $\rho_\beta$ for the one-dimensional harmonic oscillator and want to associate with it a diffusion process on the real line, using stochastic mechanics \cite{Nel85,Gol87}. Since stochastic mechanics associates a process with every wave function, they achieve this by finding a measure $\mu_\beta$ on $\sphere(L^2(\RRR))$ whose density matrix is $\rho_\beta$. They propose the following measure $\mu_\beta$, supported by coherent states. With every point $(q,p)$ in the classical phase space $\RRR^2$ of the harmonic oscillator there is associated a coherent state \begin{equation}\label{coherent} \psi_{q,p} (x) = (2\pi\sigma^2)^{-1/4} \, \exp \biggl( - \frac{(x-q)^2} {4\sigma^2} + \frac{i}{\hbar} xp - \frac{i}{2\hbar} pq \biggr) \end{equation} with $\sigma^2 = \hbar/2m\omega$, thus defining a mapping $C : \RRR^2 \to \sphere(L^2(\RRR))$, $C(q,p) = \psi_{q,p}$. Let $H(q,p) = p^2/2m + \tfrac{1}{2}m \omega^2 q^2$ be the classical Hamiltonian function, and consider the classical canonical distribution at inverse temperature $\beta'$, \begin{equation}\label{rhocl} \rho^\mathrm{class}_{\beta'} (dq \times dp) = \frac{1}{Z'} e^{-\beta' H(q,p)} \, dq \, dp\,, \quad Z' = \int_{\RRR^2} dq\, dp\, e^{-\beta' H(q,p)}\,. \end{equation} Let $\beta' = \frac{e^{\beta \hbar \omega} -1}{\hbar \omega} \,.$ Then $\mu_\beta = C_* \rho^\mathrm{class}_{\beta'}$ is the distribution on coherent states arising from $ \rho^\mathrm{class}_{\beta'}$. The density matrix of $\mu_\beta$ is $\rho_\beta$ \cite{GL81}. This measure is concentrated on a 2-dimensional submanifold of $\sphere(L^2(\RRR))$, namely on the set of coherent states (the image of $C$). Note also that not every density matrix $\rho$ on $L^2(\RRR)$ can arise as the density matrix of a distribution on the set of coherent states; for example, a pure state $\rho = |\psi \rangle \langle \psi|$ can arise in this way if and only if $\psi$ is a coherent state. \subsubsection{The Distribution Maximizing an Entropy Functional} In a similar spirit, one may consider, on a finite-dimensional Hilbert space $\Hilbert$, the distribution $\gamma(d\psi) = f(\psi) \, u_{\sphere (\Hilbert)} (d\psi)$ that maximizes the Gibbs entropy functional \begin{equation}\label{Gibbs} \mathscr{G}[f] = - \int\limits_{\sphere (\Hilbert)} \!\!\! u(d\psi) \:\: f(\psi) \, \log f(\psi) \end{equation} under the constraints that $\gamma$ be a probability distribution with mean 0 and covariance $\rho_{\Hilbert, H, \beta}$: \begin{subequations} \begin{align} f &\geq 0\\ \int\limits_{\sphere (\Hilbert)} \!\!\! u(d\psi) \, f(\psi) &= 1 \label{fnorm}\\ \int\limits_{\sphere (\Hilbert)} \!\!\! u(d\psi) \, f(\psi) \, | \psi \rangle &= 0\\ \int\limits_{\sphere (\Hilbert)} \!\!\! u(d\psi) \, f(\psi) \, |\psi \rangle \langle \psi | &= \rho_{\Hilbert, H, \beta}\,. \label{fcov} \end{align} \end{subequations} A standard calculation using Lagrange multipliers leads to \begin{equation}\label{fform} f(\psi) = \exp \langle \psi | L | \psi \rangle \end{equation} with $L$ a self-adjoint matrix determined by \eqref{fnorm} and \eqref{fcov}; comparison with \eqref{mupowerlaw} shows that $\gamma$ is not a GAP measure. (We remark, however, that another Gibbs entropy functional, $\mathscr{G}'[f] = -\int_\Hilbert \lambda(d\psi) \: f(\psi) \, \log f(\psi)$, based on the Lebesgue measure $\lambda$ on $\Hilbert$ instead of $u_{\sphere (\Hilbert)}$, is maximized, under the constraints that the mean be 0 and the covariance be $\rho$, by the Gaussian measure, $f(\psi) \, \lambda(d\psi) = \G{\rho} (d\psi)$.) There is no apparent reason why the family of $\gamma$ measures should be hereditary. The situation is different for the microcanonical ensemble: here, the distribution $u_{E, \delta} = \GAP{\rho_{E,\delta}}$ that we propose is in fact the maximizer of the appropriate Gibbs entropy functional $\mathscr{G}''$. Which functional is that? Since any measure $\gamma (d\psi)$ on $\sphere(\Hilbert)$ whose covariance matrix is the projection $\rho_{E,\delta} = \mathrm{const.} \, 1_{[E,E+\delta]}(H)$ must be concentrated on the subspace $\Hilbert_{E,\delta}$ and thus cannot be absolutely continuous (possess a density) relative to $u_{\sphere (\Hilbert)}$, we consider instead its density relative to $u_{\sphere (\Hilbert_{E, \delta})} = u_{E, \delta}$, that is, we consider $\gamma (d\psi) = f(\psi) \, u_{E, \delta} (d\psi)$ and set \begin{equation} \mathscr{G}'' [f] = - \int\limits_{\sphere (\Hilbert_{E, \delta})} u_{E, \delta} (d\psi) \, f(\psi) \, \log f(\psi) \,. \end{equation} Under the constraints that the probability measure $\gamma$ have mean 0 and covariance $\rho_{E, \delta}$, $\mathscr{G}'' [f]$ is maximized by $f \equiv 1$, or $\gamma = u_{E, \delta}$; in fact even without the constraints on $\gamma$, $\mathscr{G}'' [f]$ is maximized by $f \equiv 1$. \subsubsection{The Distribution of Brody and Hughston} \label{sec:Brody} Brody and Hughston \cite{Brody} have proposed the following distribution $\mu$ to describe thermal equilibrium. They observe that the projective space arising from a finite-dimensional Hilbert space, endowed with the dynamics arising from the unitary dynamics on Hilbert space, can be regarded as a classical Hamiltonian system with Hamiltonian function $H(\CCC \psi) = \langle \psi| H | \psi \rangle / \langle \psi | \psi \rangle$ (and symplectic form arising from the Hilbert space structure). They then define $\mu$ to be the classical canonical distribution of this Hamiltonian system, i.e., to have density proportional to $\exp(-\beta H(\CCC \psi))$ relative to the uniform volume measure on the projective space (which can be obtained from the symplectic form or, alternatively, from $u_{\sphere(\Hilbert)}$ by projection from the sphere to the projective space). However, this distribution leads to a density matrix, different from the usual one $\rho_\beta$ given by \eqref{rhobetaH}, that does not describe the canonical ensemble. \subsection{A Uniqueness Result for $\GAP{\rho}$} \label{sec:uniqueness} As $\EIG{\rho}$ is a family of measures satisfying the properties of Properties~1--3 for \emph{most} density matrices $\rho$, the question arises whether there is any family of measures, besides $\GAP{\rho}$, satisfying these properties for \emph{all} density matrices. We expect that the answer is no, and formulate the following uniqueness conjecture: \textit{Given, for every Hilbert space $\Hilbert$ and every density matrix $\rho$ on $\Hilbert$, a probability measure $\mu(\rho)$ on $\sphere(\Hilbert)$ such that Properties~1--3 remain true when $\GAP{\rho}$ is replaced by $\mu(\rho)$, then $\mu(\rho) = \GAP{\rho}$.} In other words, we conjecture that $\mu = \GAP{\rho}$ is the only hereditary covariant inverse of \eqref{rhomupsi}. This is in fact true when we assume in addition that the mapping $\mu: \rho \mapsto \mu(\rho)$ is suitably continuous. Here is the argument: When $\rho$ is a multiple of a projection, $\rho = (\dim \Hilbert')^{-1} P_{\Hilbert'}$ for a subspace $\Hilbert' \subseteq \Hilbert$, then $\mu(\rho)$ must be, by covariance $U_* \mu(\rho) = \mu(U\rho U^*)$, the uniform distribution on $\sphere(\Hilbert')$, and thus $\mu(\rho) = \GAP{\rho}$ in this case. Consider now a composite of a system (system 1) and a large heat bath (system 2) with Hilbert space $\Hilbert = \Hilbert_1 \otimes \Hilbert_2$ and Hamiltonian $H = H_1 \otimes I_2 + I_1 \otimes H_2$, and consider the microcanonical density matrix $\rho_{E,\delta}$ for this system. By equivalence of ensembles, we have for suitable $\beta>0$ that $\rho_{E,\delta} \approx \rho_{\Hilbert, H, \beta} = \rho^{(1)}_\beta \otimes \rho^{(2)}_\beta$ where $\rho^{(i)}_\beta = \rho_{\Hilbert_i, H_i, \beta}$. By the continuity of $\mu$ and $GAP$, \[ \mu \bigl( \rho^{(1)}_\beta \otimes \rho^{(2)}_\beta \bigr) \approx \mu(\rho_{E,\delta}) = \GAP{\rho_{E,\delta}} \approx \GAP{ \rho^{(1)}_\beta \otimes \rho^{(2)}_\beta}\,. \] Now consider, for a wave function $\Psi$ with distribution $\mu \bigl( \rho^{(1)}_\beta \otimes \rho^{(2)}_\beta \bigr)$ respectively $\GAP{\rho^{(1)}_\beta \otimes \rho^{(2)}_\beta}$, the distribution of the conditional wave function $\Psi_1$: by heredity, this is $\mu(\rho^{(1)}_\beta)$ respectively $\GAP{\rho^{(1)}_\beta}$. Since the distribution of $\Psi_1$ is a continuous function of the distribution of $\Psi$, we thus have that $\mu(\rho^{(1)}_\beta) \approx \GAP{\rho^{(1)}_\beta}$. Since we can make the degree of approximation arbitrarily good by making the heat bath sufficiently large, we must have that $\mu(\rho^{(1)}_\beta) = \GAP{\rho^{(1)}_\beta}$. For any density matrix $\rho$ on $\Hilbert_1$ that does not have zero among its eigenvalues, there is an $H_1$ such that $\rho = \rho^{(1)}_\beta = Z^{-1} \, \exp(-\beta H_1)$ for $\beta =1$, and thus we have that $\mu(\rho) = \GAP{\rho}$ for such a $\rho$; since these are dense, we have that $\mu(\rho) = \GAP{\rho}$ for all density matrices $\rho$ on $\Hilbert_1$. Since $\Hilbert_1$ is arbitrary we are done. \subsection{Dynamics of the Conditional Wave Function} Markov processes in Hilbert space have long been considered (see \cite{BP02} for an overview), particularly diffusion processes and piecewise deterministic (jump) processes. This is often done for the purpose of numerical simulation of a master equation for the density matrix, or as a model of continuous measurement or of spontaneous wave function collapse. Such processes could arise as follows. Since the conditional wave function $\Psi_1$ arises from the wave function $\sp{q_2}{\psi}$ by inserting a random coordinate $Q_2$ for the second variable (and normalizing), any dynamics (i.e., time evolution) for $Q_2$, described by a curve $t \mapsto Q_2(t)$ and preserving the quantum probability distribution of $Q_2$, for example, as given by Bohmian mechanics \cite{DGZ}, gives rise to a dynamics for the conditional wave function, $t \mapsto \Psi_1(t) = \mathcal{N}(t) \, \sp{Q_2(t)}{\psi(t)}$, where $\psi(t)$ evolves according to Schr\"odinger's equation and $\mathcal{N}(t)= \|\sp{Q_2(t)}{\psi(t)} \|^{-1}$ is the normalizing factor. In this way one obtains a stochastic process (a random path) in $\sphere(\Hilbert_1)$. In the case considered in Section~\ref{sec:hb1}, in which $\Hilbert_2$ corresponds to a large heat bath, this process must have $\GAP{\rho_{\Hilbert_1, H_1,\beta}}$ as an invariant measure. It would be interesting to know whether this process is approximately a simple process in $\sphere(\Hilbert_1)$, perhaps a diffusion process, perhaps one of the Markov processes on Hilbert space considered already in the literature. \section{The Two-Level System as a Simple Example} \label{sec:two} In this last section, we consider a two-level system, with $\Hilbert = \CCC^2$ and \begin{equation}\label{H2} H = E_1 |1\rangle \langle 1| + E_2 |2\rangle \langle 2|, \end{equation} and calculate the joint distribution of the energy coefficients $Z_1 = \sp{1}{\Psi}$ and $Z_2 = \sp{2}{\Psi}$ for a $\GAP{\rho _\beta}$-distributed $\Psi$ as explicitly as possible. We begin with a general finite-dimensional system, $\Hilbert = \CCC^k$, and specialize to $k=2$ later. One way of describing the distribution of $\Psi$ is to give its density relative to the hypersurface area measure $u$ on $\sphere(\CCC^k)$; this we did in \eqref{mudensity}. Another way of describing the joint distribution of the $Z_n$ is to describe the joint distribution of their moduli $|Z_n|$, or of $|Z_n|^2$, as the phases of the $Z_n$ are independent (of each other and of the moduli) and uniformly distributed, see \eqref{independ}. Before we determine the distribution of $|Z_n|^2$, we repeat that its expectation can be computed easily. In fact, for any $\phi \in \Hilbert$ we have \[ \EEE \bigl| \sp{\phi}{\Psi} \bigr|^2 = \int\limits_{\sphere(\Hilbert)} \!\! \GAP{\rho_\beta} (d\psi) \, \bigl| \sp{\phi}{\psi} \bigr|^2 \stackrel{\eqref{rhomupsi}}{=} \sp{\phi}{\rho_\beta | \phi} \stackrel{\eqref{rhobetaH}}{=} \frac{1}{Z(\beta)} \sp{\phi}{e^{-\beta H} | \phi}. \] Thus, for $|\phi\rangle = |n\rangle$, we obtain $\EEE |Z_n|^2 = e^{-\beta E_n}/\tr\, e^{-\beta H}$. For greater clarity, from now on we write $Z_n^{GAP}$ instead of $Z_n$. A relation similar to that between $\GAP{\rho}$, $\GA{\rho}$, and $\G{\rho}$ holds between the joint distributions of the $|Z_n^{GAP}|^2$, of the $|Z_n^{GA}|^2$, and of the $|Z_n^G|^2$. The joint distribution of the $|Z_n^G|^2$ is very simple: they are independent and exponentially distributed with means $p_n = e^{-\beta E_n} / Z(\beta)$. Since the density of $GA$ relative to $G$, $dGA/dG = \sum_n |z_n|^2$, is a function of the moduli alone, and since, according to \eqref{independ}, $GA = GA_\mathrm{phases} \times GA_\mathrm{moduli}$, we have that \[ GA_\mathrm{moduli} = \sum_n |z_n|^2 \, G_\mathrm{moduli}. \] Thus, \begin{equation}\label{jointZN} \PPP \bigl( |Z_1^{GA}|^2 \in ds_1 , \ldots, |Z_k^{GA}|^2 \in ds_k \bigr) = \frac{s_1+ \ldots + s_k} {p_1 \cdots p_k} \exp \Bigl( -\sum_{n=1}^k \frac{s_n}{p_n} \Bigr) ds_1 \cdots ds_k, \end{equation} where each $s_n \in (0,\infty)$. Finally, the $|Z_n^{GAP}|^2$ arise by normalization, \begin{equation}\label{ZZN} |Z_n^{GAP}|^2 = \frac{|Z_n^{GA}|^2}{\sum\limits_{n'} |Z_{n'}^{GA}|^2}. \end{equation} We now specialize to the two-level system, $k=2$. Since $|Z_1^{GAP}|^2 +|Z_2^{GAP}|^2 =1$, it suffices to determine the distribution of $|Z_1^{GAP}|^2$, for which we give an explicit formula in \eqref{Z12dist} below. We want to obtain the marginal distribution of \eqref{ZZN} from the joint distribution of the $|Z_n^{GA}|^2$ in $(0,\infty)^2$, the first quadrant of the plane, as given by \eqref{jointZN}. To this end, we introduce new coordinates in the first quadrant: \[%\begin{equation} s=\frac{s_1}{s_1 + s_2}, \quad \lambda = s_1 +s_2, \]%\end{equation} where $\lambda>0$ and $00, \]%\end{equation} we obtain \begin{subequations} \begin{align} \PPP \Bigl( |Z_1^{GAP}|^2 \in ds \Bigr) &= ds \int\limits_0^\infty d \lambda \, \frac{e^{\beta \tr\, H}}{Z(\beta)} \lambda^2 \, \exp \Bigl( -\lambda \bigl( e^{\beta E_1} s + e^{\beta E_2} (1-s) \bigr) \Bigr) =\\ &= \frac{2e^{\beta \tr\, H}}{Z(\beta)} \bigl( e^{\beta E_1} s + e^{\beta E_2} (1-s) \bigr)^{-3} ds = \\ &= \bigl( \alpha_1 s + \alpha_2 (1-s) \bigr)^{-3} ds \label{Z12dist} =: f(s) \, ds\,, \quad 01$, we have $\alpha_2 = \delta \alpha_1 > \alpha_1$, so that $\alpha_1 s + \alpha_2 (1-s)$ is decreasing monotonically from $\alpha_2$ at $s=0$ to $\alpha_1$ at $s=1$; hence, $f$ is increasing monotonically from $\alpha_2^{-3}$ to $\alpha_1^{-3}$. For $\delta<1$, we have $\alpha_2< \alpha_1$, and hence $f$ is decreasing monotonically from $\alpha_2^{-3}$ to $\alpha_1^{-3}$. In all cases $f$ is convex since $f'' \geq 0$. \bigskip \noindent \textit{Acknowledgments.} We thank Andrea Viale (Universit\`a di Genova, Italy) for preparing the figure, Eugene Speer (Rutgers University, USA) for comments on an earlier version, and Eric Carlen (Georgia Institute of Technology, USA), Detlef D\"urr (LMU M\"unchen, Germany), Raffaele Esposito (Universit\`a di L'Aquila, Italy), Rossana Marra (Universit\`a di Roma ``Tor Vergata'', Italy), and Herbert Spohn (TU M\"unchen, Germany) for suggesting references. We are grateful for the hospitality of the Institut des Hautes \'Etudes Scientifiques (Bures-sur-Yvette, France), where part of the work on this paper was done. The work of J.~Lebowitz was supported by NSF Grant DMR 01-279-26 and AFOSR Grant AF 49620-01-1-0154. The work of R.~Tumulka was supported by INFN and by the European Commission through its 6th Framework Programme ``Structuring the European Research Area'' and the contract Nr. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action. The work of N.~Zangh\`\i\ was supported by INFN. \begin{thebibliography}{28} \bibitem{LAG75} Aizenman, M., Goldstein, S., Lebowitz, J. L.: On the stability of equilibrium states of finite classical systems. \textit{J. Math. Phys.} \textbf{16}, 1284--1287 (1975) \bibitem{ergodic} Arnol'd, V. I., Avez, A.: \textit{Ergodic problems of classical mechanics} (Benjamin, New York, 1968). \bibitem{BP02} Breuer, H.-P., Petruccione, F.: \textit{Open Quantum Systems} (Oxford University Press, 2002) \bibitem{Brody} Brody, D. C., Hughston, L. P.: The quantum canonical ensemble. \textit{J. Math. 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Manuscript and notes of Felix Bloch} (Stanford University Press, Stanford, CA, 1989). \end{thebibliography} \end{document} ---------------0504130819417 Content-Type: application/postscript; name="thermo1f1.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="thermo1f1.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: figure.dvi %%BoundingBox: 126 239 560 508 %%DocumentFonts: Courier Math2Mono %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -E -f figure.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.04.05:1839 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 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ED moveto n 1 sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore } def /Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1 sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0 y0 x0 2 mul x1 sub y1 ] def } def /Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 } if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if /dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub 0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def /Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def } if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def } { /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } 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gt {z1 z2 add neg} {z1} ifelse h 0 gt {z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g exch t L stroke /i i h add def } repeat grestore } def /ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0 1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c mul neg d } def /Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def /Rot { CP CP translate 3 -1 roll neg rotate NET } def /RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 } def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle dup a add ] cvx def } def /RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ] cvx def } def /PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def /PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def /PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def /Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } { h1 abs h sub dup s mul abs } ifelse } def /UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q { x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add exch } def /BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible /IfVisible true def } if } { IfVisible { Invisible /IfVisible false def } if } ifelse } def /InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def /Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def /IfVisible true def } def end % END pstricks.pro %%EndProcSet %%BeginProcSet: pst-dots.pro %!PS-Adobe-2.0 %%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07. %%Creator: Timothy Van Zandt %%Creation Date: May 7, 1993 10 dict dup begin /FontType 3 def /FontMatrix [ .001 0 0 .001 0 0 ] def /FontBBox [ 0 0 0 0 ] def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for Encoding dup (b) 0 get /Bullet put dup (c) 0 get /Circle put dup (C) 0 get /BoldCircle put dup (u) 0 get /SolidTriangle put dup (t) 0 get /Triangle put dup (T) 0 get /BoldTriangle put dup (r) 0 get /SolidSquare put dup (s) 0 get /Square put dup (S) 0 get /BoldSquare put dup (q) 0 get /SolidPentagon put dup (p) 0 get /Pentagon put (P) 0 get /BoldPentagon put /Metrics 13 dict def Metrics begin /Bullet 1000 def /Circle 1000 def /BoldCircle 1000 def /SolidTriangle 1344 def /Triangle 1344 def /BoldTriangle 1344 def /SolidSquare 886 def /Square 886 def /BoldSquare 886 def /SolidPentagon 1093.2 def /Pentagon 1093.2 def /BoldPentagon 1093.2 def /.notdef 0 def end /BBoxes 13 dict def BBoxes begin /Circle { -550 -550 550 550 } def /BoldCircle /Circle load def /Bullet /Circle 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def /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def /SquarePath { -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto closepath } def /SolidSquare { SquarePath fill } def /Square { SquarePath .89 .89 scale SquarePath eofill } def /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def /PentagonPath { -337.8 -465 moveto 337.8 -465 lineto 546.6 177.6 lineto 0 574.7 lineto -546.6 177.6 lineto closepath } def /SolidPentagon { PentagonPath fill } def /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def /.notdef { } def end /BuildGlyph { exch begin Metrics 1 index get exec 0 BBoxes 3 index get exec setcachedevice CharProcs begin load exec end end } def /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } bind def end /PSTricksDotFont exch definefont pop % END pst-dots.pro %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "22 June 1996", % filename = "8r.enc", % email = "kb@@mail.tug.org", % address = "135 Center Hill Rd. // Plymouth, MA 02360", % codetable = "ISO/ASCII", % checksum = "119 662 4424", % docstring = "Encoding for TrueType or Type 1 fonts to be used with TeX." % @} % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. 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/Courier-Oblique findfont def /MathPictureStart { /Mimatrix matrix currentmatrix def gsave newpath Mleft Mbottom translate /Mtmatrix matrix currentmatrix def Plain Mfontsize scalefont setfont 0 setgray 0 setlinewidth } bind def /MathPictureEnd { grestore } bind def /MathSubStart { Momatrix Mgmatrix Mtmatrix Mleft Mbottom Mwidth Mheight 9 -2 roll moveto Mtmatrix setmatrix currentpoint Mgmatrix setmatrix 11 -2 roll moveto Mtmatrix setmatrix currentpoint 2 copy translate /Mtmatrix matrix currentmatrix def /Mleft 0 def /Mbottom 0 def 3 -1 roll exch sub /Mheight exch def sub /Mwidth exch def } bind def /MathSubEnd { /Mheight exch def /Mwidth exch def /Mbottom exch def /Mleft exch def /Mtmatrix exch def dup setmatrix /Mgmatrix exch def /Momatrix exch def } bind def /Mdot { moveto 0 0 rlineto stroke } bind def /Mtetra { moveto lineto lineto lineto fill } bind def /Metetra { moveto lineto lineto lineto closepath gsave fill grestore 0 setgray stroke } bind def /Mistroke { flattenpath 0 0 0 { 4 2 roll pop pop } { 4 -1 roll 2 index sub dup mul 4 -1 roll 2 index sub dup mul add sqrt 4 -1 roll add 3 1 roll } { stop } { stop } pathforall pop pop currentpoint stroke moveto currentdash 3 -1 roll add setdash } bind def /Mfstroke { stroke currentdash pop 0 setdash } bind def /Mrotsboxa { gsave dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def grestore Msboxa 3 -1 roll /Mtmatrix exch def /Mrot 0 def } bind def /Msboxa { newpath 5 -1 roll Mvboxa pop Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Msboxa1 5 -3 roll Msboxa1 Mboxrot [ 7 -2 roll 2 copy [ 3 1 roll 10 -1 roll 9 -1 roll ] 6 1 roll 5 -2 roll ] } bind def /Msboxa1 { sub 2 div dup 2 index 1 add mul 3 -1 roll -1 add 3 -1 roll mul } bind def /Mvboxa { Mfixwid { Mvboxa1 } { dup Mwidthcal 0 exch { add } forall exch Mvboxa1 4 index 7 -1 roll add 4 -1 roll pop 3 1 roll } ifelse } bind def /Mvboxa1 { gsave newpath [ true 3 -1 roll { Mbbox 5 -1 roll { 0 5 1 roll } { 7 -1 roll exch sub (m) stringwidth pop .3 mul sub 7 1 roll 6 -1 roll 4 -1 roll Mmin 3 -1 roll 5 index add 5 -1 roll 4 -1 roll Mmax 4 -1 roll } ifelse false } forall { stop } if counttomark 1 add 4 roll ] grestore } bind def /Mbbox { 0 0 moveto false charpath flattenpath pathbbox newpath } bind def /Mmin { 2 copy gt { exch } if pop } bind def /Mmax { 2 copy lt { exch } if pop } bind def /Mrotshowa { dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def Mgmatrix setmatrix Mshowa /Mtmatrix exch def /Mrot 0 def } bind def /Mshowa { 4 -2 roll moveto 2 index Mtmatrix setmatrix Mvboxa 7 1 roll Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Mshowa1 4 1 roll Mshowa1 rmoveto currentpoint Mfixwid { Mshowax } { Mshoway } ifelse pop pop pop pop Mgmatrix setmatrix } bind def /Mshowax { 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get show } for } bind def /Mshoway { 3 index Mwidthcal 5 1 roll 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get [ 6 index aload length 2 add -1 roll { pop Strform stringwidth pop neg exch add 0 rmoveto } exch kshow cleartomark } for pop } bind def /Mwidthcal { [ exch { Mwidthcal1 } forall ] [ exch dup Maxlen -1 add 0 1 3 -1 roll { [ exch 2 index { 1 index Mget exch } forall pop Maxget exch } for pop ] Mreva } bind def /Mreva { [ exch aload length -1 1 {1 roll} for ] } bind def /Mget { 1 index length -1 add 1 index ge { get } { pop pop 0 } ifelse } bind def /Maxlen { [ exch { length } forall Maxget } bind def /Maxget { counttomark -1 add 1 1 3 -1 roll { pop Mmax } for exch pop } bind def /Mwidthcal1 { [ exch { Strform stringwidth pop } forall ] } bind def /Strform { /tem (x) def tem 0 3 -1 roll put tem } bind def /Mshowa1 { 2 copy add 4 1 roll sub mul sub -2 div } bind def /MathScale { Mwidth Mheight Mlp translate scale 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