--- trunk/mattDisertation/Introduction.tex	2004/01/30 21:47:22	1000
+++ trunk/mattDisertation/Introduction.tex	2004/01/31 22:10:21	1001
@@ -30,11 +30,155 @@ which method best suits that objective.
 thermodynamic properties of the system are being probed, then chose
 which method best suits that objective.
 
-\subsection{\label{introSec:statThermo}Statistical Thermodynamics}
+\subsection{\label{introSec:statThermo}Statistical Mechanics}
 
-ergodic hypothesis
+The following section serves as a brief introduction to some of the
+Statistical Mechanics concepts present in this dissertation.  What
+follows is a brief derivation of Blotzman weighted statistics, and an
+explanation of how one can use the information to calculate an
+observable for a system.  This section then concludes with a brief
+discussion of the ergodic hypothesis and its relevance to this
+research.
 
-enesemble averages
+\subsection{\label{introSec:boltzman}Boltzman weighted statistics}
+
+Consider a system, $\gamma$, with some total energy,, $E_{\gamma}$.
+Let $\Omega(E_{gamma})$ represent the number of degenerate ways
+$\boldsymbol{\Gamma}$, the collection of positions and conjugate
+momenta of system $\gamma$, can be configured to give
+$E_{\gamma}$. Further, if $\gamma$ is in contact with a bath system
+where energy is exchanged between the two systems, $\Omega(E)$, where
+$E$ is the total energy of both systems, can be represented as
+\begin{equation}
+eq here
+\label{introEq:SM1}
+\end{equation}
+Or additively as 
+\begin{equation}
+eq here
+\label{introEq:SM2}
+\end{equation}
+
+The solution to Eq.~\ref{introEq:SM2} maximizes the number of
+degenerative configurations in $E$. \cite{fix} 
+This gives
+\begin{equation}
+eq here
+\label{introEq:SM3}
+\end{equation}
+Where $E_{\text{bath}}$ is $E-E_{\gamma}$, and
+$\frac{partialE_{\text{bath}}}{\partial E_{\gamma}}$ is
+$-1$. Eq.~\ref{introEq:SM3} becomes
+\begin{equation}
+eq here
+\label{introEq:SM4}
+\end{equation}
+
+At this point, one can draw a relationship between the maximization of
+degeneracy in Eq.~\ref{introEq:SM3} and the second law of
+thermodynamics.  Namely, that for a closed system, entropy wil
+increase for an irreversible process.\cite{fix} Here the
+process is the partitioning of energy among the two systems.  This
+allows the following definition of entropy:
+\begin{equation}
+eq here
+\label{introEq:SM5}
+\end{equation}
+Where $k_B$ is the Boltzman constant.  Having defined entropy, one can
+also define the temperature of the system using the relation
+\begin{equation}
+eq here
+\label{introEq:SM6}
+\end{equation}
+The temperature in the system $\gamma$ is then
+\begin{equation}
+eq here
+\label{introEq:SM7}
+\end{equation}
+Applying this to Eq.~\ref{introEq:SM4} gives the following
+\begin{equation}
+eq here
+\label{introEq:SM8}
+\end{equation}
+Showing that the partitioning of energy between the two systems is
+actually a process of thermal equilibration. \cite{fix}
+
+An application of these results is to formulate the form of an
+expectation value of an observable, $A$, in the canonical ensemble. In
+the canonical ensemble, the number of particles, $N$, the volume, $V$,
+and the temperature, $T$, are all held constant while the energy, $E$,
+is allowed to fluctuate. Returning to the previous example, the bath
+system is now an infinitly large thermal bath, whose exchange of
+energy with the system $\gamma$ holds teh temperature constant.  The
+partitioning of energy in the bath system is then related to the total
+energy of both systems and the fluctuations in $E_{\gamma}}$:
+\begin{equation}
+eq here
+\label{introEq:SM9}
+\end{equation}
+As for the expectation value, it can be defined
+\begin{equation}
+eq here
+\label{introEq:SM10}
+\end{eequation}
+Where $\int_{\boldsymbol{\Gamma}} d\Boldsymbol{\Gamma}$ denotes an
+integration over all accessable phase space, $P_{\gamma}$ is the
+probability of being in a given phase state and
+$A(\boldsymbol{\Gamma})$ is some observable that is a function of the
+phase state.
+
+Because entropy seeks to maximize the number of degenerate states at a
+given energy, the probability of being in a particular state in
+$\gamma$ will be directly proportional to the number of allowable
+states the coupled system is able to assume. Namely,
+\begin{equation}
+eq here
+\label{introEq:SM11}
+\end{equation}
+With $E_{\gamma} \lE$, $\ln \Omega$ can be expanded in a Taylor series:
+\begin{equation}
+eq here
+\label{introEq:SM12}
+\end{equation}
+Higher order terms are omitted as $E$ is an infinite thermal
+bath. Further, using Eq.~\ref{introEq:SM7}, Eq.~\ref{introEq:SM11} can
+be rewritten:
+\begin{equation}
+eq here
+\label{introEq:SM13}
+\end{equation}
+Where $\ln \Omega(E)$ has been factored out of the porpotionality as a
+constant.  Normalizing the probability ($\int_{\boldsymbol{\Gamma}}
+d\boldsymbol{\Gamma} P_{\gamma} =1$ gives 
+\begin{equation}
+eq here
+\label{introEq:SM14}
+\end{equation}
+This result is the standard Boltzman statistical distribution.
+Applying it to Eq.~\ref{introEq:SM10} one can obtain the following relation for ensemble averages:
+\begin{equation}
+eq here
+\label{introEq:SM15}
+\end{equation}
+
+\subsection{\label{introSec:ergodic}The Ergodic Hypothesis}
+
+One last important consideration is that of ergodicity. Ergodicity is
+the assumption that given an infinite amount of time, a system will
+visit every available point in phase space.\cite{fix} For most
+systems, this is a valid assumption, except in cases where the system
+may be trapped in a local feature (\emph{i.~e.~glasses}). When valid,
+ergodicity allows the unification of a time averaged observation and
+an ensemble averged one. If an observation is averaged over a
+sufficiently long time, the system is assumed to visit all
+appropriately available points in phase space, giving a properly
+weighted statistical average. This allows the researcher freedom of
+choice when deciding how best to measure a given observable.  When an
+ensemble averaged approach seems most logical, the Monte Carlo
+techniques described in Sec.~\ref{introSec:MC} can be utilized.
+Conversely, if a problem lends itself to a time averaging approach,
+the Molecular Dynamics techniques in Sec.~\ref{introSec:MD} can be
+employed.
 
 \subsection{\label{introSec:monteCarlo}Monte Carlo Simulations}
 
@@ -243,7 +387,7 @@ researcher is interested.  If the observabvles depend 
 
 The choice of when to use molecular dynamics over Monte Carlo
 techniques, is normally decided by the observables in which the
-researcher is interested.  If the observabvles depend on momenta in
+researcher is interested.  If the observables depend on momenta in
 any fashion, then the only choice is molecular dynamics in some form.
 However, when the observable is dependent only on the configuration,
 then most of the time Monte Carlo techniques will be more efficent.
@@ -306,12 +450,12 @@ the surface. \cite{fix} For a cubic system of 1000 par
 
 Another consideration one must resolve, is that in a given simulation
 a disproportionate number of the particles will feel the effects of
-the surface. \cite{fix} For a cubic system of 1000 particles arranged
+the surface.\cite{fix} For a cubic system of 1000 particles arranged
 in a $10x10x10$ cube, 488 particles will be exposed to the surface.
 Unless one is simulating an isolated particle group in a vacuum, the
 behavior of the system will be far from the desired bulk
 charecteristics.  To offset this, simulations employ the use of
-periodic boundary images. \cite{fix}
+periodic boundary images.\cite{fix}
 
 The technique involves the use of an algorithm that replicates the
 simulation box on an infinite lattice in cartesian space.  Any given
@@ -329,7 +473,7 @@ calculation. \cite{fix} In a simultation with periodic
 cutoff radius be employed.  Using a cutoff radius improves the
 efficiency of the force evaluation, as particles farther than a
 predetermined distance, $fix$, are not included in the
-calculation. \cite{fix} In a simultation with periodic images, $fix$
+calculation.\cite{fix} In a simultation with periodic images, $fix$
 has a maximum value of $fix$.  Fig.~\ref{fix} illustrates how using an
 $fix$ larger than this value, or in the extreme limit of no $fix$ at
 all, the corners of the simulation box are unequally weighted due to
@@ -383,7 +527,7 @@ with a velocity reformulation of teh Verlet method. \c
 order of $\Delta t^4$.
 
 In practice, however, the simulations in this research were integrated
-with a velocity reformulation of teh Verlet method. \cite{allen87:csl}
+with a velocity reformulation of teh Verlet method.\cite{allen87:csl}
 \begin{equation}
 eq here
 \label{introEq:MDvelVerletPos}
@@ -406,7 +550,7 @@ It can be shown, \cite{Frenkel1996} that although the 
 therefore increases, but does not guarantee the ``correctness'' or the
 integrated trajectory.
 
-It can be shown, \cite{Frenkel1996} that although the Verlet algorithm
+It can be shown,\cite{Frenkel1996} that although the Verlet algorithm
 does not rigorously preserve the actual Hamiltonian, it does preserve
 a pseudo-Hamiltonian which shadows the real one in phase space.  This
 pseudo-Hamiltonian is proveably area-conserving as well as time
@@ -572,11 +716,59 @@ symplectic and time reversible.
 unitary and symplectic, the entire integration scheme is also
 symplectic and time reversible.
 
-\section{\label{introSec:chapterLayout}Chapter Layout}
+\section{\label{introSec:layout}Dissertation Layout}
 
-\subsection{\label{introSec:RSA}Random Sequential Adsorption}
+This dissertation is divided as follows:Chapt.~\ref{chapt:RSA}
+presents the random sequential adsorption simulations of related
+pthalocyanines on a gold (111) surface. Chapt.~\ref{chapt:OOPSE}
+is about the writing of the molecular dynamics simulation package
+{\sc oopse}, Chapt.~\ref{chapt:lipid} regards the simulations of
+phospholipid bilayers using a mesoscale model, and lastly,
+Chapt.~\ref{chapt:conclusion} concludes this dissertation with a
+summary of all results. The chapters are arranged in chronological
+order, and reflect the progression of techniques I employed during my
+research.  
 
-\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package}
+The chapter concerning random sequential adsorption
+simulations is a study in applying the principles of theoretical
+research in order to obtain a simple model capaable of explaining the
+results.  My advisor, Dr. Gezelter, and I were approached by a
+colleague, Dr. Lieberman, about possible explanations for partial
+coverge of a gold surface by a particular compound of hers. We
+suggested it might be due to the statistical packing fraction of disks
+on a plane, and set about to simulate this system.  As the events in
+our model were not dynamic in nature, a Monte Carlo method was
+emplyed.  Here, if a molecule landed on the surface without
+overlapping another, then its landing was accepted.  However, if there
+was overlap, the landing we rejected and a new random landing location
+was chosen.  This defined our acceptance rules and allowed us to
+construct a Markov chain whose limiting distribution was the surface
+coverage in which we were interested.
 
-\subsection{\label{introSec:bilayers}A Mesoscale Model for
-Phospholipid Bilayers}
+The following chapter, about the simulation package {\sc oopse},
+describes in detail the large body of scientific code that had to be
+written in order to study phospholipid bilayer.  Although there are
+pre-existing molecular dynamic simulation packages available, none
+were capable of implementing the models we were developing.{\sc oopse}
+is a unique package capable of not only integrating the equations of
+motion in cartesian space, but is also able to integrate the
+rotational motion of rigid bodies and dipoles.  Add to this the
+ability to perform calculations across parallel processors and a
+flexible script syntax for creating systems, and {\sc oopse} becomes a
+very powerful scientific instrument for the exploration of our model.
+
+Bringing us to Chapt.~\ref{chapt:lipid}. Using {\sc oopse}, I have been
+able to parametrize a mesoscale model for phospholipid simulations.
+This model retains information about solvent ordering about the
+bilayer, as well as information regarding the interaction of the
+phospholipid head groups' dipole with each other and the surrounding
+solvent.  These simulations give us insight into the dynamic events
+that lead to the formation of phospholipid bilayers, as well as
+provide the foundation for future exploration of bilayer phase
+behavior with this model.  
+
+Which leads into the last chapter, where I discuss future directions
+for both{\sc oopse} and this mesoscale model.  Additionally, I will
+give a summary of results for this dissertation.
+
+