--- trunk/mattDisertation/Introduction.tex	2004/01/22 17:34:20	976
+++ trunk/mattDisertation/Introduction.tex	2004/01/22 21:13:55	977
@@ -53,10 +53,10 @@ The equation can be recast as:
 \end{equation}
 The equation can be recast as:
 \begin{equation}
-I = (b-a)<f(x)>
+I = (b-a)\langle f(x) \rangle
 \label{eq:MCex2}
 \end{equation}
-Where $<f(x)>$ is the unweighted average over the interval
+Where $\langle f(x) \rangle$ 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
@@ -66,16 +66,18 @@ integrals of the form:
 However, in Statistical Mechanics, one is typically interested in
 integrals of the form:
 \begin{equation}
-<A> = \frac{A}{exp^{-\beta}}
+\langle A \rangle = \frac{\int d^N \mathbf{r}~A(\mathbf{r}^N)% 
+	e^{-\beta V(\mathbf{r}^N)}}%
+	{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}}
 \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
+Where $\mathbf{r}^N$ stands for the coordinates of all $N$ particles
+and $A$ is some observable that is only dependent on
+position. $\langle A \rangle$ 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
@@ -86,92 +88,126 @@ Eq.~\ref{eq:MCex1} rewritten to be:
 efficiently calculate the integral.\cite{Frenkel1996} Consider again
 Eq.~\ref{eq:MCex1} rewritten to be:
 \begin{equation}
-EQ Here
+I = \int^b_a \frac{f(x)}{\rho(x)} \rho(x) dx
+\label{introEq:Importance1}
 \end{equation}
-Where $fix$ is an arbitrary probability distribution in $x$.  If one
-conducts $fix$ trials selecting a random number, $fix$, from the
-distribution $fix$ on the interval $[a,b]$, then Eq.~\ref{fix} becomes
+Where $\rho(x)$ is an arbitrary probability distribution in $x$.  If
+one conducts $\tau$ trials selecting a random number, $\zeta_\tau$,
+from the distribution $\rho(x)$ on the interval $[a,b]$, then
+Eq.~\ref{introEq:Importance1} becomes
 \begin{equation}
-EQ Here
+I= \biggl \langle \frac{f(x)}{\rho(x)} \biggr \rangle_{\text{trials}}
+\label{introEq:Importance2}
 \end{equation}
-Looking at Eq.~ref{fix}, and realizing
+Looking at Eq.~\ref{eq:mcEnsAvg}, and realizing
 \begin {equation}
-EQ Here
+\rho_{kT}(\mathbf{r}^N) = 
+	\frac{e^{-\beta V(\mathbf{r}^N)}}
+	{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}}
+\label{introEq:MCboltzman}
 \end{equation}
-The ensemble average can be rewritten as
+Where $\rho_{kT}$ is the boltzman distribution.  The ensemble average
+can be rewritten as
 \begin{equation}
-EQ Here
+\langle A \rangle = \int d^N \mathbf{r}~A(\mathbf{r}^N) 
+	\rho_{kT}(\mathbf{r}^N)
+\label{introEq:Importance3}
 \end{equation}
-Appllying Eq.~ref{fix} one obtains
+Applying Eq.~\ref{introEq:Importance1} one obtains
 \begin{equation}
-EQ Here
+\langle A \rangle = \biggl \langle
+	\frac{ A \rho_{kT}(\mathbf{r}^N) }
+	{\rho(\mathbf{r}^N)} \biggr \rangle_{\text{trials}}
+\label{introEq:Importance4}
 \end{equation}
-By selecting $fix$ to be $fix$ Eq.~ref{fix} becomes
+By selecting $\rho(\mathbf{r}^N)$ to be $\rho_{kT}(\mathbf{r}^N)$
+Eq.~\ref{introEq:Importance4} becomes
 \begin{equation}
-EQ Here
+\langle A \rangle = \langle A(\mathbf{r}^N) \rangle_{\text{trials}}
+\label{introEq:Importance5}
 \end{equation}
-The difficulty is selecting points $fix$ such that they are sampled
-from the distribution $fix$.  A solution was proposed by Metropolis et
-al.\cite{fix} which involved the use of a Markov chain whose limiting
-distribution was $fix$.
+The difficulty is selecting points $\mathbf{r}^N$ such that they are
+sampled from the distribution $\rho_{kT}(\mathbf{r}^N)$.  A solution
+was proposed by Metropolis et al.\cite{metropolis:1953} which involved
+the use of a Markov chain whose limiting distribution was
+$\rho_{kT}(\mathbf{r}^N)$.
 
-\subsection{Markov Chains}
+\subsubsection{\label{introSec:markovChains}Markov Chains}
 
 A Markov chain is a chain of states satisfying the following
-conditions:\cite{fix}
-\begin{itemize}
+conditions:\cite{leach01:mm}
+\begin{enumerate}
 \item The outcome of each trial depends only on the outcome of the previous trial.
 \item Each trial belongs to a finite set of outcomes called the state space.
-\end{itemize}
-If given two configuartions, $fix$ and $fix$, $fix$ and $fix$ are the
-probablilities of being in state $fix$ and $fix$ respectively.
-Further, the two states are linked by a transition probability, $fix$,
-which is the probability of going from state $m$ to state $n$.
+\end{enumerate}
+If given two configuartions, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$,
+$\rho_m$ and $\rho_n$ are the probablilities of being in state
+$\mathbf{r}^N_m$ and $\mathbf{r}^N_n$ respectively.  Further, the two
+states are linked by a transition probability, $\pi_{mn}$, which is the
+probability of going from state $m$ to state $n$.
 
+\newcommand{\accMe}{\operatorname{acc}}
+
 The transition probability is given by the following:
 \begin{equation}
-EQ Here
-\end{equation}
-Where $fix$ is the probability of attempting the move $fix$, and $fix$
-is the probability of accepting the move $fix$.  Defining a
-probability vector, $fix$, such that
+\pi_{mn} = \alpha_{mn} \times \accMe(m \rightarrow n)
+\label{introEq:MCpi}
+\end{equation}
+Where $\alpha_{mn}$ is the probability of attempting the move $m
+\rightarrow n$, and $\accMe$ is the probability of accepting the move
+$m \rightarrow n$.  Defining a probability vector,
+$\boldsymbol{\rho}$, such that
 \begin{equation}
-EQ Here
+\boldsymbol{\rho} = \{\rho_1, \rho_2, \ldots \rho_m, \rho_n, 
+	\ldots \rho_N \}
+\label{introEq:MCrhoVector}
 \end{equation}
-a transition matrix $fix$ can be defined, whose elements are $fix$,
-for each given transition.  The limiting distribution of the Markov
-chain can then be found by applying the transition matrix an infinite
-number of times to the distribution vector.
+a transition matrix $\boldsymbol{\Pi}$ can be defined,
+whose elements are $\pi_{mn}$, for each given transition.  The
+limiting distribution of the Markov chain can then be found by
+applying the transition matrix an infinite number of times to the
+distribution vector.
 \begin{equation}
-EQ Here
+\boldsymbol{\rho}_{\text{limit}} = 
+	\lim_{N \rightarrow \infty} \boldsymbol{\rho}_{\text{initial}}
+	\boldsymbol{\Pi}^N
+\label{introEq:MCmarkovLimit}
 \end{equation}
-
 The limiting distribution of the chain is independent of the starting
 distribution, and successive applications of the transition matrix
 will only yield the limiting distribution again.
 \begin{equation}
-EQ Here
+\boldsymbol{\rho}_{\text{limit}} = \boldsymbol{\rho}_{\text{initial}}
+	\boldsymbol{\Pi}
+\label{introEq:MCmarkovEquil}
 \end{equation}
 
-\subsection{fix}
+\subsubsection{\label{introSec:metropolisMethod}The Metropolis Method}
 
-In the Metropolis method \cite{fix} Eq.~ref{fix} is solved such that
-$fix$ matches the Boltzman distribution of states.  The method
-accomplishes this by imposing the strong condition of microscopic
-reversibility on the equilibrium distribution.  Meaning, that at
-equilibrium the probability of going from $m$ to $n$ is the same as
-going from $n$ to $m$.
+In the Metropolis method\cite{metropolis:1953}
+Eq.~\ref{introEq:MCmarkovEquil} is solved such that
+$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzman distribution
+of states.  The method accomplishes this by imposing the strong
+condition of microscopic reversibility on the equilibrium
+distribution.  Meaning, that at equilibrium the probability of going
+from $m$ to $n$ is the same as going from $n$ to $m$.
 \begin{equation}
-EQ Here
+\rho_m\pi_{mn} = \rho_n\pi_{nm}
+\label{introEq:MCmicroReverse}
 \end{equation}
-Further, $fix$ is chosen to be a symetric matrix in the Metropolis
-method.  Using Eq.~\ref{fix}, Eq.~\ref{fix} becomes
+Further, $\boldsymbol{\alpha}$ is chosen to be a symetric matrix in
+the Metropolis method.  Using Eq.~\ref{introEq:MCpi},
+Eq.~\ref{introEq:MCmicroReverse} becomes
 \begin{equation}
-EQ Here
+\frac{\accMe(m \rightarrow n)}{\accMe(n \rightarrow m)} =
+	\frac{\rho_n}{\rho_m}
+\label{introEq:MCmicro2}
 \end{equation}
-For a Boltxman limiting distribution
+For a Boltxman limiting distribution,
 \begin{equation}
-EQ Here
+\frac{\rho_n}{\rho_m} = e^{-\beta[\mathcal{U}(n) - \mathcal{U}(m)]}
+	= e^{-\beta \Delta \mathcal{U}}
+\label{introEq:MCmicro3}
 \end{equation}
 This allows for the following set of acceptance rules be defined:
 \begin{equation}
@@ -193,7 +229,7 @@ distribution is the Boltzman distribution.
 the ensemble averages, as this method ensures that the limiting
 distribution is the Boltzman distribution.
 
-\subsection{\label{introSec:md}Molecular Dynamics Simulations}
+\subsection{\label{introSec:MD}Molecular Dynamics Simulations}
 
 The main simulation tool used in this research is Molecular Dynamics.
 Molecular Dynamics is when the equations of motion for a system are
@@ -216,7 +252,7 @@ making molecular dynamics key in the simulation of tho
 centered around the dynamic properties of phospholipid bilayers,
 making molecular dynamics key in the simulation of those properties.
 
-\subsection{Molecular dynamics Algorithm}
+\subsubsection{Molecular dynamics Algorithm}
 
 To illustrate how the molecular dynamics technique is applied, the
 following sections will describe the sequence involved in a
@@ -225,7 +261,7 @@ discussion with the integration of the equations of mo
 calculation of the forces.  Sec.~\ref{fix} concludes the algorithm
 discussion with the integration of the equations of motion. \cite{fix}
 
-\subsection{initialization}
+\subsubsection{initialization}
 
 When selecting the initial configuration for the simulation it is
 important to consider what dynamics one is hoping to observe.
@@ -256,7 +292,7 @@ kinetic energy from energy stored in potential degrees
 first few initial simulation steps due to either loss or gain of
 kinetic energy from energy stored in potential degrees of freedom.
 
-\subsection{Force Evaluation}
+\subsubsection{Force Evaluation}
 
 The evaluation of forces is the most computationally expensive portion
 of a given molecular dynamics simulation.  This is due entirely to the
