text:
\section{Introduction}
\label{sect:Introduction}

The emergence of determinism in the macroscopic variables of a system from its underlying microscopic dynamics has been a subject of great importance in the field of statistical mechanics.  If one supposes the microscopic variables are rigidly deterministic, then the time evolution of the macroscopic variables as a function of the initial \emph{microstate} is of course rigidly deterministic as well.  Considered as a function of the initial \emph{macrostate}, however, its time evolution fails to be deterministic due to the multiplicity of microstates consistent with the given macrostate. This leads to the familiar ensemble description, as described by a conditional \textit{a priori\/} measure.  Given the highly irregular behavior of many macroscopic systems on the microscopic level, however, it may appear that all deterministic behavior is utterly lost on the macroscopic level.

By contrast, many macroscopic systems are described quite well by deterministic models, even if they are not deterministic in the strict sense.  Well known examples include thermal conduction, diffusion, hydrodynamics, and chemical reactions. Characteristic of many such systems is the presence of extensive macroscopic variables which are sums or averages over a great many microscopic quantities.  The task of deriving differential equations for such variables has been taken up by several researchers \cite{vanKampen1961,Kurtz1972,Kubo1973}, and here Markov processes have played an important role. In this approach, one derives a coupled set of differential equations for the moments of the macroscopic variable, where the deterministic behavior is given by the first moment when the dispersion goes to zero.  Since few physical observables are truly Markovian, delicate scaling between vanishing interactions and dilating time scales must be used to obtain an asymptotically Markovian process \cite{Spohn,Davies}.  However, the consistency of such assumptions with the underlying microscopic dynamics and the relevance of strong interactions have been repeatedly called into question \cite{Sklar,Mehra1972}. In particular, the relaxation of macroscopic systems to a state of equilibrium may seem inconsistent with the underlying reversible microscopic dynamics.

We take a somewhat different and more general view upon this problem. Instead of attempting to derive differential equations of motion for a class of macroscopic variables, we consider instead the existence and character of an emergent form of determinism for large systems, which we call macroscopic determinism. By this we mean that if the macrostate is initially constrained to be near a given value, $y$, then there exists a map $\psi_t$ such that the probability that the macrostate is near $\psi_t(y)$ at time $t$ approaches one as the number of particles approaches infinity.  For simplicity, we restrict attention to systems of dynamically noninteracting particles and macroscopic variables which take the form of  averages over these microscopic quantities. The mathematical theory of large deviations, an extension of the law of large numbers, provides an useful tool for addressing such questions and is used in \Sec \ref{sec:LDTA} to obtain the main result.  Our primary contribution has been to apply well-known equilibrium results to systems initially out of equilibrium.

The collection of macrostates $\set{\psi_t(y) : t \in T}$ for a given $y$ and set of times $T$ constitutes a collection of highly probable states, which we call the deterministic curve, akin to the concentration curve of P. and T. Ehrenfest in their discussion of Boltzmann's H-theorem \cite{Ehrenfest}. In \Sec \ref{sec:DC} we show that the macroscopic average converges to this curve in probability for any given finite, and in some cases infinite, set of times, though large deviations from this curve will persist whenever the number of particles is finite. For certain reversible microscopic dynamics which preserve the \textit{a priori\/} measure, the deterministic curve of a restricted class of macroscopioc variables is symmetric in time about the initial specification of the macroscopic variable. Furthermore, if the microscopic dynamical law is mixing, then the deterministic curve converges as $t \rightarrow \pm\infty$ to a value independent of $y$, a situation corresponding to equilibration of the macrostate.

\section{Problem Description}
\label{sec:PD}

We begin with a mathematical description of the relevant physical quantities. The microstate space of a single particle is denoted by $X$, while the microstate space of the total $n$-particle system is given by the Cartesian product $X^n = X \times \stackrel{n}{\cdots} \times X$. The projection map $\pi_i$ is defined such that, if $(x_1, \ldots, x_n) \in X^n$ is the microstate of the system, then $\pi_i(x_1, \ldots, x_n) = x_i$ is the microstate of the $i^{\mathrm{th}}$ particle.  To apply the techniques of large deviation theory we shall need to make the mild assumption that $(X, d)$ is a Polish space, i.e.\ a complete separable metric space, with respect to some given metric $d$, and denote by $\mathcal{A}$ the set of Borel subsets of $X$ generated by the metric topology.

Let $\mathcal{P}(X)$ denote the set of Borel probability measures on $X$ and note that $\mathcal{P}(X)$ is itself a Polish space under the Prohorov metric $\rho$ induced by the metric $d$ on $X$ \cite[p.\ 317]{Dudley}. Denote by $\mu \in \mathcal{P}(X)$ the \textit{a priori\/} probability measure for the initial microstate of a given particle. In other words, for $C \in \mathcal{A}$, $\mu[C]$ is the probability that a given particle's initial microstate is in $C \subseteq X$, \emph{before} any conditioning on the macrostate has taken place. All particle microstates therefore have equal \textit{a priori\/} probability in the sense that they are uniformly distributed with respect to $\mu$.  Usually, $\mu$ is taken to be an invariant measure. The system microstates are assumed to be \textit{a priori} independent and identically distributed (i.i.d.\/) with marginal $\mu$. Thus, the \textit{a priori} distribution on $X^n$ is given by the product measure $\mu^n = \mu \times \stackrel{n}{\cdots} \times \mu$.

Let $g: X \rightarrow Y$ be a measurable function which is bounded and continuous $\mu$-almost everywhere (a.e.\/).  For a given particle microstate $x_i \in X$, $g(x_i)$ gives the corresponding single particle macrostate. For a collection of $n$ particles, the macroscopic average $G: X^n \rightarrow Y$ of $g$ is given by
\begin{equation}
G := \frac{1}{n} \sum_{i=1}^{n} g \circ \pi_i.
\end{equation}
It is to this macroscopic variable that we will focus our attention.  We shall take $Y$ to be a set of real numbers equipped with the Euclidean norm. To accommodate all possible values of $G$ as well as any accumulation points, we shall assume $Y = [\ymin,\; \ymax]$, where $\ymin = \inf g(X)$, $\ymax = \sup g(X)$, and $g(X)$ is the image of $X$ under $g$. Note that $G$ is indeed a measurable function since it is a finite sum of measurable functions.

The dynamics of the system are described by a family of measurable transformations $\Phi_{t}$ on $X^n$ indexed by the time parameter $t \in T \subseteq \mathbb{R}$, where $\Phi_{0}$ is the identity map. The macroscopic average at time $t$ is thus given by $G_{t} := G \circ \Phi_{t}$. We shall suppose that the particles are dynamically independent and identical in the sense that
\begin{equation}
\Phi_{t} = (\varphi_t \circ \pi_1, \ldots, \varphi_t \circ \pi_n),
\end{equation}
where $\varphi_t$ is a measurable transformation on $X$.  We shall further suppose that $\varphi_t$ is continuous $\mu$-a.e.\ on $X$ for each $t \in T$, though not necessarily continuous on $T$ for $\mu$-a.e.\ $x \in X$.

The macrostate $G_{t}$ may be considered a sample mean over the \textit{a priori\/} i.i.d.\ random variables $\set{\varphi_t \circ \pi_i}_{i=1}^{n}$ with common marginal $\mu \circ (g \circ \varphi_t)^{-1}$. Since $g$ is bounded, the weak law of large numbers implies $G_{t}$ converges in probability to the expectation value $\int_{X} g \circ \varphi_t \;\d \mu$ as $n \rightarrow \infty$; in other words,
\begin{equation}
\lim_{n \rightarrow \infty} \mu^{n}[G_{t} \in B] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\int_{X} g \circ \varphi_t \;\d \mu \not\in \closure{B}$,} \\
    1, & \mbox{if $\int_{X} g \circ \varphi_t \;\d \mu \in \interior{B}$} \\
  \end{array}
  \right.
\end{equation}
for any Borel set $B \subseteq Y$, where $\interior{B}$ is the interior of $B$ and $\closure{B}$ is its closure.  No limit is specified for points on the boundary, $\boundary{B}$, of $B$.

If the initial microstates are restricted so as to satisfy some initial macroscopic constraint, then the variables $\set{\varphi_t \circ \pi_i}_{i=1}^{n}$ may no longer be independent due to the correlations imposed by this conditioning.  A direct application of the law of large numbers is therefore no longer valid.  Suppose, in particular, that the initial macrostate $G_{0} = G$ is constrained to be in a small region $B_{\delta}$ containing a given macrostate $y$. We will show that $G_{t}$ converges in conditional probability to some $\psi_t(y)$; in other words,
\begin{equation}
\lim_{n \rightarrow \infty} \mu^{n}[G_{t} \in B \:| G \in B_{\delta}] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\psi_t(y) \not\in \closure{B}$,} \\
    1, & \mbox{if $\psi_t(y) \in \interior{B}$}. \\
  \end{array}
  \right.
\end{equation}
where $\psi_t(y)$ is the expectation value of $g \circ \varphi_t$ with respect to a new probability measure determined by $y$.

In \Sec \ref{sec:LLNA} we shall consider three values of $y$, namely $\ymin, \ymax$, and $y_* = \int_{X} g \;\d \mu$, for which the law of large numbers alone may be used to determine $\psi_t(y)$ and prove convergence in conditional probability. Later, in \Sec \ref{sec:LDTA}, convergence is proven for $y \in (\ymin, \ymax) = \interior{Y}$ using the theory of large deviations, from which a general expression for $\psi_t(y)$ is obtained in terms of the expectation of $g \circ \varphi_t$ with respect to a suitable canonical measure.

\section{Law of Large Numbers Approach}
\label{sec:LLNA}

We consider first an approach using only the law of large numbers, which will be valid for $y = y_* := \int_{X} g \;\d \mu$ and, in certain cases, $y = \ymin$ and $y=\ymax$.  We begin with the latter two cases.

Suppose $\mu[\set{g = \ymin}] > 0$ and note that, since $\set{ G = \ymin } = \set{ g = \ymin } \times \stackrel{n}{\cdots} \times \set{ g = \ymin }$,
\begin{equation}
\mu^{n}\left[\left.A_1 \times \cdots \times A_n \;\right| \set{G = \ymin}\right] = \prod_{i=1}^{n} \mu[A_i \:| \set{g = \ymin}],
\end{equation}
for any $A_1, \ldots, A_n$ in $\mathcal{A}$.  Thus, conditioned on $\set{G = \ymin}$, $G_{t}$ is a sample mean of i.i.d.\ random variables with common marginal $\mu[(g \circ \varphi_{t})^{-1}(\;\cdot\;) \:| \set{g = \ymin}]$.  The weak law of large numbers then implies
\begin{equation}
\lim_{n \rightarrow \infty} \mu^{n}[G_{t} \in B \:|
\set{G = \ymin}] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\psi_{t}(\ymin) \not\in \closure{B}$,} \\
    1, & \mbox{if $\psi_{t}(\ymin) \in \interior{B}$} \\
  \end{array}
  \right.
\end{equation}
for any Borel set $B \subseteq Y$, where
\begin{equation}
\label{eqn:ECM}
\psi_{t}(\ymin) := \int_{X} g \circ \varphi_{t} \;\d \mu[\;\cdot\; \:| \set{g = \ymin}].
\end{equation}
A similar result holds for conditioning on $\set{G = \ymax}$, provided $\mu[\set{g = \ymax}] > 0$.

Let us turn now to the case $y = y_*$, where we suppose $y_* \in \interior{Y}$. Given a set $B_{\delta} \subseteq Y$ whose interior contains $y_*$, we wish to examine the limit
\begin{equation}
\lim_{n \rightarrow \infty} \mu^n[\set{G_{t} \in B} \:| \set{G \in B_{\delta}}],
\end{equation}
for an arbitrary Borel subset $B$ of $Y$.

To parallel the large deviation approach of the following section, we consider the convergence of the empirical measure $L_n$, defined by
\begin{equation}
L_n(x_1, \ldots, x_n) := \frac{1}{n} \sum_{i=1}^{n} \delta_{x_i},
\end{equation}
where $\delta_{x_i}$ is the unit point measure on $x_i$. Given $(x_1, \ldots, x_n) \in X^n$ and $C \subseteq X$ measurable, $L_n(x_1, \ldots, x_n) \in \mathcal{P}(X)$, and $L_n(x_1, \ldots, x_n)[C]$ is the fraction of particles whose microstate is in $C$.  A useful result is that $G_{t}$, and hence $G$, may be written in terms of $L_n$, since
\begin{equation}
G_{t}(x_1, \ldots, x_n) = \frac{1}{n} \sum_{i=1}^{n} g(\varphi_t(x_i)) = \int_X g \circ \varphi_t \;\d L_n(x_1, \ldots, x_n).
\end{equation}
Using the above relation, convergence properties of $G_{t}$ may then be deduced from those of $L_n$.

Now, since $(X,d)$ is a separable space, $L_n$ converges almost surely to $\mu$ \cite[p.\ 313]{Dudley} and hence in probability as well. Thus, for any Borel set $A \subseteq \mathcal{P}(X)$,
\begin{equation}
\label{eqn:EMCP}
  \lim_{n \rightarrow \infty} \mu^n[\set{L_n \in A}] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\mu \not\in \closure{A}$,} \\
    1, & \mbox{if $\mu \in \interior{A}$.} \\
  \end{array}
  \right.
\end{equation}
Now suppose $A_{\delta}$ is a Borel subset of $\mathcal{P}(X)$ such that $\mu \in \interior{A_{\delta}}$.  Using \Eqn (\ref{eqn:EMCP}), we have that
\begin{equation}
\label{eqn:EMCPCE}
  \lim_{n \rightarrow \infty} \mu^n[\set{L_n \in A} \:| \set{L_n \in A_{\delta}}] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\mu \not\in \closure{A}$,} \\
    1,& \mbox{if $\mu \in \interior{A}$,} \\
  \end{array}
  \right.
\end{equation}
since $\mu^n[\set{L_n \in A_{\delta}}] \rightarrow 1$ as $n \rightarrow \infty$.

To apply this result to the convergence of $G_{t}$, we define the expectation function $E_{g}(P) := \int_X g \;\d P$ for $P \in \mathcal{P}(X)$ and note that
\begin{equation}
E_{g} \circ L_n = \frac{1}{n} \sum_{i=1}^{n} g \circ \pi_i = G.
\end{equation}
Since $g$ is bounded and continuous $\mu$-a.e.\, $E_{g}$ is continuous at $\mu$ in the Prohorov metric topology \cite{Dudley,Ash}. (Of course, if $P \prec \mu$, then $E_{g}$ is continuous at $P$ as well.) Thus, $E_{g}(\mu) \in \interior{B}$ implies $\mu \in \interior{E_{g}^{-1}(B)}$, and similarly $E_{g}(\mu) \not\in \closure{B}$ implies $\mu \not\in \closure{E_{g}^{-1}(B)}$.

Now, since $g \circ \varphi_t$ is also bounded and continuous $\mu$-a.e.\/, $\psi_t(y_*) = E_{g \circ \varphi_t}(\mu) \in \interior{B}$ implies $\mu \in \interior{E_{g \circ \varphi_t}^{-1}(B)}$, and similarly $\psi_t(y_*) \not\in \closure{B}$ implies $\mu \not\in \closure{E_{g \circ \varphi_t}^{-1}(B)}$. Setting $A = E_{g \circ \varphi_t}^{-1}(B)$ and $A_{\delta} = E_{g}^{-1}(B_{\delta})$ in \Eqn (\ref{eqn:EMCPCE}), we conclude that
\begin{equation}
\label{eqn:MACPCE}
  \lim_{n \rightarrow \infty} \mu^n[\set{G_{t} \in B}
  \:| \set{G \in B_{\delta}}] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\psi_t(y_*) \not\in \closure{B}$,} \\
    1, & \mbox{if $\psi_t(y_*) \in \interior{B}$,} \\
  \end{array}
  \right.
\end{equation}
where
\begin{equation}
\label{eqn:ECE}
\psi_t(y_*) := E_{g \circ \varphi_t}(\mu) = \int g \circ \varphi_t \;\d \mu.
\end{equation}
If $\mu$ is an invariant measure, i.e.\ $\mu \circ \varphi_t^{-1} = \mu$, we obtain the rather trivial result that $G_{t}$ converges in probability to its equilibrium value, $y_*$, as $n \rightarrow \infty$.

The general problem, wherein $y \in \interior{Y} \setminus \set{y_*}$, cannot be addressed with the law of large numbers alone.  To see this, consider a set $B_{\delta} \subset \interior{Y}$ such that $y_* \not\in \closure{B_{\delta}}$ and let $B \subseteq Y$ be some Borel set. We wish to evaluate the following conditional probability as $n \rightarrow \infty$:
\[
\mu^n[\set{G_{t} \in B} \:| \set{G \in
B_{\delta}}] = \mu^n[\set{L_n \in E_{g \circ \varphi_t}^{-1}(B)} \:| \set{L_n \in E_{g}^{-1}(B_{\delta})}]
\]
\[
= \mu^n[\set{L_n \in E_{g \circ \varphi_t}^{-1}(B) \cap
E_{g}^{-1}(B_{\delta})}] \;/\; \mu^n[\set{L_n \in E_{g}^{-1}(B_{\delta})}].
\]
Since $y_* = E_{g}(\mu) \not\in \closure{B_{\delta}}$, $\mu \not\in \closure{E_{g}^{-1}(B_{\delta})}$ and the denominator goes to zero. Since $\mu \not\in \closure{E_{g \circ \varphi_t}^{-1}(B)} \cap \closure{E_{g}^{-1}(B_{\delta})} \supseteq \closure{E_{g \circ \varphi_t}^{-1}(B) \cap E_{g}^{-1}(B_{\delta})}$, we see that the numerator goes to zero as well, leaving the limit indeterminant. To proceed further requires more detailed information about the rate of convergence of each limit, which the law of large numbers alone cannot provide.  For this we turn to the theory of large deviations.

\section{Large Deviation Theory Approach}
\label{sec:LDTA}

We have seen in the previous section that the weak law of large numbers is insufficient for evaluating limiting conditional probabilities when the initial macrostate is not $y_*$. (More precisely, this is true when we condition on a \emph{set} of macrostates whose closure does not contain $y_*$.) Large deviation theory provides a tool for evaluating such limits and, through its application, provides an explicit expression for $\psi_t(y)$ even when $y \neq y_*$.  This approach is a refinement of the weak law of large numbers for cases in which the probabilities converge exponentially fast at a rate given in terms of a function $I$, the so-called ``rate function.''  The ground work for this theory was established by Boltzmann \cite{Boltzmann1877} in his study of the asymptotic properties of multinomials and later applied by Einstein \cite{Einstein1907} in his analysis of fluctuations. Recent years have seen great development of this relatively new field of mathematical probability, including applications in equilibrium statistical mechanics, stochastic processes, and mathematical statistics \cite{Deuschel_and_Stroock,Ellis,Dembo_and_Zeitouni,Dupuis_and_Ellis}.  Ruelle \cite{Ruelle} and Lanford \cite{Lanford1973} have developed similar techniques for studying the equilibrium distributions of dynamical maps, where the (negative) Kolmogorov-Sinai entropy serves as a rate function \cite{Kifer1990}.

In \Sec \ref{ssec:LDT} we define rate functions and the large deviation principle.  The main result is \Thm \ref{thm:LDCC} regarding convergence in probability for conditional probabilities.  In \Sec \ref{ssec:GC} we take up the notion of Gibbs Conditioning, which will allow us to apply \Thm \ref{thm:LDCC} to cases in which we condition on an arbitrary initial macrostate $y$.  Our results are similar to those of \cite{Dembo_and_Zeitouni}, who use the stronger $\tau$-topology on $\mathcal{P}(X)$, and follow from an approach adapted from \cite{Ellis1999}, who considers discrete macrostates.

\subsection{Large Deviation Theory}
\label{ssec:LDT}

For a topological space $(X, \mathcal{T})$, a function $I$ mapping $X$ into $[0, \infty]$ is called a \emph{rate function\/} iff $\inf I(X) = 0$ and $I$ is lower semicontinuous, i.e.\ the preimage $I^{-1}([0, \alpha])$ is closed for all $\alpha \in [0, \infty)$. A \emph{good\/} rate function is one in which these preimages are compact.  The property of being lower semicontinous guarantees that $I$ attains its infimum on any compact set, from which it follows that a good rate function attains its infimum over any closed set \cite[pp.\ 4, 308]{Dembo_and_Zeitouni}.  Any point, $x_*$, at which $I(x_*) = 0$ is called an \emph{equilibrium point\/}, since it corresponds to a state of maximum probability.

A sequence $\seq{P_n}_{n \in \mathbb{N}}$ of probability measures on the Borel subsets of $X$ is said to satisfy a \emph{large deviation principle} with rate function $I$ iff there exists a sequence $\seq{a_{n}}_{n \in \mathbb{N}}$ of positive numbers tending to infinity such that, for any Borel set $A \subseteq X$,
\begin{equation}
\label{eqn:LDP}
-\inf I(\interior{A}) \le \liminf_{n \rightarrow \infty} \frac{1}{a_{n}} \log P_n[A]
\le \limsup_{n \rightarrow \infty} \frac{1}{a_{n}} \log P_n[A] \le - \inf I(\closure{A}).
\end{equation}
A set $A$ for which $\inf I(\interior{A}) = \inf I(\closure{A})$ is called an \emph{$I$-continuity set\/}.  Clearly for such sets we have
\begin{equation}
\label{eqn:LDPCS}
\lim_{n \rightarrow \infty} \frac{1}{a_{n}} \log P_{n}[A] = -\inf I(A).
\end{equation}
This result should be compared with the Boltzmann relation, $S = k_{\mathrm{B}} \log W$, relating the thermodynamic entropy, $S$, to a volume, $W$, in phase space, where $k_{\mathrm{B}}$ is Boltzmann's constant. The quantity $-a_n I$ serves as a negative entropy, while $P_{n}$ may be viewed as a normalized volume measure.

If $0 < \inf I(A) < \infty$, \Eqn(\ref{eqn:LDPCS}) implies that $P_{n}[A]$ converges to zero at least exponentially fast.  Specifically, from \Eqn(\ref{eqn:LDP}) we may deduce the following:  Given any Borel set $A \subseteq X$ and $\varepsilon > 0$, then for all $n$ sufficiently large,
\begin{equation}
\label{eqn:LDULB}
\exp[-a_{n} (1+\varepsilon) \inf I(\interior{A})] \le P_{n}[A] \le
\exp[-a_{n} (1-\varepsilon) \inf I(\closure{A})],
\end{equation}
provided\/ $\inf I(\interior{A}) > 0$ and\/ $\inf I(\closure{A}) < \infty$. Equality for the lower bound may hold only if\/ $\inf I(\interior{A}) = \infty$, where\/ $\e^{-\infty} := 0$, while equality for the upper bound may hold only if\/ $\inf I(\closure{A}) = 0$.

If, for example, there is a unique equilibrium point $x_{*}$, then $x_{*} \not\in \closure{A}$ implies $\inf I(\closure{A}) > 0$. Provided $\inf I(\closure{A}) < \infty$ and taking $\varepsilon = 1/2$, say, this implies $P_{n}[A] \le \exp[-n \inf I(\closure{A})/2]$ for all $n$ sufficiently large. If $\inf I(\closure{A}) = \infty$, then $A$ is an $I$-continuity set and \Eqn (\ref{eqn:LDPCS}) implies $a_{n}^{-1} \log P_{n}[A] \rightarrow -\infty $. Thus, if $x_* \not\in \closure{A}$, then $P_{n}[A] \rightarrow 0$ as $n \rightarrow \infty$, and we recover the weak law of large numbers, i.e.\
\begin{equation}
\label{eqn:LDZOC}
\lim_{n \rightarrow \infty} P_{n}[A] = \left\{
\begin{array}{ll}
  0, & \mbox{if\/ $x_* \not\in \closure{A}$}, \\
  1, & \mbox{if\/ $x_* \in \interior{A}$},
\end{array} \right.
\end{equation}
for any Borel set $A \subseteq X$.

For our purposes, we would like to show a result similar to \Eqn~(\ref{eqn:LDZOC}) for the case in which we condition on a suitable initial condition.  In particular, we would like to obtain a result analogous to \Eqn (\ref{eqn:EMCPCE}). The following theorem, stated in its general form, may be used for this purpose. The proof is deferred to \App \ref{sec:LDCC}.

\begin{thm}
\label{thm:LDCC}
Suppose\/ $\seq{P_{n}}_{n \in \mathbb{N}}$ satisfies a large deviation principle with a good rate function\/ $I$ and a unique equilibrium point\/ $x_*$.  Let\/ $B \subseteq X$ be an $I$-continuity set such that\/ $x_* \not\in \boundary{B}$ and there exists a unique\/ $x_{B} \in \closure{B}$ such that\/ $I(x_{B}) = \inf I(\closure{B}) < \infty$. Given any Borel set\/ $A \subseteq X$,
\begin{equation}
\label{eqn:LDCC}
  \lim_{n \rightarrow \infty} P_{n}[A \:| B] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if\/ $x_{B} \not\in \closure{A}$,} \\
    1, & \mbox{if\/ $x_{B} \in \interior{A}$.}
  \end{array}
  \right.
\end{equation}
\end{thm}

In the next section, we shall consider a large deviation principle for the sequence $\seq{\mu^n \circ L_n^{-1}}_{n \in \mathbb{N}}$ of distributions of empirical measures.  Conditioning on $A_{\delta} = E_{g}^{-1}(B_{\delta})$ will then give rise to a new equilibrium probability measure, $P_{\lambda}$, in general different from $\mu$.  We shall show that under these conditions \Thm \ref{thm:LDCC} is satisfied, thus establishing convergence of $L_{n}$ in conditional probability to $P_{\lambda}$.

\subsection{Gibbs Conditioning}
\label{ssec:GC}

In his 1877 paper, Boltzmann proved that the asymptotically most probable configuration for a gas of $n$ particles with a finite number of macrostates is given by a multinomial distribution. Sanov's theorem \cite[p.\ 70]{Deuschel_and_Stroock}, a modern refinement of this classic result, states that the sequence $\seq{\mu^n \circ L_n^{-1}}_{n \in \mathbb{N}}$ of distributions of empirical measures satisfies a large deviation principle with rate function $I_{\mu}: \mathcal{P}(X) \rightarrow [0, \infty]$. Here $I_{\mu}(P)$ is the (negative) Gibbs entropy of $P$ with respect to $\mu$, defined by
\begin{equation}
I_{\mu}(P) := \left\{
\begin{array}{ll}
 \int_{X}   \frac{\d P}{\d \mu} \log \frac{\d P}{\d \mu} \;\d \mu, & \mbox{if $P \prec \mu$,} \\
 \infty, & \mbox{otherwise}, \\
\end{array}
\right.
\end{equation}
where $0 \log 0 := 0$. It can be shown that $I_{\mu}$ is a good, strictly convex, rate function \cite[p.\ 240]{Dembo_and_Zeitouni} which attains its infimum uniquely at $\mu$ \cite[pp.\ 32--34]{Dupuis_and_Ellis}.

Given $y \in \interior{Y}$ and $\delta > 0$, let $A_{\delta} = E_{g}^{-1}(B_{\delta})$, where, $B_{\delta} = (y - \delta, y]$ when $y < y_*$, $B_{\delta} = [y, y + \delta)$ when $y > y_*$, and $B_{\delta} = (y_* - \delta, y_* + \delta)$ when $y = y_*$. (Note that $\mu \not\in \boundary{A_{\delta}}$, since $y_* \not\in \boundary{B_{\delta}}$ and $E_{g}$ is continuous at $\mu$.) We wish to consider the asymptotic behavior of the conditional probability $(\mu^{n} \circ L_{n}^{-1})[A \:| A_{\delta}] = \mu^{n}[\set{L_{n} \in A} \:| \set{L_{n} \in A_{\delta}}]$, as $n \rightarrow \infty$, for any Borel set $A \subseteq \mathcal{P}(X)$. For example, if $G$ is the macroscopic average energy of the system, then $\mu^n [\;\cdot\; \:| \set{G \in B_{\delta}}]$ is the microcanonical distribution on the ``thickened'' energy shell with energy $y$.

We will show that, conditioned on $\set{G \in B_{\delta}}$, the empirical measures $\seq{L_n}_{n \in \mathbb{N}}$ converge in probability to the canonical Gibbs measure $P_{\lambda}$, where $\lambda$ satisfies the constraint $y = \int_{X} g \;\d P_{\lambda}$ and
\begin{equation}
\d P_{\lambda}(x) := \frac{\e^{\lambda g(x)}}{Z(\lambda)} \;\d \mu(x).
\end{equation}
The normalization factor $Z(\lambda) := \int_{X} \e^{\lambda g} \;\d \mu$ is the \emph{partition function}, and the quantity $\Psi(\lambda) := \log Z(\lambda)$ is the \emph{generalized free energy}.  Note that, if $G$ is the macroscopic average energy, then $-k_{\mathrm{B}} T \: \Psi(-1/(k_{\mathrm{B}} T))$ is the familiar Helmholtz free energy at temperature $T$.

Denote by $\theta$ the map $\theta: [-\infty, +\infty] \rightarrow Y$ which associates a given $\lambda$ with a certain value of $y$ and is defined by $\theta(\lambda) := \int_{X} g \;\d P_{\lambda}$ for $\lambda \in \mathbb{R}$, $\theta(-\infty) := \ymin$, and $\theta(+\infty) := \ymax$.  The following lemmas will be needed.  The proofs are deferred to \App \ref{sec:GCL}.

\begin{lem}
\label{lem:FEL}
If\/ $\Psi''(\lambda)$ is nonzero for all\/ $\lambda \in (-\infty, +\infty)$, then the map\/ $\theta$ is invertible. Furthermore,\/ $y < y_*$ iff\/ $\lambda < 0$,\/ $y = y_*$ iff\/ $\lambda = 0$,\/ and $y > y_*$ iff\/ $\lambda > 0$.
\end{lem}

\begin{lem}
\label{lem:GIL}
Given\/ $y \in \interior{Y}$, let\/ $A_{\delta} = E_{g}^{-1}(B_{\delta})$ and\/ $\lambda = \theta^{-1}(y)$.  Then\/ $I_{\mu}(P_{\lambda}) = \inf I_{\mu}(\closure{A_{\delta}}) < \infty$ and\/ $I_{\mu}(P_{\lambda}) < I_{\mu}(P)$ for all\/ $P \in \closure{A_{\delta}} \setminus \set{P_{\lambda}}$,\/ where $\lambda = \theta^{-1}(y)$.
\end{lem}

\begin{lem}
\label{lem:CSL}
Given\/ $y \in \interior{Y}$, the set\/ $A_{\delta} = E_{g}^{-1}(B_{\delta})$ is an\/ $I_{\mu}$-continuity set.
\end{lem}

Using the above lemmas we may deduce that, given $y \in \interior{Y}$, $A_{\delta} = E_{g}^{-1}(B_{\delta})$ is such that $\mu \not\in \boundary{A_{\delta}}$ and, for $\lambda = \theta^{-1}(y)$, $P_{\lambda}$ is the unique measure in $\closure{A_{\delta}}$ such that $I_{\mu}(P_{\lambda}) = \inf I(\closure{A_{\delta}}) = \inf I(\interior{A_{\delta}}) < \infty$. By \Thm \ref{thm:LDCC} we conclude that for any Borel set $A \subseteq \mathcal{P}(X)$,
\begin{equation}
\label{eqn:EMCPCN}
  \lim_{n \rightarrow \infty} \mu^n[\set{L_n \in A} \:| \set{L_n \in A_{\delta}}] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $P_{\lambda} \not\in \closure{A}$,} \\
    1, & \mbox{if $P_{\lambda} \in \interior{A}$.} \\
  \end{array}
  \right.
\end{equation}
In particular, for any Borel set $B \subset Y$,
\begin{equation}
\label{eqn:MACPCN}
  \lim_{n \rightarrow \infty} \mu^n[\set{G_{t} \in B} \:| \set{G \in B_{\delta}}] =
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\psi_{t}(y) \not\in \closure{B}$,} \\
    1, & \mbox{if $\psi_{t}(y) \in \interior{B}$,} \\
  \end{array}
  \right.
\end{equation}
where
\begin{equation}
\label{eqn:DC}
\psi_{t}(y) := \int_{X} g \circ \varphi_{t} \;\d P_{\theta^{-1}(y)}.
\end{equation}
Note that $\psi_{0}(y) = y$ since $\varphi_{0}$ is the identity on $X$.  Conditioning on $\set{G_{t_0} \in B_{\delta}}$ merely shifts the time axis, in which case $G_{t}$ converges to $\psi_{t-t_0}(y)$ in probability.

\section{The Deterministic Curve}
\label{sec:DC}

We have shown that, conditioned on $G \in B_{\delta}$, the macrostate $G_{t}$ at a given time, $t$, converges in probability to the expectation value $\psi_t(y)$, where $y$ is contained in $B_{\delta}$. Hence, of the initial microstates consistent with the initial macrostate, $y$, ``most'' will be such that the actual macrostate realized will be near this value. Now, each microstate, $(x_1, \ldots, x_n)$, gives rise to a collection, $\set{G_{t}(x_1, \ldots, x_n) : t \in T}$, of macrostates constituting a single trajectory. Likewise, each macrostate, $y$, gives rise to a collection, $\set{\psi_t(y) : t \in T}$, of expected macrostates, which we shall call the \emph{deterministic curve}. This graph represents the asymptotically deterministic behavior of the macrostate. In this section we shall investigate in what sense the deterministic curve is representative of a typical trajectory and consider several properties of the deterministic curve itself, considered as a function of time.

In general, there may be striking qualitative differences between the deterministic curve and a particular macrostate trajectory.  Suppose $\mu$ is invariant under $\varphi_t$ and $\mu^n[G^{-1}(B_{\delta})] > 0$.  The Poincar\'{e} recurrence theorem tells us that for some unbounded sequence $\tau_1, \tau_2, \ldots$ of times
\begin{equation}
\mu^n[\set{G_{\tau_1} \in B_{\delta},\: G_{\tau_2} \in B_{\delta},\: \ldots} \:| \set{G \in
B_{\delta}}] =  1,
\end{equation}
i.e.\/, the macrostate returns infinitely often to a neighborhood of its initial value, $y$, almost surely. Now suppose the map $t \mapsto \psi_t(y)$ has an attracting set $A$ with domain of attraction $D$ and take $B_{\delta} \subseteq D \setminus U$, where $U$ is a neighborhood of $A$.  For all $t$ sufficiently large, $\psi_t(y)$ will be in $U$, but $G_{t}$ will almost surely fall outside $U$ on the recurrence times $\tau_1, \tau_2, \ldots$. These recurrence times will depend upon $B_{\delta}$ and $\varphi_t$, of course, but typically increase rapidly with $n$. On a time scale small compared to $\tau_1$, one then expects the deterministic curve to be quite representative of a typical trajectory.  On time scales larger than $\tau_1$, however, the deterministic curve will be qualitatively quite different from a typical trajectory; the former converges to an attracting set while the latter exhibits quasi-periodic behavior.  This highlights the importance of a clearer understanding of the correspondence between the very distinct $n < \infty$ and $n = \infty$ cases.

\subsection{Convergence to the Deterministic Curve}
\label{ssec:CDC}

Consider a finite set, $\set{t_1, \ldots, t_m}$, of times and for each time $t_i$ let $B_i \subseteq Y$ be a Borel set.  The set of microstates in which $G_{t_i} \in B_i$ for all $i$ is given by
\[
\bigcap_{i=1}^{m} \set{G_{t_i} \in B_i} = \set{ L_n \in \bigcap_{i=1}^{m} A_i},
\]
where $A_i = E_{g \circ \varphi_{t_i}}^{-1}(B_i)$.  According to \Eqn(\ref{eqn:EMCPCN}), with $A = \bigcap_{i=1}^{m} A_i$, the limiting conditional probability will be zero if $P_{\lambda} \not\in \closure{\bigcap_{i=1}^{m} A_i}$, where $\lambda = \theta^{-1}(y)$, and it will be one if $P_{\lambda} \in \interior{\left(\bigcap_{i=1}^{m} A_i\right)}$. Since the intersection is over a finite number of sets, we may use the fact that $\interior{\left(\bigcap_{i=1}^{m} A_i\right)} = \bigcap_{i=1}^{m} \interior{A_i}$ and $\closure{\bigcap_{i=1}^{m} A_i} \subseteq \bigcap_{i=1}^{m} \closure{A_i}$ to obtain the following:
\begin{equation}
\lim_{n \rightarrow \infty} \mu^{n}\left[\left. \bigcap_{i=1}^{m} \set{G_{t_i} \in B_i} \;\right| \set{G \in B_{\delta}} \right]
=
  \left\{
  \begin{array}{ll}
    0, & \mbox{if $\psi_{t_j}(y) \not\in \closure{B}_j$ for some $j$,} \\
    1, & \mbox{if $\psi_{t_i}(y) \in \interior{B}_i$ for all $i$.} \\
  \end{array}
  \right.
\end{equation}

For a countably infinite set of times, we may again conclude that the conditional probability goes to zero if $\psi_{t_j}(y) \not\in \closure{B_j}$ for some value of $j$, since $\bigcap_{i=1}^{\infty} \set{G_{t_i} \in B_i} \subseteq \set{G_{t_j} \in B_j}$. However, even if $\psi_{t_i}(y) \in \interior{B_i}$ for all $i$ and hence $P_{\lambda} \in \bigcap_{i=1}^{\infty} \interior{A_i}$, this does not assure us that $P_{\lambda} \in \interior{\left(\bigcap_{i=1}^{\infty} A_i\right)}$.  For the latter to be true, there must be a neighborhood of $P_{\lambda}$ which is sufficiently small so that it is contained in every $\interior{A_i}$.  Let us consider, then, conditions for which this is true.

Suppose each $B_i$ is an open interval of radius $\delta'$ centered at $\psi_{t_i}(y)$.  Now, each corresponding $A_i$ is an open set of probability measures which give an expectation of $g \circ \varphi_{t_i}$ in the open ball $B_i$.  Loosely speaking, if $g \circ \varphi_{t_i}$, and hence $E_{g \circ \varphi_{t_i}}$, varies too rapidly, then $A_i$ will be small as a result.  If, therefore, the functions $\set{g \circ \varphi_{t_i}}_{i \in \mathbb{N}}$ are somehow limited in how rapidly they may vary, then one might expect the sizes of $A_1, A_2, \ldots$ to be bounded from below, thus giving a nonempty interior for their intersection.  For a general metric space, $(X,d)$, one way to characterize how rapidly a function, $f$, may vary by its Lipschitz norm, $\|f\|_{\mathrm{L}}$, define as follows:
\begin{equation}
\|f\|_{\mathrm{L}} := \sup_{x \neq x'} \frac{|f(x)-f(x')|}{d(x, x')}.
\end{equation}
Differentiable functions will be Lipschitz if they have bounded derivatives; discontinuous functions, such as indicator functions, have infinite Lipschitz norm provided $d(x,x')$ may be made arbitrarily small.

To ensure well-defined expectations we require the functions to be bounded as well, so consider instead the bounded Lipschitz norm, $\|\cdot\|_{\mathrm{BL}}$, defined simply by $\|f\|_{\mathrm{BL}} := \|f\|_{\mathrm{L}} + \|f\|_{\infty}$.  (For a discussion of this norm, see Dudley \cite{Dudley}).  A set, $\set{f_i}_{i \in \mathbb{N}}$, of functions will be called \emph{uniformly bounded Lipschitz} if there exists a number, $K$, such that $\|f_i\|_{\mathrm{BL}} \le K$ for all $i \in \mathbb{N}$.  The following theorem states that for such observables the macroscopic trajectories converge in conditional probability to the deterministic curve on any countable set of times.

\begin{thm}
\label{thm:CIC}
If $\set{g \circ \varphi_{t_i}}_{i \in \mathbb{N}}$ are uniformly Lipschitz bounded functions, then for any $t_1, t_2, \ldots$,
\begin{equation}
\lim_{n \rightarrow \infty} \mu^{n}[ \set{\,|G_{t_i} - \psi_{t_i}(y)| < \delta',\, \forall i \in \mathbb{N}} \:| \set{G \in B_{\delta}}] = 1.
\end{equation}
\end{thm}

As noted above, this result may be proven by finding an open ball, $B_{\rho}(P_{\lambda}, \varepsilon)$, about $P_{\lambda}$ which is contained in every $A_i$.  Consider an arbitrary probability measure, $P$, in  $B_{\rho}(P_{\lambda}, \varepsilon)$ and observe that, since we have assumed $\|g \circ \varphi_{t_i}\|_{\mathrm{BL}} \le K$,
\begin{eqnarray*}%
|E_{g \circ \varphi_{t_i}}(P) - \psi_{t_i}(y)| &\le& \sup \set{|E_{f}(P) - E_{f}(P_{\lambda})| : \|f\|_{\mathrm{BL}} \le K} \\
&=& K \sup \set{|E_{f}(P) - E_{f}(P_{\lambda})| : \|f\|_{\mathrm{BL}} \le 1} \\
&\le& 2 K \rho(P, P_{\lambda}) < 2 K \varepsilon,
\end{eqnarray*}%
where the last inequality follows from \cite[pp.\ 310, 322]{Dudley}, since $(X,d)$ is a separable metric space.  Taking $\varepsilon \le \delta'/(2 K)$ shows that $E_{g \circ \varphi_{t_i}}(P) \in B_i$ for all $P \in B_{\rho}(P_{\lambda}, \varepsilon)$ and hence that $B_{\rho}(P_{\lambda}, \varepsilon) \subseteq A_i$.  Since $\varepsilon$ is independent of $i$, we conclude that $P_{\lambda} \in B_{\rho}(P_{\lambda}, \varepsilon) \subseteq \interior{\left(\bigcap_{i=1}^{\infty} A_i\right)}$.

For discrete time maps with suitable observables, \Thm \ref{thm:CIC} shows that the macrostate trajectories converge in conditional probability \emph{everywhere} to the deterministic curve.  This result may seem surprising, since we have seen that the long-time behavior of a typical trajectory may differ radically from that of the deterministic curve.  However, this simply means that the time scale on which the macroscopic behavior appears deterministic grows rapidly with the number of particles.

The case in which the set of times, $T$, takes on a continuum of values is somewhat more problematic owing to the fact that an arbitrary intersection of measurable sets need not be measurable.  If, however, $\set{G_{t} : t \in T}$ is sample continuous on an interval, $T$, i.e.\ if $t \mapsto G_{t}(x_1, \ldots, x_n)$ is continuous for every $(x_1, \ldots, x_n)$, then we may consider $\set{G_{t} : t \in T}$ to be a random process on the metric space of bounded continuous functions with the supremum norm.  For physical observables this is quite reasonable to suppose.  Much as in the theory of Brownian motion, we may then consider events of the form $\set{\sup_{t \in T} |G_{t} - \psi_t(y)| \ge \delta'}$ and their corresponding conditional probabilities as $n \rightarrow \infty$.  It may then be possible to derive a large deviation principle on the set of bounded continuous functions on $T$, much as is done for Brownian motion in Schilder's theorem \cite{Dembo_and_Zeitouni}, but here we do not pursue this matter further.

\subsection{Properties of the Deterministic Curve}
\label{ssec:PDC}

In the previous section we considered the probabilistic convergence of macrostate trajectories to the asymptotic deterministic curve.  In this section we consider properties of the deterministic curve, $\set{\psi_t(y) : t \in T}$ as a function of $t$ in its own right.  Of course, we do not expect these properties to necessarily carry over to those of typical trajectories.  Nevertheless, they do give a clue to the behavior of these trajectories for large $n$ and relatively small $t$.

If $g$ is a discontinuous function, then specific realizations of $\set{G_{t} : t \in T}$ will also be discontinuous.  Since $G$ is the average of a bounded function, however, the size of these discontinuities will vanish as $n \rightarrow \infty$.  It is then reasonable to suppose that the deterministic curve, $\set{\psi_t(y) : t \in T}$, will be continuous in $t$.

Recall that $g$ and $\varphi_t$ are continuous $\mu$-almost everywhere.  Since $g$ is bounded $\mu$-a.e.\/, clearly $g \circ \varphi_t$ is so as well.  If we suppose $t \mapsto g(\varphi_t(x))$ is continuous for $\mu$-a.e.\ $x \in X$, then of course $g(\varphi_{t'}(x)) \rightarrow g(\varphi_t(x))$ as $t' \rightarrow t$ for $\mu$-a.e.\ $x \in X$.  By \Eqn~(\ref{eqn:DC}) and Lebesgue dominated convergence, this implies
\begin{equation}
\lim_{t' \rightarrow t} \psi_{t'}(y) = \lim_{t' \rightarrow t} \int_X g \circ \varphi_{t'} \;\d P_{\lambda} = \int_X g \circ \varphi_t \;\d P_{\lambda} = \psi_t(y),
\end{equation}
where $\lambda = \theta^{-1}(y)$.  Thus, $t \mapsto \psi_t(y)$ will be continuous provided $y \in \interior{Y}$ and $t \mapsto g(\varphi_t(x))$ is continuous for $\mu$-a.e.\ $x \in X$.

Now suppose only that $t \mapsto \varphi_t(x)$ is continuous for $\mu$-a.e.\ $x \in X$ and that $\varphi_t$ is $\mu$-nonsingular, i.e.\ $\mu \circ \varphi_t^{-1} \prec \mu$, for all $t \in T$.  Since $g$ is continuous $\mu$-a.e.\/, there exists an open set, $A$, with null complement such that $g$, restricted to $A$, is continuous.  Furthermore, since $\varphi_t$ is continuous $\mu$-a.e.\/, there is a set, $B$, with null complement such that $t \mapsto \varphi_t(x)$ is continuous for all $x \in B$. Thus, if $x \in \varphi_t^{-1}(A) \cap B$, then the composite map $t \mapsto \varphi_t(x) \mapsto g(\varphi_t(x))$ is continuous.  Since $\varphi_t$ is $\mu$-nonsingular,
\[
\mu[X \setminus (\varphi_t^{-1}(A) \cap B)]
     \le \mu[X \setminus \varphi_t^{-1}(A)] + \mu[X \setminus B]
     = \mu[\varphi_t^{-1}(X \setminus A)] = 0.
\]
Thus, $t \mapsto g(\varphi_t(x))$ is continuous for $\mu$-a.e.\ $x \in X$, and we conclude the following:

\begin{thm}
If $t \mapsto \varphi_t(x)$ is continuous for $\mu$-a.e.\ $x \in X$ and $\varphi_t$ is $\mu$-nonsingular for all $t \in T$, then $t \mapsto \psi_t(y)$ is continuous for all $y \in \interior{Y}$.
\end{thm}

For many physical systems, the dynamical law $\varphi_t$ is not only invertible but also time reversible in the sense that $\varphi_t^{-1} = \varphi_{-t}$.  In such cases, we shall say that $\varphi_t$ is \emph{time reversible\/}. Time reversibility often appears in physical systems which have the added property that $\varphi_t \circ R = R \circ \varphi_{-t}$ for some involution $R = R^{-1}$, a property referred to as \emph{time reversal invariance\/}.  When such an $R$ exists, every particle microstate $x$ has a mirror point $R(x)$ such that $\varphi_t(x) = R(\varphi_{-t}(R(x)))$; hence, the trajectory of $x$ is mirrored by the trajectory of $R(x)$ with the direction of time reversed. We may then partition $X$ into disjoint sets $X_0 = R(X_0)$, $X_1$ and $R(X_1)$. If the observable and \textit{a priori\/} measure are invariant under $R$, i.e.\ $g = g \circ R$ and $\mu \circ R^{-1} = \mu$, then a typical initial microstate $(x_1, \ldots, x_n) \in \set{G \in B_{\delta}}$ will include roughly equal numbers of points from $X_1$ and $R(X_1)$. On this basis, one expects the trajectories $\set{G_{t}(x_1, \ldots, x_n) : t \ge 0}$ and $\set{G_{t}(x_1, \ldots, x_n) : t \le 0}$ to be similar, though not identical, when $n$ is large.  This suggests that the deterministic curve should be perfectly symmtric in time.  Indeed, it is easy to see that, if $\varphi_t$ is time reversal invariant under $R$ and both $g$ and $\mu$ are invariant under $R$, then
\begin{eqnarray*}%
\psi_t(y)
&=& \int_{X} (g \circ R \circ \varphi_t) \frac{\e^{\lambda g \circ R}}{Z(\lambda)} \;\d \mu \\
&=& \int_{X} (g \circ \varphi_{-t} \circ R) \frac{\e^{\lambda g \circ R}}{Z(\lambda)} \;\d (\mu \circ R^{-1}) \\
&=& \psi_{-t}(y).
\end{eqnarray*}%
Thus, time reversal invariance is sufficient for time symmetry of the deterministic curve, provided both the observable and the \textit{a priori} measure are invariant under $R$.

Suppose $g$ is a simple function of the form $g = \sum_{i=1}^{m} a_{i} 1_{C_i}$, where each $a_i$ is distinct and $C_1, \ldots, C_m$ form a partition of $X$.  Now, the deterministic curve for this $g$ is given by
\begin{equation}
\psi_t(y) = \sum_{i=1}^{m} \sum_{j=1}^{m} a_i \e^{\lambda a_j} \mu[\varphi_t^{-1}(C_i) \cap C_j] \;/\; \sum_{k=1}^{m} \e^{\lambda a_k} \mu[C_k],
\end{equation}
using \Eqn~(\ref{eqn:DC}).  Since we have supposed $g \circ R = g$, notice that $R^{-1}(C_i) = C_i$ for each $i$.  Furthermore, since $\mu$ is invariant under $R$,
\begin{eqnarray}%
\mu[\varphi_{t}^{-1}(C_i) \cap C_j]
&=& \mu[R^{-1}(\varphi_{t}^{-1}(C_i)) \cap R^{-1}(C_j)] \nonumber \\
&=& \mu[\varphi_{t}(R^{-1}(C_i)) \cap C_j] \nonumber \\
&=& \mu[\varphi_{t}(C_i) \cap C_j].
\end{eqnarray}%
Furthermore, if $\mu$ is invariant under $\varphi_t$, then
\begin{equation}
\mu[\varphi_{t}^{-1}(C_i) \cap C_j] = \mu[C_i \cap \varphi_{t}(C_j)].
\end{equation}
The system therefore exhibits \emph{strong detailed balance} in the sense that
\begin{equation}
\label{eqn:SDB}
\mu[\varphi_t^{-1}(C_i) \cap C_j] = \mu[C_i \cap \varphi_t^{-1}(C_j)] \;\;\;\mbox{for all $i$, $j$, and $t$}.
\end{equation}

Conversely, if we suppose only that $\varphi_t^{-1} = \varphi_{-t}$ exists and preserves $\mu$, then \Eqn~(\ref{eqn:SDB}) implies time symmetry of the deterministic curve when $g$ is simple.  For example, suppose $g = a_1 1_{C_1} + a_2 1_{C_2}$ has only two possible states, with $C_1 = C$ and $C_2 = X \setminus C$.  For any $\varphi_t$ which preserves $\mu$,
\begin{eqnarray*}%
\mu[\varphi_t^{-1}(X \setminus C) \cap C]
&=& \mu[C] - \mu[\varphi_t^{-1}(C) \cap C] \\
&=& \mu[\varphi_t^{-1}(C)] - \mu[C \cap \varphi_t^{-1}(C)] \\
&=& \mu[(X \setminus C) \cap \varphi_t^{-1}(C)].
\end{eqnarray*}%
Thus, $g$ exhibits strong detailed balance. The corresponding deterministic curve will be time symmetric provided $\varphi_{t}^{-1} = \varphi_{-t}$ exists.  In general, macroscopic averages of two-state single-particle observables are always time symmetric, provided the dynamics are time reversible and preserve the \textit{a priori} measure.  A system which exhibits strong detailed balance need not, however, be time reversal invariant.  (Consider $C = [0,1]$ and $\varphi_t(x) = x+t$ on $\mathbb{R}$.  The only possible $R$ is $R(x) = -x$, yet clearly $R(C) \neq C$.)  Thus, time reversal invariant systems form a proper subset of all systems exhibiting time symmetry.

Finally, let us consider the asymptotic behavior of the deterministic curve for large $t$.  Suppose once more that $\mu$ is invariant under $\varphi_{t}$. The collection, $\set{\varphi_t : t \in T}$,  of maps is said to be \emph{mixing} with respect to $\mu$ if, for any measurable subsets $A$ and $B$ of $X$,
\begin{equation}
\lim_{t \rightarrow \infty} \mu[\varphi_t^{-1}(A) \cap B] = \mu[A] \; \mu[B].
\end{equation}
From this definition and the fact that $P_{\lambda} \prec \mu$, it follows that \cite[p.\ 72]{Lasota_and_Mackey}
\begin{equation}
\lim_{t \rightarrow \infty} \int_{X} g \circ \varphi_t \;\d P_{\lambda} = \int_{X} g
\;\d \mu = y_*.
\end{equation}
If $\varphi_t$ is time reversible, then this result clearly holds in the limit $t \rightarrow -\infty$ as well. By its definition, mixing is both a necessary and sufficient condition for $\psi_t(y)$ to converge to $E_{g}(\mu) = y_*$ for \emph{any\/} $g$. In cases where the dynamics are not mixing, however, one may still have convergence in time for a restricted set of macroscopic functions.

As we have seen in our discussion of Poincar\'{e} recurrence, convergence in time for $\psi_t(y)$ need not imply convergence in time for $G_{t}$. In fact, such behavior is often quite unlikely. Nevertheless, on a short enough time scale, a scale which increases with $n$, and for large enough $n$, both $\psi_t(y)$ and $G_{t}$ will appear to converge to the same limit along the same trajectory.

\section{Fractional Occupations}
\label{sec:FO}

Consider the case $g = 1_{C}$, for which $G_{t}$ is the fraction of points in $C$ at time $t$.  Since $\e^{\lambda g} = 1_{X \setminus C} + \e^{\lambda}1_{C}$, the corresponding partition function is
\begin{equation}
\label{eqn:FOPF}
Z(\lambda) = \mu[X \setminus C] + \e^{\lambda}\mu[C].
\end{equation}
Recall that $\theta(\lambda) = \Psi'(\lambda) = \e^{\lambda} \mu[C] / Z(\lambda)$ for $\lambda \in (-\infty, \infty)$; thus, for $y \in (0,1)$,
\begin{equation}
\label{eqn:FOTI}
\theta^{-1}(y) = \log \left[ \frac{y}{\mu[C]} \frac{\mu[X \setminus C]}{1-y}
\right].
\end{equation}
Using \Eqns (\ref{eqn:DC}), (\ref{eqn:FOPF}), and (\ref{eqn:FOTI}), we find
\begin{equation}
\label{eqn:FODC}
\psi_{t}(y) = y \: \mu[\varphi_{t}^{-1}(C) \:| C] + (1-y) \:
\mu[\varphi_{t}^{-1}(C) \:| X \setminus C].
\end{equation}
This result is easily understood as follows:  The expected number of points in $C$ at time $t$ will be the number of points starting in $C$ times the fraction of those points expected to be in $C$ at time $t$ plus the number initially outside $C$ times the fraction of those points expected to be in $C$ at time $t$.

The derivation of \Eqn (\ref{eqn:FODC}) was valid for $y \in (0,1)$; using \Eqn (\ref{eqn:ECM}) we can see that it is valid for $y \in \set{0, 1}$ as well. Consider conditioning on $\set{G=1}$ and note that this is equivalent to conditioning on $C^{n}$, since $G(x_1, \ldots, x_n)=1$ iff $x_i \in C$ for all $i$. The conditional distribution of $G_{t}$ is therefore binomial with parameters $n$ and $\mu[\varphi_{t}^{-1}(C) \:| C]$. By the strong law of large numbers, $G_{t}$, conditioned on $\set{G=1}$, converges almost surely, hence in probability, to $\mu[\varphi_{t}^{-1}(C) \:| C]$.  A similar argument shows that $G_{t}$, conditioned on $\set{G=0}$, converges to $\mu[\varphi_{t}^{-1}(C) \:| X \setminus C]$.  We note in passing that the distribution of $\sqrt{n} G_{t}$ is asymptotically Gaussian by the central limit theorem, provided these transition probabilities are neither 0 nor 1.  Thus, deviations from the deterministic curve go to zero as $1/\sqrt{n}$ when $y \in \set{0,1}$.

In \Fig \ref{fig:theta} we have plotted $\theta$ versus $\lambda$ for three different values of $\mu[C] = \theta(0)$. The strict monotonicity of the curve implies $\theta$ is invertible, in accordance with \Lem \ref{lem:FEL}. If, for example, $g = 1_{C}$ is the energy of a two state particle, with energies 0 and 1, then $\lambda = -(k_{\mathrm{B}} T)^{-1}$, where $T$ is the absolute temperature and $k_{\mathrm{B}}$ is Boltzmann's constant. An initial macrostate with energy density, $y$, near zero corresponds to a heavily populated low energy state and hence a low, positive temperature ($\lambda \ll 0$). If $y = \mu[C]$, then the initial macrostate is at its \textit{a priori\/} most likely state; if $\mu$ is invariant under $\varphi_t$, this corresponds to macroscopic equilibrium. Note that $\lambda \rightarrow 0^-$ corresponds to $T \rightarrow +\infty$; at high temperatures the particles are uniformly distributed, with respect to $\mu$, between the two energy states.  For macrostates beginning above equilibrium, i.e.\ $y > \mu[C]$, there is an effective population inversion similar to that found in systems of weakly coupled magnetic dipoles. Systems with small negative temperatures ($\lambda \gg 0$) tend to have more densely populated high energy states, while large negative temperatures ($\lambda \to 0^{+}$) again correspond to near equilibrium conditions. The single parameter $\mu[C]$ determines the asymmetry between states below equilibrium and those above. Thus, we see that $\lambda$ plays a role in defining initial macrostates which is analogous to that of temperature in defining equilibrium states.

Notice that the function $\theta$ is associated only with the initial macrostate.  The time-evolved behavior of fractional occupations is contained in the two transition probabilities in \Eqn (\ref{eqn:FODC}).  In general, these may be difficult to determine.  For the baker map, $\phi$, a discrete time map on the unit square \cite{Arnold_and_Avez}, these may be computed for rectangular cells with Lebesgue measure \cite{LaCour1999}.  This allows one to calculate $\psi_t(y)$ for several time iterations and to compare this with Monte Carlo simulations.  Although an abstract map, the baker map shares many of the relevant features of more realistic Hamiltonian dynamical systems.  In particular, it is Lebesgue measure preserving, mixing, and time reversal invariant in the sense that $\phi \circ R = R \circ \phi^{-1}$ for $R(x,y)=(y,x)$.

In \Fig \ref{fig:baker} we have plotted $\psi_t(y)$ for $t = -10, \ldots, 10$ and $y = 0.4$ using the baker map.  (Since the map is invertible, negative times refer to iterations of the inverse map.) The cell was chosen arbitrarily to be $C = [0.2,\, 0.6) \times [0.0,\, 0.5)$, for which $\mu[C] = 0.2$. The values of $\psi_t(y)$ are connected by straight solid lines in the figure. For comparison, a single realization of an ensemble of $n = 50,000$ points was generated which satiosfied the initial macrostate $y = 0.4$. This was done by drawing the first ${\lfloor n y \rfloor}$ points uniformly from $C$ and then drawing the rest from outside $C$.  (Here, ``uniformly'' means with respect to Lebesgue measure.) Once generated, the known form of the map $\varphi_t := \phi^t$ was used to time evolve the initial ensemble for each value of $t$. The fractional occupation, $G_{t}$, was then computed for each time-evolution of the initial ensemble and is indicated by a solid dot in the figure.

The qualitative behavior of $\psi_t(y)$ in \Fig \ref{fig:baker} is particularly notable in two regards.  First, it is readily observed that the plot is symmetric about $t = 0$; in particular, $\psi_{-t}(y) = \psi_{t}(y)$ exactly, while $G_{-t}$ and $G_{t}$ are only approximately equal.  This, as was shown in \Sec \ref{ssec:PDC}, is a general property of two-state systems for which $\varphi_t$ is $\mu$-measure preserving and time reversible.  Hence, there is no distinction between the forward and reverse time directions.  The second observation is that $\psi_t(y) \rightarrow \mu[C]$ as $t \rightarrow \pm \infty$, which is a direct consequence of the mixing property.  Thus, the baker map provides a simple model of an equilibrating macroscopic quantity.

A second comment is that, while at each \emph{given\/} time, $t$, the most probable macrostate is $\psi_t(y)$, for any finite $n$ the set $\set{\psi_t(y) : t \in \mathbb{Z}}$ is itself an improbable realization of $\set{G_{t} : t \in \mathbb{Z}}$. This may be understood by observing that, given $\varepsilon > 0$, we have $|\psi_t(y) - \mu[C]| < \varepsilon$ for all $|t|$ sufficiently large, yet, by Poincar\'{e} recurrence theorem, $|\psi_t(y) - \mu[C]| > \varepsilon$ for infinitely many values of $t$, almost surely.

The family of macroscopic maps, $\set{\psi_t: t \in \mathbb{Z}}$, does not form a group, or even a semigroup, in contrast to the family of microscopic maps, $\set{\varphi^t: t \in \mathbb{Z}}$.  Thus, while $x = \varphi_{-t}(\varphi_{t}(x))$, in general $y \neq \psi_{-t}(\psi_{t}(y)) =\psi_{t}(\psi_{t}(y))$, since $\psi_{t}$ is time symmetric.  Furthermore, $\set{\psi_{t}: t \ge 0}$ does not even form a semigroup, since this would imply $|\psi_{t}(y) - y_{*}| \ge |\psi_{t+s}(y) - y_{*}|$ for all $s, t \ge 0$, i.e.\ that all future macrostates are closer to equilibrium than their predecessors.  To understand this note that, while $\psi_{t+s}(y)$ describes the state of an observable at time $t+s$ whose value was $y$ at time zero, $\psi_{s}(\psi_{t}(y))$ describes the state of an observable at time $t+s$ whose value was $\psi_t(y)$ at time $t$.  The latter corresponds to a rerandomization of the original distribution, which removes correlations that would otherwise be preserved by the dynamics and causes disagreement with the actual time evolution of the observable.

The baker map is a discrete time map, whereas the dynamics of physical systems are given by continuous time flows.  A simple example of this type of system is the rotation map on the unit square, which is given by
\begin{equation}
\varphi_t(x_1, x_2) = (x_1 + \omega_1 t,\: x_2 + \omega_2 t) \bmod 1.
\end{equation}
A plot of $\psi_t(y)$ for $\omega_1 = \sqrt{2}$, $\omega_2 = \sqrt{3}$, and $y = 0.4$ is given in \Fig \ref{fig:rotation}.  We again note the perfect symmetry about $t = 0$ found previously in the baker map. Unlike the baker map, however, the expectation value does not converge to an asymptotic value but varies quasi periodically in time. (The flat portions of the graph occur when $\varphi_t^{-1}(C) \cap C$ is empty and hence only $0.6 \times (0.2/0.8) = 0.15$ of the remaining points are expected to be in $C$.) A particular realization using $n = 5,000$ is plotted for comparison.

Near $t = 0$, the deterministic curve is linear with a slope pointing toward the equilibrium value, $y_* = 0.2$, as $|t|$ increases, a behavior which holds generally for any value of $y$.  Thus, the initial tendency of the system is to move monotonically toward the equilibrium value.  Such behavior has been ascribed to Boltzmann's H-function \cite{Ehrenfest}, though extrapolated to include times far from zero as well.  As we have seen from the maps considered here, this extrapolation need not be valid.  Since the H-function is computed from fractional occupations, though, it seems reasonable that monotonicity toward equilibrium should hold at least for $t$ near zero.

\section*{Acknowledgements}

The authors would like to thank D. Driebe and H. Hasagawa for their helpful comments and constructive criticisms.

\section{Discussion}

We have considered a general class of systems composed of identical constituents, here called ``particles,'' that are dynamically noninteracting.  For the microstates of the collective system, we supposed there is an \textit{a priori\/} measure, typically an invariant measure, that describes the distribution of these microstates in the absence of any restrictions based on the given macrostate.  We further supposed that the particles are statistically independent, that any correlations among them arise only by the need to satisfy the given macroscopic constraint.  No attempt was made to justify these assumptions at a more fundamental level, though we believe they are quite reasonable for many physical systems.

What we have shown is that the time evolution of a particular macroscopic variable, namely the average over certain real-valued single-particle functions, is such that it converges in a probabilistic sense to a well defined curve as the number of particles tends to infinity.  Specifically, we have derived a map $\psi_t$ such that, if the macrostate at time 0 is constrained to be near a value $y$, then the macrostate at time $t$ will be in a given neighborhood of $\psi_t(y)$ with a probability approaching one as the number of particles tends to infinity.  The map $\psi_t$ was defined in terms of an expectation with respect to a canonical distribution in which $y$ plays the role of an average energy in the familiar thermodynamic formalism.  The restrictions on the single-particle function were that it be bounded and continuous almost everywhere in a sense specified by the \textit{a priori\/} measure.  We found that the family of macroscopic maps, $\set{\psi_{t}: t \in T}$, in general forms neither a group nor a semigroup, even if the family of microscopic maps, $\set{\varphi_{t}: t \in T}$, has this property.

Having established this basic convergence result for a given time, we then considered how well the deterministic curve, the graph of $\psi_t(y)$ versus $t$, represented the behavior of a typical realization of the macrostates over all time.  We found that the two may differ qualitatively quite substantially; while there may be good agreement on a finite set of selected times, there will typically be times at which they differ substantially.  This was particularly true of mixing systems, for which $\psi_t(y)$ always converges in the long time limit, while a typical trajectory exhibits recurrences. Under some more restrictive conditions we proved convergence on any countably infinite set of times, but even then recurrences are possible when $n$ is finite.

With these caveats on the correspondence between the finite and infinite particles cases, we considered some general properties of the expectation curve as a function of time.  We found that, despite the fact that the macrostates may evolve discontinuously, the deterministic curve may be continuous in time.  We also found that, for systems which are time reversal invariant, the deterministic curve is symmetric in time about $t=0$, the point at which conditioning of initial macrostates takes place.  These properties were then related to familiar geometric properties attributed to Boltzmann's H-curve.

We have not considered extensions of these results to macroscopic variables in, say, $\mathbb{R}^d$, which would involve issues of convexity that make the extension nontrivial.  The general problem of interacting particles poses a  greater difficulty and requires a significant change of methodology, though we conjecture that similar results will hold if $\psi_t(y)$ is defined as a limit of $n$-particle expectations.

redpajama_set_name: RedPajamaArXiv


text:
Swim with a Tiger Cub at One Florida Zoo
By JuJu Kim @jujkimOct. 10, 2012
China Photos / Getty Images
A Siberian tiger cub is fed from a bottle.
Swim With A Tiger Cub at Fla. Private Zoo ABC
Dade City Zoo Offers Chance to Swim with Tiger Cub Bay News 9
There's a very clear reason why most zoos keep their tigers in a cage. But one private Florida zoo is pioneering a new trend in how visitors can get close to the animals. At Dade City's Wild Things, you can get in a pool with one big cat — who happens to be quite small right now.
It's an experience that will set you back $200. But that's a small price to take a plunge with a 6-week-old Siberian tiger cub named Tony, ABC News reported.
You'll have to act quickly if you want the 30-minute experience. The price isn't going up — but Tony's weight is. State law prohibits the public from swimming with Tony after he reaches more than 40 pounds. He's still got some growing up to do, though, tipping the scales right now at just 8 pounds. When the cub weighs 25 pounds, he'll have to wear a leash when interacting with visitors, according to the New York Daily News.
(WATCH: Why Tigers Make Lousy Pets)
So what are the perks of the attraction? For starters, you'll have a great party story. "If you come and encounter little Tony now when he's a baby, you can come back in a few years when he is full grown and tell people, 'I swam with that guy,'" Randy Stearns, the president of Wild Things, told Bay News 9.
The attraction, which also includes the opportunity to bottle-feed and cuddle with the tiger, has drawn visitors from around the world, even Iceland, ABC reported. "I never thought I'd get to touch one, much less swim with one," Bertha Cruz, who took a dip with Tony, told ABC.
Stearns told FOX affiliate WSVN-TV the opportunity is uncommon because many parks aren't equipped to handle cub interaction. "They're very strong, but they're also very well trained," said Stearns, adding that tigers are natural swimmers thanks to their webbed feet. "That's why a lot of other facilities don't let you interact with the cubs because they don't have the staff or the knowledge to do it."
The zoo also offers a $100 swim with Ally, an alligator whose mouth is taped shut, Bay News 9 reported.
(MORE: Too Close for Comfort: Thailand's Tiger Temple)
The animal swims are facing heat from critics, though, and the Florida Fish and Wildlife Conservation Commission (FWC) is investigating the 20-minute alligator swim at Wild Things. There are other gator encounters offered in the state, including the Madeira Beach Alligator Attraction, which the FWC has already investigated, according to Bay News 9.
Bob Barrett, who runs Alligator Attraction, even started a business that brought the reptiles to kids' birthday parties. "It just took off like wildfire," Barrett told ABC. "Everybody loves having the gator pool parties." But the FWC decided Barrett's business was illegal, so it shut down early this month, ABC reported.
Wild Things owner Kathy Stearns is confident her zoo's animal encounters will remain open. She told Bay News 9 the tiger and gator swims are "perfectly safe and perfectly legal."
"We make sure all of the things are in line, plus a keeper is always present within reach at all times," she said.
And the price is comparable to swimming with dolphins — markedly tamer animals. Though the therapy you might need after being in the pool with a tiger could set you back a lot more.
MORE: Ang Lee's 'Life of Pi': The Next Avatar

redpajama_set_name: RedPajamaCommonCrawl


text:
Action Alert, Human Rights, Queer Politics
Ugandan Anti-Homosexuality Bill: What you can do
Africa LGBTIQ
Twitterview with Bisi Alimi – Living Positively with HIV for 10 Years
Diriye Osman: Why we must tell our own stories
US Evangelists are involved in the Anti-Homosexuality Bill – as per this report
Sadly, this witch-hunt has the blood stained fingerprints of leading American evangelicals. The Fellowship, (aka The Family) one of America's most powerful and secretive fundamentalist organization's, converted Uganda's President Yoweri Museveni (pictured top) to its anti-gay brand of Christianity, which is the "intellectual" impetus behind the anti-gay crackdown. The clandestine organization's leader, Doug Coe, calls Museveni The Fellowship's "key man" in Africa. Jeff Sharlet, author of "The Family", writes of the African strongman's conversion:
"So," Doug Coe told us, "my friend said to the president, 'why don't you come and pray with me in America? I have a good group of friends–senators, congressmen–who I like to pray with, and they'd like to pray with you.' And that president came to the Cedars (a religious retreat), and he met Jesus. And his name is Yoweri Museveni…And he is a good friend of the Family."
These people are using Uganda as a proxy site for their war against LGBTI people – Whether you are inside Uganda or outside, where ever you are there is something that you can do to stop this Bill going through but we need to act fast. You can contact your own local parliamentary representative or senator and ask them to intervene – if you are in the US you can contact the President – maybe he can take time out from assembling thousands of his army to look at what is happening in his own yard – and ask him to directly intervene. IGLHRC has posted the facts around the Bill and a list of names and addresses of Ugandan ministers plus a sample letter below. Please take the time to speak out because once passed every LGBTI person in Uganda or anyone who provides a home, job or any kind of support will be criminalised and face between 7 years in prison and the death sentence. That is how horrible this thing is.
The Ugandan Parliament is now considering a homophobic law that would reaffirm penalties for homosexuality and criminalize the "promotion of homosexuality." The Anti-Homosexuality Bill of 2009 targets lesbian, gay, bisexual, and transgender (LGBT) Ugandans, their defenders and anyone else who fails to report them to the authorities whether they are inside or outside of Uganda. The International Gay and Lesbian Human Rights Commission (IGLHRC) and Sexual Minorities of Uganda (SMUG) are calling for the swift dismissal of the bill and human rights protections for all Ugandans, regardless of their sexual orientation or gender identity.
Join the International Gay and Lesbian Human Rights Commission (IGLHRC) and Sexual Minorities of Uganda (SMUG) in calling for the swift dismissal of the Anti-Homosexuality Bill of 2009 and the protection of all Ugandans, regardless of their sexual orientation or gender identity.
Continue to Contacts and Sample Letter
President Yoweri Kaguta Museveni
State House Nakasero
Fax: 256-41-346-102
info@statehouse.go.ug
Prime Minister Apollo Nsibambi
Post Office Building, Yusuf Lule Road
Tel: 256-41-423-6252
Fax: 256-41-434-1139
ps@opm.go.ug
Speaker of the Parliament Edward Ssekandi Kiwanuka
P.O. Box 7178, Parliamentary Avenue
Tel: 256-414-377-100
Fax: 256-414-231-296
speaker@parliament.go.ug
Minister of Gender, Labour, and Social Affairs
Honorable Opio Gabriel
ps@mglsd.go.ug
Med Kaggwa, Chair of the Uganda Human Rights Commission
Plot 20/22/24 Buganda Road
Tel: 256-414-348-007/8
uhrc@uhrc.ug
Directorate for Ethics and Integrity
Tel: 256-(0)41-430-1600
Fax: 256-(0)41-434-3177
info@dei.go.ug
Mathisen Gørild
Chair of the Uganda Diplomatic Human Rights Working Groups
gorild.mathisen@mfa.no
Send a copy to:
Jerry P. Lanier, Ambassador to the Republic of Uganda
Embassy of the United States of America
Tel: 256-414-259-791/2/3/5
kampalawebcontact@state.gov
IGLHRC
communications+action.alert@iglhrc.org
Send an email and fax to:
Perezi K. Kamunanwire, Ambassador to the US
pkamunanwire@ugandaembassyus.org
Dr. Ruhakana Rugunda
Permanent Representative of the Republic of Uganda to the United Nations
336 East 45 Street
ugandaunny@un.int
Sample Letter:
Your Excellencies:
I am writing to express concern about legislation that would severely restrict the rights of Ugandan citizens, including lesbian, gay, bisexual, and transgender (LGBT) people and their defenders, in direct contravention of domestic and international law. The Anti-Homosexuality Bill of 2009 would not only reaffirm penalties for homosexuality, but would criminalize the "promotion of homosexuality," including funding and sponsoring LGBT organizations and broadcasting, publishing, or marketing materials on homosexuality. Any person in authority who fails to report known violations of the law within 24 hours will also be subject to a significant fine and up to 3 years in prison — even when this means turning in their colleagues, family, or friends.
The negative repercussions of the bill in Uganda will be immediate and severe. It effectively bans the free association and expression that are necessary for a flourishing civil society, and creates a climate of fear and hostility that undermines the citizenship and solidarity of all Ugandans. It will lend itself to misapplication and abuse, and implicitly encourages persecution of LGBT people by private actors. Effective HIV prevention activities in Uganda, which rely on an ability to talk frankly about sexuality and provide condoms and other safer-sex materials, will be difficult, if not impossible.
The Anti-Homosexuality Bill violates National Objective 5(2) of the Ugandan Constitution, which provides that "the State shall guarantee and respect the independence of non-governmental organizations which protect and promote human rights." Moreover, it directly violates the right to equality and freedom from discrimination (Article 21), the right to privacy (Article 27), the right to freedoms of speech, expression, association, and assembly (Article 29), the protection of minorities (Article 36), and the protection of civic rights and activities (Article 38) to which all Ugandans are entitled. It also violates the African Charter on Human and People's Rights, the International Covenant on Civil and Political Rights (ICCPR) and other international human rights treaties to which Uganda is a party. This bill undermines Uganda's commitment to the international human rights regime and threatens the basic human rights of all its citizens.
The Bill's revocation of fundamental rights would also seriously undermine the country's reputation and credibility in the international arena. Because it claims jurisdiction over Ugandans who violate its provisions while outside of the country, the Bill will strain Uganda's relations with regional and international partners.
While people may hold differing opinions about sexual orientation and gender identity, the legislation before Parliament is an ineffective and fundamentally illegal way to express opposition to a minority group. In recognition of the importance of a diverse, dynamic civil society and the domestic and international commitments that Uganda has made, I urge you to swiftly dismiss the Anti-Homosexuality Bill of 2009 and reaffirm the rights and responsibilities of all Ugandans.
Action AlertAnti-Homosexuality BillHuman RightsQueer PoliticsUganda
Why didn't this fundamentalist group indoctrinate Musevini to be a tolerant, loving President and not a dictator?


Avène Hydrance Riche helps restore and maintain on-going hydration in sensitive skin through the synergistic action of the active ingredients it contains. It is ideal for dry to very dry sensitive skin dehydrated by various stress factors (irritants, sun, wind etc). Hydrance Riche has a gentle, creamy texture that provides ultimate skin comfort and immediate soothing benefits. With its original balanced restored, the skin regains a softer, smoother, and more supple feel. The unique and stable composition of Avène thermal spring water (eau Avène thermale) makes the Avène skincare range ideal for normal, sensitive and allergy prone skins.


56 pages featuring poetry, personal prose, full-colour illustrations and colour photography.
Contributors include: Faith Arkorful, Doretta Lau, Jen Sookfong Lee, Dacy Lim, Furqan Mohamed, Jenny Elizabeth Scoones, Erin Klassen, Najla Nubyanluv, Elizabeth Polanco and Erin Pehlivan.
Illustrated by Maggie Li, photography by Fehn Foss. Art direction by Erin McPhee.
There are as many paths to healing as there are ways to be wounded, and as many unique stories as there are people. But whether pain is cultural, ancestral, familial, corporeal, romantic, or any category in between, there is a common truth: healing begins with the act of acknowledging our hurt.
Turning our most vulnerable experiences into art gives us power, strength and a better appreciation of how to care for ourselves and each other. Through making art, our pain becomes tangible, and in that way, healing can be shared.
The contributions in this collection offer a window into what healing can look like when we are brave enough to name our suffering.


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Top 5 Bike-Friendly Cities in the Philippines
| Cycling in the Philippines
Emily A.
The call for declaring 2015 as the Air Pollution Awareness Year by doctors and scientists urges the government to address the worsening air quality in the Philippines. Air pollution can expose people to risks of getting cancer as pollutants are proven to be carcinogenic by the WHO International Agency for Research on Cancer. In the Philippines, 80% of the pollution load are emitted by motor vehicles, which greatly contributes to the warming of our environment. Therefore, aside from optimizing anti-smoke belching campaigns, it is also important to raise our awareness on bicycles as a mode of transportation.
Fortunately, it didn't take long for these cities below to realize the importance of having a highway network system where motorists can share the lane with bikers. And to give you an idea on bike-able cities in the Philippines, here's our top 5!
Bike along Diversion Road to get to SM City Iloilo from Esplanade or vice versa. The bike lane is filled up with plant boxes, providing a good air quality in the environment. The city also has recently played host to the 2nd Bike Festival which is about promoting bicycles as a mode of transportation and to raise awareness about road safety among Ilonggos.
The City of Love is on a quest for being the Philippine's Bike Capital by installing more bike lanes on its busier roads like Diversion Road in Mandurriao, the city Business Park, and thoroughfares around the local tourist spots. Walkways are also in progress in an effort to secure the safety of pedestrians and joggers.
Marikina City
It is no secret that Marikina is on its way to becoming a major bike-friendly city on the country. When the Marikina River got rehabilitated in the 1990's, the local government had installed a 10-km jogging and bike lane, too. People have then begun to spend some quiet time by the river and the site became a recreational destination for families.
In 2007, Marikina bid to become a bicycle-friendly and walkable city by improving its roadways and walkways. The local government unit had offered a bicycle for rent program for its employees and held in-school activities to educate the youth about the benefits of cycling and its how-to's. Today, Marikina is proud of its clean river and 70-km bike lanes, where they can also hold fun run events.
Metro Cebu doesn't get lag behind other cities when it comes to developing sustainable programs. The city did even conduct the so-called Green City Loop Experiment in 2014, through which only a limited number of motor vehicles were allowed to use half of each of the four major roads under a certain period of time. Meanwhile, the other half of those roads was given to joggers, locals, and bikers, to freely use it all to themselves. The experiment lasted for a day, from 3PM to 7PM, but this served as a springboard to Cebu's long-term sustainable development programs under the Green Loop Plan.
Vigan City
If you haven't been to Calle Crisologo in Vigan City, do note that it is now a car-less street. It is a heritage zone that tourists may enjoy marveling at while on a bike or on foot. Local government officials believe that turning Calle Crisologo into a walkable and bike-able zone is crucial for the city's economic growth and preservation. No wonder the same plans are now underway to develop the touristy Governor Reyes Street into a bike-able zone.
Pasig is one of the first cities in Metro Manila to close off its roads on Sundays to share it with bikers and pedestrians from 6AM to 12PM. But part of the long term sustainable development plan is to really turn a section of C. Raymundo Avenue, Caruncho Avenue, and MRR Pineda Road into a bike lane, and then establish an elevated walkway from Ortigas MRT to Opal Street so that pedestrians can easily and safely get to the Central Business District.
In summary, the worsening traffic situation and air pollution in bigger cities cannot be ignored. Now that some LGU's are actually willing to take action on it is a good start, and you can also contribute to this initiative by coming to work by means of biking. On weekends, why not tag some of your friends from the cycling community along to enjoy this exclusive privilege given to bikers and pedestrians. Not only you are able to save up on your gas, but you have also helped the city minimize its level of greenhouse gas emission.
(Photo credits: www.mb.com.ph, www.onwardplanet.com, cebudailynews.inquirer.net, www.businessmirror.com.ph www.gmanetwork.com)
ROBOTO REG 13PT / FEBRUARY 7, 2013
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Manga Reviews Reviews Yen Press
A Witch's Printing Office Volume 4
By Siddharth Gupta
Story by Mochinchi
Art by Yasuhiro Miyama
Translated by Amber Tamosaitis
Lettered by Erin Hickman
A Witch's Printing Office continues to play with fantasy archetypes in its fourth volume, twisting their roles to fit into the context of fandom. Witches creating spells are analogous to artists drawing comics, a band of warriors who secretly covet magic is akin to sporty jocks hiding their nerdy hobbies, and attendees falling ill at Magiket basically just suffer more exaggerated versions of ailments people can suffer from at an actual convention, like malnutrition and exhaustion. Printing Office uses the veneer of its fantasy trappings in service of its satire of fan culture. The characters grapple with grounded, relatable situations like becoming disillusioned with your career path, being unable to openly embrace what you like, and overworking yourself to the point of collapse. By having said characters be the likes of magicians and swordsmen, the series adds a layer of separation that allows readers to both appreciate the story's emotional truthfulness while indulging in the absurdity of fantasy archetypes facing these relatively mundane dilemmas with such difficulty.
Despite its satirical nature, Printing Office genuinely cares about the emotional well-being of its characters and tells heartfelt stories through them. Claire's backstory is a particularly relatable metaphor for not succeeding in your chosen career but finding personal fulfillment in another field. Her relationship with Mika also illuminates the power of having a mentor that believes in their student and cultivates their growth, even if they're struggling for a long time. Mika was rejected by other tutors for not being able to grasp magic immediately, but Claire accepted her and encouraged her even though her talents weren't quick to bloom. However, giving Mika the opportunity and time she needed to find what she was good at allowed her to hone a specialty that she'd not only make a career out of, but revolutionize the field of magic itself. In turn, Claire realizes that she probably wouldn't have met or helped Mika if she hadn't failed as a magical researcher first, recognizing that her own failures have ultimately guided her to find her own true calling. Claire and Mika's story serves as an encouraging reminder to not be discouraged by failure; there is more than one measure of success, even if it takes a little bit longer for some to grow into their own.
The series also has some strong commentary on the consequences of overwork in a couple of stories. In one chapter, a grateful fairy tries to help Mika rest by giving the workaholic an involuntary good night's rest. Unfortunately, Mika's accumulated lack of sleep karmically results in her being conked out for a week, awakening to the complaints of angry customers and an overdue workload. It's a funny moral about the necessity of getting proper sleep lest you risk crashing and oversleeping at the worst possible time. Similarly, a subsequent chapter finds Hera, the head healing mage at Magiket, also overworking herself to the point of passing out despite knowing better and lecturing others about taking care of their health. As a result, she leaves Mika and her students with the arduous task of taking care of all the sick attendees without her guidance, fumbling about until they're able to find workable solutions. Hera's recklessness pushes more of a workload on her ill-prepared staff, causing more stress and chaos, leading to Mika becoming exhausted herself trying to fix the mess left behind. By overworking, Hera causes more problems for her staff than she would've if she'd not tried to juggle so much on her own. It's very easy for workaholics to lose themselves in their work, pulling all-nighters and stretching themselves thin. A Witch's Printing Office humorously illustrates the snowballing effect that neglecting your health can have, showing that in the long run, you'll only cause more problems for yourself and the people relying on you.
The highlight of this volume, however, is the manga's direct satire of the scheduling conflict between Comiket and the Tokyo 2020 Olympics. Hindsight is 2020 and obviously, neither event ended up happening anyway, but the battle for space between the two events was fertile ground for the series to spin into a hilarious conflict between Magiket and the Battle Olympia, an international tournament for warriors. The allusions to the real-life conflict are humorously on the nose, with the manga highlighting all sides of the argument and ultimately showing the fecklessness of Magiket's resistance. The story also depicts the consequences of suddenly canceling a con, with Mika left to figure out how to refund attendees after already blowing the registration money on supplies, a situation many cons, unfortunately, found themselves in last year. Luckily for Mika and the reader, the manga then delves into a fantastic parody of tournament arc tropes as Mika unwittingly enters the Olympia with her invincible sword in the stone. The series dispatches all the tournament arc cliches gleefully and sardonically, complete with your typical rivals and even "denizens of darkness" plotting behind the scenes. One page is a devoted montage of all the tournament arc story beats packed together, rife with parodies of somewhat obscure but still recognizable moments from famous manga tournament arcs. This storyline is the perfect marriage of A Witch's Printing Office's marriage of social commentary and pop culture parody, crafting a hilarious storyline motivated by real problems and resolved by literally cutting down manga cliches.
Yasuhiro Miyama deserves so much credit for Printing Office's comedic versatility. Their ability to blend styles, seamlessly transitioning from serious manga action to goofy slapstick, often recontextualizing badass characters as goofballs, complements the absurd range of the story so well. Their compositions are really well done too, with panels blocked in a clean and easy to read manner that always makes the reading experience feel effortless and engaging. My one criticism of their art is their tendency to needlessly sexualize female characters in strange ways. This isn't a fanservice-laden manga, but most of the female characters are very voluptuous with absurdly pronounced breasts, and they often find themselves exposed in fetishy situations. The one that really sticks out is the fairy, who is introduced being squeezed in a compromising pose inside a carnivorous plant, its walls enveloping her as mucus drips all over her body. Stuff like this feels unnecessarily gratuitous, and while the characters are well-written, the art could really stand to sexualize them less.
A Witch's Printing Office's fourth volume is one of its most eclectic collections of stories yet. Determined to subvert fantasy, isekai, and shonen tropes alike, the stories in this volume run a gamut of tones. If you were to give a different chapter in this volume to different people unfamiliar with the series I'm confident they'd all get a different impression of what it's about. Never let it be said that A Witch's Printing Office is formulaic, because it fully uses the potential laden in its world, exploring it to its fullest in a wonderful variety of stories. Yet, through all its disparate ridiculousness, lies a sincere love for creative passion, and deep respect for people endeavoring to share their art with the world at its heart. Pick up this book and you'll be spellbound from the start.
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a witch's printing officeamber tamosaitiserin hickmanmochinchiyasuhiro miyamaYen Press
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About The Author Siddharth Gupta
Siddharth Gupta is an illustrator, animator, and writer based in Minnesota. They graduated with a Bachelor's degree in Animation from the School of Visual Arts, and have worked on projects for the University of Minnesota and the Shreya R. Dixit Foundation. An avid animation and comics fan since childhood, they've turned their passion towards being both a creator and a critic. They credit their love for both mediums to Akira Toriyama's Dragon Ball, which has also defined their artistic and comedic sensibilities. A frequent visitor to their local comic book shop, they are an avid reader and collector, particularly fond of manga. Their favorite comics include The Adventures of Tintin by Herge, Bloom County by Berkeley Breathed, and pretty much anything and everything by Rumiko Takahashi.
Fire Force, Vol 1 – 10


Trying to find someone to train or assess you? This link lists all the RTOs that are currently registered to deliver CPCPCM2039A, 'Carry out interactive workplace communication'.
Visual communicationis used that follows accepted industry practice or social convention.
Apply written communication and signage.


X1 Power Hammer running at short stroke (3.2" next to the lowest power) setting.
At the beginning of the video I am running the hammer just fast enough that the dies are not striking or just barely tapping so that you can watch the spring and toggle action. The piece of stock being forged is 1" (25mm) square. This large stock would forge better at the long stroke setting (4.5").
At the long stroke setting the hammer hits much harder but is also a little harder to control. There are four stroke settings. The shortest is suitable for smaller stock and detail forging, the longest for heavy forging. We have only forged with the machine for a few minutes so we are stilling learning the machine.
This is a 110 pound (50kg) hammer with a small foot print running without being bolted down - thus the rocking back and forth. More video coming in a few weeks . . .


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confidence podcast Season 4
Kelly Denithorne on Transgender Advocacy, Education, and Fashion (Episode 189 and 204)
Kelly Denithorne is an educator, transgender advocate, and the blogger behind "Unordinary Style." Kelly started her blog in 2010, and began it as a place to explore with her love of style while she was exploring women's fashion. Born as a male, Kelly tried on several different titles for herself, from transvestite, to cross dresser, and then to transgender non-binary. In the last year, Kelly has begun teaching in her middle school as a woman, and has started the process of transitioning.
The thing that I love about Kelly and our conversation is the humor and grace with which she shares her journey. In the last few months, she's become an advocate for the transgender community, and had several speaking engagements where she gets to share her story, as well. I also have the unique experience of having gone to high school with Kelly, when she was known as Bob, and so it was a true joy to reconnect with her and learn about where her life has taken her.
The true inspiration that I see in Kelly is her dedication to being curious, and to "always be humble enough to learn." As she and I spoke, it was clear that she is a powerful example of someone who is a lifelong educator, and learner. Inspired by another trans woman, Jessica Lynn, Kelly has started sharing her story so that other people can learn more about transgendered individuals, and to spread awareness.
Kelly has a wide knowledge of topics concerning the transgender community, including the increased need for rights and protections for transgender sex workers. Transgender people often turn to adult entertainment careers on websites such as shemalehd.sex, and Kelly wants to raise awareness about other career options outside of the adult industry for people within the trans community.
In this episode, Kelly Denithorne and I talk about:
Her earliest sparks of joy, hanging out with her sister
Her experience of growing up as a male, and the confusion that she experienced around having unidentified gender variance
Being a teacher, and transitioning from dressing as a male to dressing female this school year
Seeking therapy with someone who specializes in gender variance
The reactions of her students and parents to her presenting as a woman
Her path to discovering her true self
The role being a fashion blogger played in her journey
Her inspiration in starting to take on speaking opportunities
How to jumpstart your joy: learn to accept who you are and how you work with it, sharing your story, and always be humble enough to learn
Kelly Denithorne's blog: Unordinary Style
Kelly Denithorne's Speakering Website
How to Create Cleavage, as found on Kelly's blog
Kelly (as Nadine Spirit) on Instagram
Jessica Lynn's website
Episode 11: Hope, Love, and Serious Creatures with Amber and Brady Black
Ep47: Cameron Adair on Living a Life of Purpose after Video Game Addiction
Ep57: What Failure Can Teach You About Balance


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Q: Using template tag in ModelAdmin.readonly_fields method I use this pattern:
class PersonAdmin(admin.ModelAdmin):
    readonly_fields = ('address_report',)

    def address_report(self, instance):
        return format_html(...)

Source: docs about readonly_fields
Now I would like to use a custom templatetag in the Python method address_report().
What is the best way to call it?
I tried to called my templatetag directly, but this just returns a dictionary, not html.

A: I think you'll have to create a template from string. That way you can use any template tags or variables you want.
Example:
from django.template import Template, Context

def address_report(...):
    # create template
    t = Template("""
       {% load custom_tag_module %}

            Hello, {{ name }}!

            {% custom_tag %}

        """)

    # create context
    c = Context({'name': 'World'})

    # render template, mark safe and return
    return mark_safe(t.render(c))

redpajama_set_name: RedPajamaStackExchange