% intro.tex

\section{Introduction}

Annealing is a formal term for the ancient art of heating and/or
cooling materials to forge pottery, tools, weapons, and works of
art.  It is the process of subjecting a substance to changes in
pressure or temperature to achieve desired material properties.
Annealing has played a key role in human history:  entire nations have
been won and lost on the abilities of craftsmen to produce
fit materials.

A contemporary example of annealing is the manufacture of glass
products from raw silica.  The silica is first heated from an ordinary
solid phase to a liquid.  The liquid silica is then cooled according to
a precise {\it cooling schedule\/} to achieve a desired solid phase; for
example, a solid with better optical properties~\cite{boyd,pincus-holmes}.
%page 27.

A depiction of an annealing process is given in figure~\ref{fig-phases}.
Note that the path from A through B to C is not the shortest
path in phase space from A to C, but may nonetheless represent the most time and
cost efficient path.  Although it might be desirable to
move directly from A to C, it is sometimes the case that a direct
path is too costly, too time-intensive, or physically intractable.

Modern research in annealing techniques includes both the ``art'' of
experimental material science~\cite{tms-proc-92,riso} and theoretical
work based on statistical
thermodynamics~\cite{mcquarrie,statistical-thermodynamics}.  The latter
describes the relationship between {\it macrostates\/} of
the material and the {\it states\/} of the atoms or molecules that
compose it.  A practical application of this theory is to determine
what annealing schedule should be observed to effect desired changes in
the properties of a given material.

\begin{figure}
\putfig{phases.eps}{1.00}
\caption{\label{fig-phases} Example temperature-pressure phases of a material.  The dotted line shows an annealing path from phase A through phase B to phase C.}
\vspace*{1 em}
\end{figure}


\subsection{Fundamentals of Temperature}

A material is composed of a set $S$ of atoms or molecules.
The {\it states\/} $\omega \in \Omega$
of atoms or molecules in $S$ include measures of their
position, orientation, rotation, velocity, etc.
The energy $E(\omega)$ of any particular atom or molecule
is a function of its state at some instant in time.
The set of $E(\omega)$'s can be expressed by
$\cal{E}$ $=$ $\{ E(\omega) : \omega \in \Omega\}$.
In real materials these energies occur in quantized
intervals\footnote{We consider only bound states of $\Omega$}
so that $\#\cal{E}$ $\ll$ $\#\Omega$.

A material is said to be at temperature ${\Theta}(\Omega) = T$ when the
individual energies $E(\omega)$ are Boltzmann-distributed
at temperature $T$.
For example, the informal
statement ``the room temperature is $68 ^\circ$F'' is equivalent
to ``the individual energies of the gas molecules in this room are
Boltzmann-distributed at temperature $T = 293 ^\circ$K''.
For systems of non-interacting particles, this can be
mathematically stated:

\vspace{1 em}
\noindent
$ {\Theta}(\Omega) \, = \, T  \; \; \Leftrightarrow \; \; \forall \, \omega \in \Omega \, , $
\vspace*{-0.5em}
\begin{eqnarray}
p(\omega \, | \, T) & = &
  \frac{
      e^{-E(\omega)/kT}
    }{
      \sum_{\omega \in \Omega} \, e^{-E(\omega)/kT}
    } \, .
  \label{eqn-boltzmann1}
\end{eqnarray}
The Boltzmann constant $k$ may be
set equal to $1$ by using the same units for energy and temperature.

In real materials,
changes in temperature are always due to adding or subtracting energy
through conductive, convective, mechanical, or radiative means.  
It is possible for the energy distribution of a set of atoms or molecules to
be in a transition state where temperature is not defined (i.e., not
Boltzmann distributed).  A system composed of ice cubes dropped into a hot
drink are one example.

To extend equation~\ref{eqn-boltzmann1} to systems of interacting particles,
Gibbs~\cite{mcquarrie} considered an ensemble $\cal{G}$ composed of many copies
of $S$.  
Each $\gamma \in {\cal{G}}$ represents one possible configuration
of states $\{ \omega \}$ attainable by $S$.  Each $\gamma$ has an associated
bulk state $u$; e.g., one glass state vs. another.
The entire collection of these states $\cal{X}$ enumerate
the combinatorally large number of possible bulk states available to $S$.
The elements ${u} \in {\cal{X}}$ are termed {\it microstates\/}.
In this theoretical framework,
the probability of attaining some microstate ${u}$ is:
\begin{eqnarray}
p({u} \, | \, T) & = &
  \frac{
      e^{-E({u})/T}
    }{
      \sum_{{u} \in {\cal{X}}} \, e^{-E({u})/T}
    } \, .
  \label{eqn-boltzmann2}
\end{eqnarray}

For example, the microstate energies which have high probability
in material phase A of figure~\ref{fig-phases} will have low
probability in phase C.  This is true even when states in each
phase are equilibrated at the same temperature T.
The range of attainable ${u} \in {\cal{X}}$ are physically
constrained by the properties of the constituent atoms or molecules and
laws of energy conservation.  
Microstate statistics for a given material can be
obtained by laboratory measurements or derived from theoretical
predictions of desirable material phases.

