trunk/mattDisertation/Introduction.tex



\chapter{\label{chapt:intro}Introduction and Theoretical Background}


\section{\label{introSec:theory}Theoretical Background}

The techniques used in the course of this research fall under the two
main classes of molecular simulation: Molecular Dynamics and Monte
Carlo. Molecular Dynamic simulations integrate the equations of motion
for a given system of particles, allowing the researher to gain
insight into the time dependent evolution of a system. Diffusion
phenomena are readily studied with this simulation technique, making
Molecular Dynamics the main simulation technique used in this
research. Other aspects of the research fall under the Monte Carlo
class of simulations. In Monte Carlo, the configuration space
available to the collection of particles is sampled stochastichally,
or randomly. Each configuration is chosen with a given probability
based on the Maxwell Boltzman distribution. These types of simulations
are best used to probe properties of a system that are only dependent
only on the state of the system. Structural information about a system
is most readily obtained through these types of methods.

Although the two techniques employed seem dissimilar, they are both
linked by the overarching principles of Statistical
Thermodynamics. Statistical Thermodynamics governs the behavior of
both classes of simulations and dictates what each method can and
cannot do. When investigating a system, one most first analyze what
thermodynamic properties of the system are being probed, then chose
which method best suits that objective.

\subsection{\label{introSec:statThermo}Statistical Thermodynamics}

ergodic hypothesis

enesemble averages

\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations}

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 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

time integrating schemes

time reversible

symplectic methods

Extended ensembles (NVT NPT)

constrained dynamics

\section{\label{introSec:chapterLayout}Chapter Layout}

\subsection{\label{introSec:RSA}Random Sequential Adsorption}

\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package}

\subsection{\label{introSec:bilayers}A Mesoscale Model for Phospholipid Bilayers}
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#	User	Rev	Content
1	mmeineke	914
2
3			\chapter{\label{chapt:intro}Introduction and Theoretical Background}
4
5
6
7			\section{\label{introSec:theory}Theoretical Background}
8
9	mmeineke	953	The techniques used in the course of this research fall under the two
10			main classes of molecular simulation: Molecular Dynamics and Monte
11			Carlo. Molecular Dynamic simulations integrate the equations of motion
12			for a given system of particles, allowing the researher to gain
13			insight into the time dependent evolution of a system. Diffusion
14			phenomena are readily studied with this simulation technique, making
15			Molecular Dynamics the main simulation technique used in this
16			research. Other aspects of the research fall under the Monte Carlo
17			class of simulations. In Monte Carlo, the configuration space
18			available to the collection of particles is sampled stochastichally,
19			or randomly. Each configuration is chosen with a given probability
20			based on the Maxwell Boltzman distribution. These types of simulations
21			are best used to probe properties of a system that are only dependent
22			only on the state of the system. Structural information about a system
23			is most readily obtained through these types of methods.
24	mmeineke	914
25	mmeineke	953	Although the two techniques employed seem dissimilar, they are both
26			linked by the overarching principles of Statistical
27			Thermodynamics. Statistical Thermodynamics governs the behavior of
28			both classes of simulations and dictates what each method can and
29			cannot do. When investigating a system, one most first analyze what
30			thermodynamic properties of the system are being probed, then chose
31			which method best suits that objective.
32	mmeineke	914
33			\subsection{\label{introSec:statThermo}Statistical Thermodynamics}
34
35			ergodic hypothesis
36
37			enesemble averages
38
39			\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations}
40
41	mmeineke	953	The Monte Carlo method was developed by Metropolis and Ulam for their
42			work in fissionable material.\cite{metropolis:1949} The method is so
43	mmeineke	955	named, because it heavily uses random numbers in its
44			solution.\cite{allen87:csl} The Monte Carlo method allows for the
45			solution of integrals through the stochastic sampling of the values
46			within the integral. In the simplest case, the evaluation of an
47			integral would follow a brute force method of
48			sampling.\cite{Frenkel1996} Consider the following single dimensional
49			integral:
50			\begin{equation}
51			I = f(x)dx
52			\label{eq:MCex1}
53			\end{equation}
54			The equation can be recast as:
55			\begin{equation}
56			I = (b-a)<f(x)>
57			\label{eq:MCex2}
58			\end{equation}
59			Where $<f(x)>$ is the unweighted average over the interval
60			$[a,b]$. The calculation of the integral could then be solved by
61			randomly choosing points along the interval $[a,b]$ and calculating
62			the value of $f(x)$ at each point. The accumulated average would then
63			approach $I$ in the limit where the number of trials is infintely
64			large.
65	mmeineke	914
66	mmeineke	955	However, in Statistical Mechanics, one is typically interested in
67			integrals of the form:
68			\begin{equation}
69			<A> = \frac{A}{exp^{-\beta}}
70			\label{eq:mcEnsAvg}
71			\end{equation}
72			Where $r^N$ stands for the coordinates of all $N$ particles and $A$ is
73			some observable that is only dependent on position. $<A>$ is the
74			ensemble average of $A$ as presented in
75			Sec.~\ref{introSec:statThermo}. Because $A$ is independent of
76			momentum, the momenta contribution of the integral can be factored
77			out, leaving the configurational integral. Application of the brute
78			force method to this system would yield highly inefficient
79			results. Due to the Boltzman weighting of this integral, most random
80			configurations will have a near zero contribution to the ensemble
81			average. This is where a importance sampling comes into
82			play.\cite{allen87:csl}
83	mmeineke	914
84	mmeineke	955	Importance Sampling is a method where one selects a distribution from
85			which the random configurations are chosen in order to more
86			efficiently calculate the integral.\cite{Frenkel1996} Consider again
87			Eq.~\ref{eq:MCex1} rewritten to be:
88
89
90
91	mmeineke	914	\subsection{\label{introSec:md}Molecular Dynamics Simulations}
92
93			time averages
94
95			time integrating schemes
96
97			time reversible
98
99			symplectic methods
100
101			Extended ensembles (NVT NPT)
102
103			constrained dynamics
104
105			\section{\label{introSec:chapterLayout}Chapter Layout}
106
107			\subsection{\label{introSec:RSA}Random Sequential Adsorption}
108
109			\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package}
110
111			\subsection{\label{introSec:bilayers}A Mesoscale Model for Phospholipid Bilayers}