--- trunk/mattDisertation/Introduction.tex	2004/01/18 04:12:08	954
+++ trunk/mattDisertation/Introduction.tex	2004/01/18 05:26:10	955
@@ -40,10 +40,54 @@ work in fissionable material.\cite{metropolis:1949} Th
 
 The Monte Carlo method was developed by Metropolis and Ulam for their
 work in fissionable material.\cite{metropolis:1949} The method is so
-named, because it heavily uses random numbers in the solution of the
-problem.
+named, because it heavily uses random numbers in its
+solution.\cite{allen87:csl} The Monte Carlo method allows for the
+solution of integrals through the stochastic sampling of the values
+within the integral. In the simplest case, the evaluation of an
+integral would follow a brute force method of
+sampling.\cite{Frenkel1996} Consider the following single dimensional
+integral:
+\begin{equation}
+I = f(x)dx
+\label{eq:MCex1}
+\end{equation}
+The equation can be recast as:
+\begin{equation}
+I = (b-a)<f(x)>
+\label{eq:MCex2}
+\end{equation}
+Where $<f(x)>$ is the unweighted average over the interval
+$[a,b]$. The calculation of the integral could then be solved by
+randomly choosing points along the interval $[a,b]$ and calculating
+the value of $f(x)$ at each point. The accumulated average would then
+approach $I$ in the limit where the number of trials is infintely
+large.
 
+However, in Statistical Mechanics, one is typically interested in
+integrals of the form:
+\begin{equation}
+<A> = \frac{A}{exp^{-\beta}}
+\label{eq:mcEnsAvg}
+\end{equation}
+Where $r^N$ stands for the coordinates of all $N$ particles and $A$ is
+some observable that is only dependent on position. $<A>$ is the
+ensemble average of $A$ as presented in
+Sec.~\ref{introSec:statThermo}. Because $A$ is independent of
+momentum, the momenta contribution of the integral can be factored
+out, leaving the configurational integral. Application of the brute
+force method to this system would yield highly inefficient
+results. Due to the Boltzman weighting of this integral, most random
+configurations will have a near zero contribution to the ensemble
+average. This is where a importance sampling comes into
+play.\cite{allen87:csl}
 
+Importance Sampling is a method where one selects a distribution from
+which the random configurations are chosen in order to more
+efficiently calculate the integral.\cite{Frenkel1996} Consider again
+Eq.~\ref{eq:MCex1} rewritten to be:
+
+
+
 \subsection{\label{introSec:md}Molecular Dynamics Simulations}
 
 time averages
