| 3 |
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\chapter{\label{chapt:intro}Introduction and Theoretical Background} |
| 4 |
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|
| 5 |
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|
| 6 |
– |
|
| 7 |
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\section{\label{introSec:theory}Theoretical Background} |
| 8 |
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| 6 |
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The techniques used in the course of this research fall under the two |
| 7 |
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main classes of molecular simulation: Molecular Dynamics and Monte |
| 8 |
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Carlo. Molecular Dynamic simulations integrate the equations of motion |
| 9 |
< |
for a given system of particles, allowing the researher to gain |
| 9 |
> |
for a given system of particles, allowing the researcher to gain |
| 10 |
|
insight into the time dependent evolution of a system. Diffusion |
| 11 |
|
phenomena are readily studied with this simulation technique, making |
| 12 |
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Molecular Dynamics the main simulation technique used in this |
| 13 |
|
research. Other aspects of the research fall under the Monte Carlo |
| 14 |
|
class of simulations. In Monte Carlo, the configuration space |
| 15 |
< |
available to the collection of particles is sampled stochastichally, |
| 15 |
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available to the collection of particles is sampled stochastically, |
| 16 |
|
or randomly. Each configuration is chosen with a given probability |
| 17 |
< |
based on the Maxwell Boltzman distribution. These types of simulations |
| 17 |
> |
based on the Maxwell Boltzmann distribution. These types of simulations |
| 18 |
|
are best used to probe properties of a system that are only dependent |
| 19 |
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only on the state of the system. Structural information about a system |
| 20 |
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is most readily obtained through these types of methods. |
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thermodynamic properties of the system are being probed, then chose |
| 28 |
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which method best suits that objective. |
| 29 |
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|
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\subsection{\label{introSec:statThermo}Statistical Thermodynamics} |
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\section{\label{introSec:statThermo}Statistical Mechanics} |
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|
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ergodic hypothesis |
| 32 |
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The following section serves as a brief introduction to some of the |
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Statistical Mechanics concepts present in this dissertation. What |
| 34 |
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follows is a brief derivation of Boltzmann weighted statistics, and an |
| 35 |
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explanation of how one can use the information to calculate an |
| 36 |
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observable for a system. This section then concludes with a brief |
| 37 |
> |
discussion of the ergodic hypothesis and its relevance to this |
| 38 |
> |
research. |
| 39 |
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|
| 40 |
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enesemble averages |
| 40 |
> |
\subsection{\label{introSec:boltzman}Boltzmann weighted statistics} |
| 41 |
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|
| 42 |
< |
\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations} |
| 42 |
> |
Consider a system, $\gamma$, with some total energy,, $E_{\gamma}$. |
| 43 |
> |
Let $\Omega(E_{\gamma})$ represent the number of degenerate ways |
| 44 |
> |
$\boldsymbol{\Gamma}$, the collection of positions and conjugate |
| 45 |
> |
momenta of system $\gamma$, can be configured to give |
| 46 |
> |
$E_{\gamma}$. Further, if $\gamma$ is in contact with a bath system |
| 47 |
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where energy is exchanged between the two systems, $\Omega(E)$, where |
| 48 |
> |
$E$ is the total energy of both systems, can be represented as |
| 49 |
> |
\begin{equation} |
| 50 |
> |
\Omega(E) = \Omega(E_{\gamma}) \times \Omega(E - E_{\gamma}) |
| 51 |
> |
\label{introEq:SM1} |
| 52 |
> |
\end{equation} |
| 53 |
> |
Or additively as |
| 54 |
> |
\begin{equation} |
| 55 |
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\ln \Omega(E) = \ln \Omega(E_{\gamma}) + \ln \Omega(E - E_{\gamma}) |
| 56 |
> |
\label{introEq:SM2} |
| 57 |
> |
\end{equation} |
| 58 |
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|
| 59 |
< |
The Monte Carlo method was developed by Metropolis and Ulam for their |
| 60 |
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work in fissionable material.\cite{metropolis:1949} The method is so |
| 61 |
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named, because it heavily uses random numbers in the solution of the |
| 62 |
< |
problem. |
| 59 |
> |
The solution to Eq.~\ref{introEq:SM2} maximizes the number of |
| 60 |
> |
degenerative configurations in $E$. \cite{Frenkel1996} |
| 61 |
> |
This gives |
| 62 |
> |
\begin{equation} |
| 63 |
> |
\frac{\partial \ln \Omega(E)}{\partial E_{\gamma}} = 0 = |
| 64 |
> |
\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
| 65 |
> |
+ |
| 66 |
> |
\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
| 67 |
> |
\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}} |
| 68 |
> |
\label{introEq:SM3} |
| 69 |
> |
\end{equation} |
| 70 |
> |
Where $E_{\text{bath}}$ is $E-E_{\gamma}$, and |
| 71 |
> |
$\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}}$ is |
| 72 |
> |
$-1$. Eq.~\ref{introEq:SM3} becomes |
| 73 |
> |
\begin{equation} |
| 74 |
> |
\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} = |
| 75 |
> |
\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
| 76 |
> |
\label{introEq:SM4} |
| 77 |
> |
\end{equation} |
| 78 |
|
|
| 79 |
+ |
At this point, one can draw a relationship between the maximization of |
| 80 |
+ |
degeneracy in Eq.~\ref{introEq:SM3} and the second law of |
| 81 |
+ |
thermodynamics. Namely, that for a closed system, entropy will |
| 82 |
+ |
increase for an irreversible process.\cite{chandler:1987} Here the |
| 83 |
+ |
process is the partitioning of energy among the two systems. This |
| 84 |
+ |
allows the following definition of entropy: |
| 85 |
+ |
\begin{equation} |
| 86 |
+ |
S = k_B \ln \Omega(E) |
| 87 |
+ |
\label{introEq:SM5} |
| 88 |
+ |
\end{equation} |
| 89 |
+ |
Where $k_B$ is the Boltzmann constant. Having defined entropy, one can |
| 90 |
+ |
also define the temperature of the system using the relation |
| 91 |
+ |
\begin{equation} |
| 92 |
+ |
\frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V} |
| 93 |
+ |
\label{introEq:SM6} |
| 94 |
+ |
\end{equation} |
| 95 |
+ |
The temperature in the system $\gamma$ is then |
| 96 |
+ |
\begin{equation} |
| 97 |
+ |
\beta( E_{\gamma} ) = \frac{1}{k_B T} = |
| 98 |
+ |
\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
| 99 |
+ |
\label{introEq:SM7} |
| 100 |
+ |
\end{equation} |
| 101 |
+ |
Applying this to Eq.~\ref{introEq:SM4} gives the following |
| 102 |
+ |
\begin{equation} |
| 103 |
+ |
\beta( E_{\gamma} ) = \beta( E_{\text{bath}} ) |
| 104 |
+ |
\label{introEq:SM8} |
| 105 |
+ |
\end{equation} |
| 106 |
+ |
Showing that the partitioning of energy between the two systems is |
| 107 |
+ |
actually a process of thermal equilibration.\cite{Frenkel1996} |
| 108 |
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|
| 109 |
< |
\subsection{\label{introSec:md}Molecular Dynamics Simulations} |
| 109 |
> |
An application of these results is to formulate the form of an |
| 110 |
> |
expectation value of an observable, $A$, in the canonical ensemble. In |
| 111 |
> |
the canonical ensemble, the number of particles, $N$, the volume, $V$, |
| 112 |
> |
and the temperature, $T$, are all held constant while the energy, $E$, |
| 113 |
> |
is allowed to fluctuate. Returning to the previous example, the bath |
| 114 |
> |
system is now an infinitely large thermal bath, whose exchange of |
| 115 |
> |
energy with the system $\gamma$ holds the temperature constant. The |
| 116 |
> |
partitioning of energy in the bath system is then related to the total |
| 117 |
> |
energy of both systems and the fluctuations in $E_{\gamma}$: |
| 118 |
> |
\begin{equation} |
| 119 |
> |
\Omega( E_{\gamma} ) = \Omega( E - E_{\gamma} ) |
| 120 |
> |
\label{introEq:SM9} |
| 121 |
> |
\end{equation} |
| 122 |
> |
As for the expectation value, it can be defined |
| 123 |
> |
\begin{equation} |
| 124 |
> |
\langle A \rangle = |
| 125 |
> |
\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} |
| 126 |
> |
P_{\gamma} A(\boldsymbol{\Gamma}) |
| 127 |
> |
\label{introEq:SM10} |
| 128 |
> |
\end{equation} |
| 129 |
> |
Where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma}$ denotes |
| 130 |
> |
an integration over all accessible phase space, $P_{\gamma}$ is the |
| 131 |
> |
probability of being in a given phase state and |
| 132 |
> |
$A(\boldsymbol{\Gamma})$ is some observable that is a function of the |
| 133 |
> |
phase state. |
| 134 |
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|
| 135 |
< |
time averages |
| 135 |
> |
Because entropy seeks to maximize the number of degenerate states at a |
| 136 |
> |
given energy, the probability of being in a particular state in |
| 137 |
> |
$\gamma$ will be directly proportional to the number of allowable |
| 138 |
> |
states the coupled system is able to assume. Namely, |
| 139 |
> |
\begin{equation} |
| 140 |
> |
P_{\gamma} \propto \Omega( E_{\text{bath}} ) = |
| 141 |
> |
e^{\ln \Omega( E - E_{\gamma})} |
| 142 |
> |
\label{introEq:SM11} |
| 143 |
> |
\end{equation} |
| 144 |
> |
With $E_{\gamma} \ll E$, $\ln \Omega$ can be expanded in a Taylor series: |
| 145 |
> |
\begin{equation} |
| 146 |
> |
\ln \Omega ( E - E_{\gamma}) = |
| 147 |
> |
\ln \Omega (E) - |
| 148 |
> |
E_{\gamma} \frac{\partial \ln \Omega }{\partial E} |
| 149 |
> |
+ \ldots |
| 150 |
> |
\label{introEq:SM12} |
| 151 |
> |
\end{equation} |
| 152 |
> |
Higher order terms are omitted as $E$ is an infinite thermal |
| 153 |
> |
bath. Further, using Eq.~\ref{introEq:SM7}, Eq.~\ref{introEq:SM11} can |
| 154 |
> |
be rewritten: |
| 155 |
> |
\begin{equation} |
| 156 |
> |
P_{\gamma} \propto e^{-\beta E_{\gamma}} |
| 157 |
> |
\label{introEq:SM13} |
| 158 |
> |
\end{equation} |
| 159 |
> |
Where $\ln \Omega(E)$ has been factored out of the proportionality as a |
| 160 |
> |
constant. Normalizing the probability ($\int\limits_{\boldsymbol{\Gamma}} |
| 161 |
> |
d\boldsymbol{\Gamma} P_{\gamma} = 1$) gives |
| 162 |
> |
\begin{equation} |
| 163 |
> |
P_{\gamma} = \frac{e^{-\beta E_{\gamma}}} |
| 164 |
> |
{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}} |
| 165 |
> |
\label{introEq:SM14} |
| 166 |
> |
\end{equation} |
| 167 |
> |
This result is the standard Boltzmann statistical distribution. |
| 168 |
> |
Applying it to Eq.~\ref{introEq:SM10} one can obtain the following relation for ensemble averages: |
| 169 |
> |
\begin{equation} |
| 170 |
> |
\langle A \rangle = |
| 171 |
> |
\frac{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} |
| 172 |
> |
A( \boldsymbol{\Gamma} ) e^{-\beta E_{\gamma}}} |
| 173 |
> |
{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}} |
| 174 |
> |
\label{introEq:SM15} |
| 175 |
> |
\end{equation} |
| 176 |
|
|
| 177 |
< |
time integrating schemes |
| 177 |
> |
\subsection{\label{introSec:ergodic}The Ergodic Hypothesis} |
| 178 |
|
|
| 179 |
< |
time reversible |
| 179 |
> |
One last important consideration is that of ergodicity. Ergodicity is |
| 180 |
> |
the assumption that given an infinite amount of time, a system will |
| 181 |
> |
visit every available point in phase space.\cite{Frenkel1996} For most |
| 182 |
> |
systems, this is a valid assumption, except in cases where the system |
| 183 |
> |
may be trapped in a local feature (\emph{e.g.}~glasses). When valid, |
| 184 |
> |
ergodicity allows the unification of a time averaged observation and |
| 185 |
> |
an ensemble averaged one. If an observation is averaged over a |
| 186 |
> |
sufficiently long time, the system is assumed to visit all |
| 187 |
> |
appropriately available points in phase space, giving a properly |
| 188 |
> |
weighted statistical average. This allows the researcher freedom of |
| 189 |
> |
choice when deciding how best to measure a given observable. When an |
| 190 |
> |
ensemble averaged approach seems most logical, the Monte Carlo |
| 191 |
> |
techniques described in Sec.~\ref{introSec:monteCarlo} can be utilized. |
| 192 |
> |
Conversely, if a problem lends itself to a time averaging approach, |
| 193 |
> |
the Molecular Dynamics techniques in Sec.~\ref{introSec:MD} can be |
| 194 |
> |
employed. |
| 195 |
|
|
| 196 |
< |
symplectic methods |
| 196 |
> |
\section{\label{introSec:monteCarlo}Monte Carlo Simulations} |
| 197 |
|
|
| 198 |
< |
Extended ensembles (NVT NPT) |
| 198 |
> |
The Monte Carlo method was developed by Metropolis and Ulam for their |
| 199 |
> |
work in fissionable material.\cite{metropolis:1949} The method is so |
| 200 |
> |
named, because it heavily uses random numbers in its |
| 201 |
> |
solution.\cite{allen87:csl} The Monte Carlo method allows for the |
| 202 |
> |
solution of integrals through the stochastic sampling of the values |
| 203 |
> |
within the integral. In the simplest case, the evaluation of an |
| 204 |
> |
integral would follow a brute force method of |
| 205 |
> |
sampling.\cite{Frenkel1996} Consider the following single dimensional |
| 206 |
> |
integral: |
| 207 |
> |
\begin{equation} |
| 208 |
> |
I = f(x)dx |
| 209 |
> |
\label{eq:MCex1} |
| 210 |
> |
\end{equation} |
| 211 |
> |
The equation can be recast as: |
| 212 |
> |
\begin{equation} |
| 213 |
> |
I = (b-a)\langle f(x) \rangle |
| 214 |
> |
\label{eq:MCex2} |
| 215 |
> |
\end{equation} |
| 216 |
> |
Where $\langle f(x) \rangle$ is the unweighted average over the interval |
| 217 |
> |
$[a,b]$. The calculation of the integral could then be solved by |
| 218 |
> |
randomly choosing points along the interval $[a,b]$ and calculating |
| 219 |
> |
the value of $f(x)$ at each point. The accumulated average would then |
| 220 |
> |
approach $I$ in the limit where the number of trials is infinitely |
| 221 |
> |
large. |
| 222 |
|
|
| 223 |
< |
constrained dynamics |
| 224 |
< |
|
| 225 |
< |
\section{\label{introSec:chapterLayout}Chapter Layout} |
| 226 |
< |
|
| 227 |
< |
\subsection{\label{introSec:RSA}Random Sequential Adsorption} |
| 228 |
< |
|
| 229 |
< |
\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package} |
| 223 |
> |
However, in Statistical Mechanics, one is typically interested in |
| 224 |
> |
integrals of the form: |
| 225 |
> |
\begin{equation} |
| 226 |
> |
\langle A \rangle = \frac{\int d^N \mathbf{r}~A(\mathbf{r}^N)% |
| 227 |
> |
e^{-\beta V(\mathbf{r}^N)}}% |
| 228 |
> |
{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}} |
| 229 |
> |
\label{eq:mcEnsAvg} |
| 230 |
> |
\end{equation} |
| 231 |
> |
Where $\mathbf{r}^N$ stands for the coordinates of all $N$ particles |
| 232 |
> |
and $A$ is some observable that is only dependent on position. This is |
| 233 |
> |
the ensemble average of $A$ as presented in |
| 234 |
> |
Sec.~\ref{introSec:statThermo}, except here $A$ is independent of |
| 235 |
> |
momentum. Therefore the momenta contribution of the integral can be |
| 236 |
> |
factored out, leaving the configurational integral. Application of the |
| 237 |
> |
brute force method to this system would yield highly inefficient |
| 238 |
> |
results. Due to the Boltzmann weighting of this integral, most random |
| 239 |
> |
configurations will have a near zero contribution to the ensemble |
| 240 |
> |
average. This is where importance sampling comes into |
| 241 |
> |
play.\cite{allen87:csl} |
| 242 |
|
|
| 243 |
< |
\subsection{\label{introSec:bilayers}A Mesoscale Model for Phospholipid Bilayers} |
| 243 |
> |
Importance Sampling is a method where one selects a distribution from |
| 244 |
> |
which the random configurations are chosen in order to more |
| 245 |
> |
efficiently calculate the integral.\cite{Frenkel1996} Consider again |
| 246 |
> |
Eq.~\ref{eq:MCex1} rewritten to be: |
| 247 |
> |
\begin{equation} |
| 248 |
> |
I = \int^b_a \frac{f(x)}{\rho(x)} \rho(x) dx |
| 249 |
> |
\label{introEq:Importance1} |
| 250 |
> |
\end{equation} |
| 251 |
> |
Where $\rho(x)$ is an arbitrary probability distribution in $x$. If |
| 252 |
> |
one conducts $\tau$ trials selecting a random number, $\zeta_\tau$, |
| 253 |
> |
from the distribution $\rho(x)$ on the interval $[a,b]$, then |
| 254 |
> |
Eq.~\ref{introEq:Importance1} becomes |
| 255 |
> |
\begin{equation} |
| 256 |
> |
I= \biggl \langle \frac{f(x)}{\rho(x)} \biggr \rangle_{\text{trials}} |
| 257 |
> |
\label{introEq:Importance2} |
| 258 |
> |
\end{equation} |
| 259 |
> |
Looking at Eq.~\ref{eq:mcEnsAvg}, and realizing |
| 260 |
> |
\begin {equation} |
| 261 |
> |
\rho_{kT}(\mathbf{r}^N) = |
| 262 |
> |
\frac{e^{-\beta V(\mathbf{r}^N)}} |
| 263 |
> |
{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}} |
| 264 |
> |
\label{introEq:MCboltzman} |
| 265 |
> |
\end{equation} |
| 266 |
> |
Where $\rho_{kT}$ is the Boltzmann distribution. The ensemble average |
| 267 |
> |
can be rewritten as |
| 268 |
> |
\begin{equation} |
| 269 |
> |
\langle A \rangle = \int d^N \mathbf{r}~A(\mathbf{r}^N) |
| 270 |
> |
\rho_{kT}(\mathbf{r}^N) |
| 271 |
> |
\label{introEq:Importance3} |
| 272 |
> |
\end{equation} |
| 273 |
> |
Applying Eq.~\ref{introEq:Importance1} one obtains |
| 274 |
> |
\begin{equation} |
| 275 |
> |
\langle A \rangle = \biggl \langle |
| 276 |
> |
\frac{ A \rho_{kT}(\mathbf{r}^N) } |
| 277 |
> |
{\rho(\mathbf{r}^N)} \biggr \rangle_{\text{trials}} |
| 278 |
> |
\label{introEq:Importance4} |
| 279 |
> |
\end{equation} |
| 280 |
> |
By selecting $\rho(\mathbf{r}^N)$ to be $\rho_{kT}(\mathbf{r}^N)$ |
| 281 |
> |
Eq.~\ref{introEq:Importance4} becomes |
| 282 |
> |
\begin{equation} |
| 283 |
> |
\langle A \rangle = \langle A(\mathbf{r}^N) \rangle_{\text{trials}} |
| 284 |
> |
\label{introEq:Importance5} |
| 285 |
> |
\end{equation} |
| 286 |
> |
The difficulty is selecting points $\mathbf{r}^N$ such that they are |
| 287 |
> |
sampled from the distribution $\rho_{kT}(\mathbf{r}^N)$. A solution |
| 288 |
> |
was proposed by Metropolis \emph{et al}.\cite{metropolis:1953} which involved |
| 289 |
> |
the use of a Markov chain whose limiting distribution was |
| 290 |
> |
$\rho_{kT}(\mathbf{r}^N)$. |
| 291 |
> |
|
| 292 |
> |
\subsection{\label{introSec:markovChains}Markov Chains} |
| 293 |
> |
|
| 294 |
> |
A Markov chain is a chain of states satisfying the following |
| 295 |
> |
conditions:\cite{leach01:mm} |
| 296 |
> |
\begin{enumerate} |
| 297 |
> |
\item The outcome of each trial depends only on the outcome of the previous trial. |
| 298 |
> |
\item Each trial belongs to a finite set of outcomes called the state space. |
| 299 |
> |
\end{enumerate} |
| 300 |
> |
If given two configurations, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$, |
| 301 |
> |
$\rho_m$ and $\rho_n$ are the probabilities of being in state |
| 302 |
> |
$\mathbf{r}^N_m$ and $\mathbf{r}^N_n$ respectively. Further, the two |
| 303 |
> |
states are linked by a transition probability, $\pi_{mn}$, which is the |
| 304 |
> |
probability of going from state $m$ to state $n$. |
| 305 |
> |
|
| 306 |
> |
\newcommand{\accMe}{\operatorname{acc}} |
| 307 |
> |
|
| 308 |
> |
The transition probability is given by the following: |
| 309 |
> |
\begin{equation} |
| 310 |
> |
\pi_{mn} = \alpha_{mn} \times \accMe(m \rightarrow n) |
| 311 |
> |
\label{introEq:MCpi} |
| 312 |
> |
\end{equation} |
| 313 |
> |
Where $\alpha_{mn}$ is the probability of attempting the move $m |
| 314 |
> |
\rightarrow n$, and $\accMe$ is the probability of accepting the move |
| 315 |
> |
$m \rightarrow n$. Defining a probability vector, |
| 316 |
> |
$\boldsymbol{\rho}$, such that |
| 317 |
> |
\begin{equation} |
| 318 |
> |
\boldsymbol{\rho} = \{\rho_1, \rho_2, \ldots \rho_m, \rho_n, |
| 319 |
> |
\ldots \rho_N \} |
| 320 |
> |
\label{introEq:MCrhoVector} |
| 321 |
> |
\end{equation} |
| 322 |
> |
a transition matrix $\boldsymbol{\Pi}$ can be defined, |
| 323 |
> |
whose elements are $\pi_{mn}$, for each given transition. The |
| 324 |
> |
limiting distribution of the Markov chain can then be found by |
| 325 |
> |
applying the transition matrix an infinite number of times to the |
| 326 |
> |
distribution vector. |
| 327 |
> |
\begin{equation} |
| 328 |
> |
\boldsymbol{\rho}_{\text{limit}} = |
| 329 |
> |
\lim_{N \rightarrow \infty} \boldsymbol{\rho}_{\text{initial}} |
| 330 |
> |
\boldsymbol{\Pi}^N |
| 331 |
> |
\label{introEq:MCmarkovLimit} |
| 332 |
> |
\end{equation} |
| 333 |
> |
The limiting distribution of the chain is independent of the starting |
| 334 |
> |
distribution, and successive applications of the transition matrix |
| 335 |
> |
will only yield the limiting distribution again. |
| 336 |
> |
\begin{equation} |
| 337 |
> |
\boldsymbol{\rho}_{\text{limit}} = \boldsymbol{\rho}_{\text{initial}} |
| 338 |
> |
\boldsymbol{\Pi} |
| 339 |
> |
\label{introEq:MCmarkovEquil} |
| 340 |
> |
\end{equation} |
| 341 |
> |
|
| 342 |
> |
\subsection{\label{introSec:metropolisMethod}The Metropolis Method} |
| 343 |
> |
|
| 344 |
> |
In the Metropolis method\cite{metropolis:1953} |
| 345 |
> |
Eq.~\ref{introEq:MCmarkovEquil} is solved such that |
| 346 |
> |
$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzmann distribution |
| 347 |
> |
of states. The method accomplishes this by imposing the strong |
| 348 |
> |
condition of microscopic reversibility on the equilibrium |
| 349 |
> |
distribution. Meaning, that at equilibrium the probability of going |
| 350 |
> |
from $m$ to $n$ is the same as going from $n$ to $m$. |
| 351 |
> |
\begin{equation} |
| 352 |
> |
\rho_m\pi_{mn} = \rho_n\pi_{nm} |
| 353 |
> |
\label{introEq:MCmicroReverse} |
| 354 |
> |
\end{equation} |
| 355 |
> |
Further, $\boldsymbol{\alpha}$ is chosen to be a symmetric matrix in |
| 356 |
> |
the Metropolis method. Using Eq.~\ref{introEq:MCpi}, |
| 357 |
> |
Eq.~\ref{introEq:MCmicroReverse} becomes |
| 358 |
> |
\begin{equation} |
| 359 |
> |
\frac{\accMe(m \rightarrow n)}{\accMe(n \rightarrow m)} = |
| 360 |
> |
\frac{\rho_n}{\rho_m} |
| 361 |
> |
\label{introEq:MCmicro2} |
| 362 |
> |
\end{equation} |
| 363 |
> |
For a Boltzmann limiting distribution, |
| 364 |
> |
\begin{equation} |
| 365 |
> |
\frac{\rho_n}{\rho_m} = e^{-\beta[\mathcal{U}(n) - \mathcal{U}(m)]} |
| 366 |
> |
= e^{-\beta \Delta \mathcal{U}} |
| 367 |
> |
\label{introEq:MCmicro3} |
| 368 |
> |
\end{equation} |
| 369 |
> |
This allows for the following set of acceptance rules be defined: |
| 370 |
> |
\begin{equation} |
| 371 |
> |
\accMe( m \rightarrow n ) = |
| 372 |
> |
\begin{cases} |
| 373 |
> |
1& \text{if $\Delta \mathcal{U} \leq 0$,} \\ |
| 374 |
> |
e^{-\beta \Delta \mathcal{U}}& \text{if $\Delta \mathcal{U} > 0$.} |
| 375 |
> |
\end{cases} |
| 376 |
> |
\label{introEq:accRules} |
| 377 |
> |
\end{equation} |
| 378 |
> |
|
| 379 |
> |
Using the acceptance criteria from Eq.~\ref{introEq:accRules} the |
| 380 |
> |
Metropolis method proceeds as follows |
| 381 |
> |
\begin{enumerate} |
| 382 |
> |
\item Generate an initial configuration $\mathbf{r}^N$ which has some finite probability in $\rho_{kT}$. |
| 383 |
> |
\item Modify $\mathbf{r}^N$, to generate configuration $\mathbf{r^{\prime}}^N$. |
| 384 |
> |
\item If the new configuration lowers the energy of the system, accept the move with unity ($\mathbf{r}^N$ becomes $\mathbf{r^{\prime}}^N$). Otherwise accept with probability $e^{-\beta \Delta \mathcal{U}}$. |
| 385 |
> |
\item Accumulate the average for the configurational observable of interest. |
| 386 |
> |
\item Repeat from step 2 until the average converges. |
| 387 |
> |
\end{enumerate} |
| 388 |
> |
One important note is that the average is accumulated whether the move |
| 389 |
> |
is accepted or not, this ensures proper weighting of the average. |
| 390 |
> |
Using Eq.~\ref{introEq:Importance4} it becomes clear that the |
| 391 |
> |
accumulated averages are the ensemble averages, as this method ensures |
| 392 |
> |
that the limiting distribution is the Boltzmann distribution. |
| 393 |
> |
|
| 394 |
> |
\section{\label{introSec:MD}Molecular Dynamics Simulations} |
| 395 |
> |
|
| 396 |
> |
The main simulation tool used in this research is Molecular Dynamics. |
| 397 |
> |
Molecular Dynamics is when the equations of motion for a system are |
| 398 |
> |
integrated in order to obtain information about both the positions and |
| 399 |
> |
momentum of a system, allowing the calculation of not only |
| 400 |
> |
configurational observables, but momenta dependent ones as well: |
| 401 |
> |
diffusion constants, velocity auto correlations, folding/unfolding |
| 402 |
> |
events, etc. Due to the principle of ergodicity, |
| 403 |
> |
Sec.~\ref{introSec:ergodic}, the average of these observables over the |
| 404 |
> |
time period of the simulation are taken to be the ensemble averages |
| 405 |
> |
for the system. |
| 406 |
> |
|
| 407 |
> |
The choice of when to use molecular dynamics over Monte Carlo |
| 408 |
> |
techniques, is normally decided by the observables in which the |
| 409 |
> |
researcher is interested. If the observables depend on momenta in |
| 410 |
> |
any fashion, then the only choice is molecular dynamics in some form. |
| 411 |
> |
However, when the observable is dependent only on the configuration, |
| 412 |
> |
then most of the time Monte Carlo techniques will be more efficient. |
| 413 |
> |
|
| 414 |
> |
The focus of research in the second half of this dissertation is |
| 415 |
> |
centered around the dynamic properties of phospholipid bilayers, |
| 416 |
> |
making molecular dynamics key in the simulation of those properties. |
| 417 |
> |
|
| 418 |
> |
\subsection{\label{introSec:mdAlgorithm}The Molecular Dynamics Algorithm} |
| 419 |
> |
|
| 420 |
> |
To illustrate how the molecular dynamics technique is applied, the |
| 421 |
> |
following sections will describe the sequence involved in a |
| 422 |
> |
simulation. Sec.~\ref{introSec:mdInit} deals with the initialization |
| 423 |
> |
of a simulation. Sec.~\ref{introSec:mdForce} discusses issues |
| 424 |
> |
involved with the calculation of the forces. |
| 425 |
> |
Sec.~\ref{introSec:mdIntegrate} concludes the algorithm discussion |
| 426 |
> |
with the integration of the equations of motion.\cite{Frenkel1996} |
| 427 |
> |
|
| 428 |
> |
\subsection{\label{introSec:mdInit}Initialization} |
| 429 |
> |
|
| 430 |
> |
When selecting the initial configuration for the simulation it is |
| 431 |
> |
important to consider what dynamics one is hoping to observe. |
| 432 |
> |
Ch.~\ref{chapt:lipid} deals with the formation and equilibrium dynamics of |
| 433 |
> |
phospholipid membranes. Therefore in these simulations initial |
| 434 |
> |
positions were selected that in some cases dispersed the lipids in |
| 435 |
> |
water, and in other cases structured the lipids into performed |
| 436 |
> |
bilayers. Important considerations at this stage of the simulation are: |
| 437 |
> |
\begin{itemize} |
| 438 |
> |
\item There are no major overlaps of molecular or atomic orbitals |
| 439 |
> |
\item Velocities are chosen in such a way as to not give the system a non=zero total momentum or angular momentum. |
| 440 |
> |
\item It is also sometimes desirable to select the velocities to correctly sample the target temperature. |
| 441 |
> |
\end{itemize} |
| 442 |
> |
|
| 443 |
> |
The first point is important due to the amount of potential energy |
| 444 |
> |
generated by having two particles too close together. If overlap |
| 445 |
> |
occurs, the first evaluation of forces will return numbers so large as |
| 446 |
> |
to render the numerical integration of the motion meaningless. The |
| 447 |
> |
second consideration keeps the system from drifting or rotating as a |
| 448 |
> |
whole. This arises from the fact that most simulations are of systems |
| 449 |
> |
in equilibrium in the absence of outside forces. Therefore any net |
| 450 |
> |
movement would be unphysical and an artifact of the simulation method |
| 451 |
> |
used. The final point addresses the selection of the magnitude of the |
| 452 |
> |
initial velocities. For many simulations it is convenient to use |
| 453 |
> |
this opportunity to scale the amount of kinetic energy to reflect the |
| 454 |
> |
desired thermal distribution of the system. However, it must be noted |
| 455 |
> |
that most systems will require further velocity rescaling after the |
| 456 |
> |
first few initial simulation steps due to either loss or gain of |
| 457 |
> |
kinetic energy from energy stored in potential degrees of freedom. |
| 458 |
> |
|
| 459 |
> |
\subsection{\label{introSec:mdForce}Force Evaluation} |
| 460 |
> |
|
| 461 |
> |
The evaluation of forces is the most computationally expensive portion |
| 462 |
> |
of a given molecular dynamics simulation. This is due entirely to the |
| 463 |
> |
evaluation of long range forces in a simulation, typically pair-wise. |
| 464 |
> |
These forces are most commonly the Van der Waals force, and sometimes |
| 465 |
> |
Coulombic forces as well. For a pair-wise force, there are $N(N-1)/ 2$ |
| 466 |
> |
pairs to be evaluated, where $N$ is the number of particles in the |
| 467 |
> |
system. This leads to the calculations scaling as $N^2$, making large |
| 468 |
> |
simulations prohibitive in the absence of any computation saving |
| 469 |
> |
techniques. |
| 470 |
> |
|
| 471 |
> |
Another consideration one must resolve, is that in a given simulation |
| 472 |
> |
a disproportionate number of the particles will feel the effects of |
| 473 |
> |
the surface.\cite{allen87:csl} For a cubic system of 1000 particles |
| 474 |
> |
arranged in a $10 \times 10 \times 10$ cube, 488 particles will be |
| 475 |
> |
exposed to the surface. Unless one is simulating an isolated particle |
| 476 |
> |
group in a vacuum, the behavior of the system will be far from the |
| 477 |
> |
desired bulk characteristics. To offset this, simulations employ the |
| 478 |
> |
use of periodic boundary images.\cite{born:1912} |
| 479 |
> |
|
| 480 |
> |
The technique involves the use of an algorithm that replicates the |
| 481 |
> |
simulation box on an infinite lattice in Cartesian space. Any given |
| 482 |
> |
particle leaving the simulation box on one side will have an image of |
| 483 |
> |
itself enter on the opposite side (see Fig.~\ref{introFig:pbc}). In |
| 484 |
> |
addition, this sets that any two particles have an image, real or |
| 485 |
> |
periodic, within $\text{box}/2$ of each other. A discussion of the |
| 486 |
> |
method used to calculate the periodic image can be found in |
| 487 |
> |
Sec.\ref{oopseSec:pbc}. |
| 488 |
> |
|
| 489 |
> |
\begin{figure} |
| 490 |
> |
\centering |
| 491 |
> |
\includegraphics[width=\linewidth]{pbcFig.eps} |
| 492 |
> |
\caption[An illustration of periodic boundary conditions]{A 2-D illustration of periodic boundary conditions. As one particle leaves the right of the simulation box, an image of it enters the left.} |
| 493 |
> |
\label{introFig:pbc} |
| 494 |
> |
\end{figure} |
| 495 |
> |
|
| 496 |
> |
Returning to the topic of the computational scale of the force |
| 497 |
> |
evaluation, the use of periodic boundary conditions requires that a |
| 498 |
> |
cutoff radius be employed. Using a cutoff radius improves the |
| 499 |
> |
efficiency of the force evaluation, as particles farther than a |
| 500 |
> |
predetermined distance, $r_{\text{cut}}$, are not included in the |
| 501 |
> |
calculation.\cite{Frenkel1996} In a simulation with periodic images, |
| 502 |
> |
$r_{\text{cut}}$ has a maximum value of $\text{box}/2$. |
| 503 |
> |
Fig.~\ref{introFig:rMax} illustrates how when using an |
| 504 |
> |
$r_{\text{cut}}$ larger than this value, or in the extreme limit of no |
| 505 |
> |
$r_{\text{cut}}$ at all, the corners of the simulation box are |
| 506 |
> |
unequally weighted due to the lack of particle images in the $x$, $y$, |
| 507 |
> |
or $z$ directions past a distance of $\text{box} / 2$. |
| 508 |
> |
|
| 509 |
> |
\begin{figure} |
| 510 |
> |
\centering |
| 511 |
> |
\includegraphics[width=\linewidth]{rCutMaxFig.eps} |
| 512 |
> |
\caption[An explanation of $r_{\text{cut}}$]{The yellow atom has all other images wrapped to itself as the center. If $r_{\text{cut}}=\text{box}/2$, then the distribution is uniform (blue atoms). However, when $r_{\text{cut}}>\text{box}/2$ the corners are disproportionately weighted (green atoms) vs the axial directions (shaded regions).} |
| 513 |
> |
\label{introFig:rMax} |
| 514 |
> |
\end{figure} |
| 515 |
> |
|
| 516 |
> |
With the use of an $r_{\text{cut}}$, however, comes a discontinuity in |
| 517 |
> |
the potential energy curve (Fig.~\ref{introFig:shiftPot}). To fix this |
| 518 |
> |
discontinuity, one calculates the potential energy at the |
| 519 |
> |
$r_{\text{cut}}$, and adds that value to the potential, causing |
| 520 |
> |
the function to go smoothly to zero at the cutoff radius. This |
| 521 |
> |
shifted potential ensures conservation of energy when integrating the |
| 522 |
> |
Newtonian equations of motion. |
| 523 |
> |
|
| 524 |
> |
\begin{figure} |
| 525 |
> |
\centering |
| 526 |
> |
\includegraphics[width=\linewidth]{shiftedPot.eps} |
| 527 |
> |
\caption[Shifting the Lennard-Jones Potential]{The Lennard-Jones potential (blue line) is shifted (red line) to remove the discontinuity at $r_{\text{cut}}$.} |
| 528 |
> |
\label{introFig:shiftPot} |
| 529 |
> |
\end{figure} |
| 530 |
> |
|
| 531 |
> |
The second main simplification used in this research is the Verlet |
| 532 |
> |
neighbor list. \cite{allen87:csl} In the Verlet method, one generates |
| 533 |
> |
a list of all neighbor atoms, $j$, surrounding atom $i$ within some |
| 534 |
> |
cutoff $r_{\text{list}}$, where $r_{\text{list}}>r_{\text{cut}}$. |
| 535 |
> |
This list is created the first time forces are evaluated, then on |
| 536 |
> |
subsequent force evaluations, pair calculations are only calculated |
| 537 |
> |
from the neighbor lists. The lists are updated if any given particle |
| 538 |
> |
in the system moves farther than $r_{\text{list}}-r_{\text{cut}}$, |
| 539 |
> |
giving rise to the possibility that a particle has left or joined a |
| 540 |
> |
neighbor list. |
| 541 |
> |
|
| 542 |
> |
\subsection{\label{introSec:mdIntegrate} Integration of the equations of motion} |
| 543 |
> |
|
| 544 |
> |
A starting point for the discussion of molecular dynamics integrators |
| 545 |
> |
is the Verlet algorithm.\cite{Frenkel1996} It begins with a Taylor |
| 546 |
> |
expansion of position in time: |
| 547 |
> |
\begin{equation} |
| 548 |
> |
q(t+\Delta t)= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 + |
| 549 |
> |
\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} + |
| 550 |
> |
\mathcal{O}(\Delta t^4) |
| 551 |
> |
\label{introEq:verletForward} |
| 552 |
> |
\end{equation} |
| 553 |
> |
As well as, |
| 554 |
> |
\begin{equation} |
| 555 |
> |
q(t-\Delta t)= q(t) - v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 - |
| 556 |
> |
\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} + |
| 557 |
> |
\mathcal{O}(\Delta t^4) |
| 558 |
> |
\label{introEq:verletBack} |
| 559 |
> |
\end{equation} |
| 560 |
> |
Where $m$ is the mass of the particle, $q(t)$ is the position at time |
| 561 |
> |
$t$, $v(t)$ the velocity, and $F(t)$ the force acting on the |
| 562 |
> |
particle. Adding together Eq.~\ref{introEq:verletForward} and |
| 563 |
> |
Eq.~\ref{introEq:verletBack} results in, |
| 564 |
> |
\begin{equation} |
| 565 |
> |
q(t+\Delta t)+q(t-\Delta t) = |
| 566 |
> |
2q(t) + \frac{F(t)}{m}\Delta t^2 + \mathcal{O}(\Delta t^4) |
| 567 |
> |
\label{introEq:verletSum} |
| 568 |
> |
\end{equation} |
| 569 |
> |
Or equivalently, |
| 570 |
> |
\begin{equation} |
| 571 |
> |
q(t+\Delta t) \approx |
| 572 |
> |
2q(t) - q(t-\Delta t) + \frac{F(t)}{m}\Delta t^2 |
| 573 |
> |
\label{introEq:verletFinal} |
| 574 |
> |
\end{equation} |
| 575 |
> |
Which contains an error in the estimate of the new positions on the |
| 576 |
> |
order of $\Delta t^4$. |
| 577 |
> |
|
| 578 |
> |
In practice, however, the simulations in this research were integrated |
| 579 |
> |
with a velocity reformulation of the Verlet method.\cite{allen87:csl} |
| 580 |
> |
\begin{align} |
| 581 |
> |
q(t+\Delta t) &= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 % |
| 582 |
> |
\label{introEq:MDvelVerletPos} \\% |
| 583 |
> |
% |
| 584 |
> |
v(t+\Delta t) &= v(t) + \frac{\Delta t}{2m}[F(t) + F(t+\Delta t)] % |
| 585 |
> |
\label{introEq:MDvelVerletVel} |
| 586 |
> |
\end{align} |
| 587 |
> |
The original Verlet algorithm can be regained by substituting the |
| 588 |
> |
velocity back into Eq.~\ref{introEq:MDvelVerletPos}. The Verlet |
| 589 |
> |
formulations are chosen in this research because the algorithms have |
| 590 |
> |
very little long term drift in energy conservation. Energy |
| 591 |
> |
conservation in a molecular dynamics simulation is of extreme |
| 592 |
> |
importance, as it is a measure of how closely one is following the |
| 593 |
> |
``true'' trajectory with the finite integration scheme. An exact |
| 594 |
> |
solution to the integration will conserve area in phase space, as well |
| 595 |
> |
as be reversible in time, that is, the trajectory integrated forward |
| 596 |
> |
or backwards will exactly match itself. Having a finite algorithm |
| 597 |
> |
that both conserves area in phase space and is time reversible, |
| 598 |
> |
therefore increases, but does not guarantee the ``correctness'' or the |
| 599 |
> |
integrated trajectory. |
| 600 |
> |
|
| 601 |
> |
It can be shown,\cite{Frenkel1996} that although the Verlet algorithm |
| 602 |
> |
does not rigorously preserve the actual Hamiltonian, it does preserve |
| 603 |
> |
a pseudo-Hamiltonian which shadows the real one in phase space. This |
| 604 |
> |
pseudo-Hamiltonian is provably area-conserving as well as time |
| 605 |
> |
reversible. The fact that it shadows the true Hamiltonian in phase |
| 606 |
> |
space is acceptable in actual simulations as one is interested in the |
| 607 |
> |
ensemble average of the observable being measured. From the ergodic |
| 608 |
> |
hypothesis (Sec.~\ref{introSec:statThermo}), it is known that the time |
| 609 |
> |
average will match the ensemble average, therefore two similar |
| 610 |
> |
trajectories in phase space should give matching statistical averages. |
| 611 |
> |
|
| 612 |
> |
\subsection{\label{introSec:MDfurther}Further Considerations} |
| 613 |
> |
|
| 614 |
> |
In the simulations presented in this research, a few additional |
| 615 |
> |
parameters are needed to describe the motions. The simulations |
| 616 |
> |
involving water and phospholipids in Ch.~\ref{chapt:lipid} are |
| 617 |
> |
required to integrate the equations of motions for dipoles on atoms. |
| 618 |
> |
This involves an additional three parameters be specified for each |
| 619 |
> |
dipole atom: $\phi$, $\theta$, and $\psi$. These three angles are |
| 620 |
> |
taken to be the Euler angles, where $\phi$ is a rotation about the |
| 621 |
> |
$z$-axis, and $\theta$ is a rotation about the new $x$-axis, and |
| 622 |
> |
$\psi$ is a final rotation about the new $z$-axis (see |
| 623 |
> |
Fig.~\ref{introFig:eulerAngles}). This sequence of rotations can be |
| 624 |
> |
accumulated into a single $3 \times 3$ matrix, $\mathbf{A}$, |
| 625 |
> |
defined as follows: |
| 626 |
> |
\begin{equation} |
| 627 |
> |
\mathbf{A} = |
| 628 |
> |
\begin{bmatrix} |
| 629 |
> |
\cos\phi\cos\psi-\sin\phi\cos\theta\sin\psi &% |
| 630 |
> |
\sin\phi\cos\psi+\cos\phi\cos\theta\sin\psi &% |
| 631 |
> |
\sin\theta\sin\psi \\% |
| 632 |
> |
% |
| 633 |
> |
-\cos\phi\sin\psi-\sin\phi\cos\theta\cos\psi &% |
| 634 |
> |
-\sin\phi\sin\psi+\cos\phi\cos\theta\cos\psi &% |
| 635 |
> |
\sin\theta\cos\psi \\% |
| 636 |
> |
% |
| 637 |
> |
\sin\phi\sin\theta &% |
| 638 |
> |
-\cos\phi\sin\theta &% |
| 639 |
> |
\cos\theta |
| 640 |
> |
\end{bmatrix} |
| 641 |
> |
\label{introEq:EulerRotMat} |
| 642 |
> |
\end{equation} |
| 643 |
> |
|
| 644 |
> |
\begin{figure} |
| 645 |
> |
\caentering |
| 646 |
> |
\includegraphics[width=\linewidth]{eulerRotFig.eps} |
| 647 |
> |
\caption[Euler rotation of Cartesian coordinates]{The rotation scheme for Euler angles. First is a rotation of $\phi$ about the $z$ axis (blue rotation). Next is a rotation of $\theta$ about the new $x\prime$ axis (green rotation). Lastly is a final rotation of $\psi$ about the new $z\prime$ axis (red rotation).} |
| 648 |
> |
\label{introFig:eulerAngles} |
| 649 |
> |
\end{figure} |
| 650 |
> |
|
| 651 |
> |
The equations of motion for Euler angles can be written down |
| 652 |
> |
as\cite{allen87:csl} |
| 653 |
> |
\begin{align} |
| 654 |
> |
\dot{\phi} &= -\omega^s_x \frac{\sin\phi\cos\theta}{\sin\theta} + |
| 655 |
> |
\omega^s_y \frac{\cos\phi\cos\theta}{\sin\theta} + |
| 656 |
> |
\omega^s_z |
| 657 |
> |
\label{introEq:MDeulerPhi} \\% |
| 658 |
> |
% |
| 659 |
> |
\dot{\theta} &= \omega^s_x \cos\phi + \omega^s_y \sin\phi |
| 660 |
> |
\label{introEq:MDeulerTheta} \\% |
| 661 |
> |
% |
| 662 |
> |
\dot{\psi} &= \omega^s_x \frac{\sin\phi}{\sin\theta} - |
| 663 |
> |
\omega^s_y \frac{\cos\phi}{\sin\theta} |
| 664 |
> |
\label{introEq:MDeulerPsi} |
| 665 |
> |
\end{align} |
| 666 |
> |
Where $\omega^s_i$ is the angular velocity in the lab space frame |
| 667 |
> |
along Cartesian coordinate $i$. However, a difficulty arises when |
| 668 |
> |
attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and |
| 669 |
> |
Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in |
| 670 |
> |
both equations means there is a non-physical instability present when |
| 671 |
> |
$\theta$ is 0 or $\pi$. To correct for this, the simulations integrate |
| 672 |
> |
the rotation matrix, $\mathbf{A}$, directly, thus avoiding the |
| 673 |
> |
instability. This method was proposed by Dullweber |
| 674 |
> |
\emph{et. al.}\cite{Dullweber1997}, and is presented in |
| 675 |
> |
Sec.~\ref{introSec:MDsymplecticRot}. |
| 676 |
> |
|
| 677 |
> |
\subsection{\label{introSec:MDliouville}Liouville Propagator} |
| 678 |
> |
|
| 679 |
> |
Before discussing the integration of the rotation matrix, it is |
| 680 |
> |
necessary to understand the construction of a ``good'' integration |
| 681 |
> |
scheme. It has been previously |
| 682 |
> |
discussed(Sec.~\ref{introSec:mdIntegrate}) how it is desirable for an |
| 683 |
> |
integrator to be symplectic, or time reversible. The following is an |
| 684 |
> |
outline of the Trotter factorization of the Liouville Propagator as a |
| 685 |
> |
scheme for generating symplectic integrators.\cite{Tuckerman92} |
| 686 |
> |
|
| 687 |
> |
For a system with $f$ degrees of freedom the Liouville operator can be |
| 688 |
> |
defined as, |
| 689 |
> |
\begin{equation} |
| 690 |
> |
iL=\sum^f_{j=1} \biggl [\dot{q}_j\frac{\partial}{\partial q_j} + |
| 691 |
> |
F_j\frac{\partial}{\partial p_j} \biggr ] |
| 692 |
> |
\label{introEq:LiouvilleOperator} |
| 693 |
> |
\end{equation} |
| 694 |
> |
Here, $q_j$ and $p_j$ are the position and conjugate momenta of a |
| 695 |
> |
degree of freedom, and $F_j$ is the force on that degree of freedom. |
| 696 |
> |
$\Gamma$ is defined as the set of all positions and conjugate momenta, |
| 697 |
> |
$\{q_j,p_j\}$, and the propagator, $U(t)$, is defined |
| 698 |
> |
\begin {equation} |
| 699 |
> |
U(t) = e^{iLt} |
| 700 |
> |
\label{introEq:Lpropagator} |
| 701 |
> |
\end{equation} |
| 702 |
> |
This allows the specification of $\Gamma$ at any time $t$ as |
| 703 |
> |
\begin{equation} |
| 704 |
> |
\Gamma(t) = U(t)\Gamma(0) |
| 705 |
> |
\label{introEq:Lp2} |
| 706 |
> |
\end{equation} |
| 707 |
> |
It is important to note, $U(t)$ is a unitary operator meaning |
| 708 |
> |
\begin{equation} |
| 709 |
> |
U(-t)=U^{-1}(t) |
| 710 |
> |
\label{introEq:Lp3} |
| 711 |
> |
\end{equation} |
| 712 |
> |
|
| 713 |
> |
Decomposing $L$ into two parts, $iL_1$ and $iL_2$, one can use the |
| 714 |
> |
Trotter theorem to yield |
| 715 |
> |
\begin{align} |
| 716 |
> |
e^{iLt} &= e^{i(L_1 + L_2)t} \notag \\% |
| 717 |
> |
% |
| 718 |
> |
&= \biggl [ e^{i(L_1 +L_2)\frac{t}{P}} \biggr]^P \notag \\% |
| 719 |
> |
% |
| 720 |
> |
&= \biggl [ e^{iL_1\frac{\Delta t}{2}}\, e^{iL_2\Delta t}\, |
| 721 |
> |
e^{iL_1\frac{\Delta t}{2}} \biggr ]^P + |
| 722 |
> |
\mathcal{O}\biggl (\frac{t^3}{P^2} \biggr ) \label{introEq:Lp4} |
| 723 |
> |
\end{align} |
| 724 |
> |
Where $\Delta t = t/P$. |
| 725 |
> |
With this, a discrete time operator $G(\Delta t)$ can be defined: |
| 726 |
> |
\begin{align} |
| 727 |
> |
G(\Delta t) &= e^{iL_1\frac{\Delta t}{2}}\, e^{iL_2\Delta t}\, |
| 728 |
> |
e^{iL_1\frac{\Delta t}{2}} \notag \\% |
| 729 |
> |
% |
| 730 |
> |
&= U_1 \biggl ( \frac{\Delta t}{2} \biggr )\, U_2 ( \Delta t )\, |
| 731 |
> |
U_1 \biggl ( \frac{\Delta t}{2} \biggr ) |
| 732 |
> |
\label{introEq:Lp5} |
| 733 |
> |
\end{align} |
| 734 |
> |
Because $U_1(t)$ and $U_2(t)$ are unitary, $G(\Delta t)$ is also |
| 735 |
> |
unitary. Meaning an integrator based on this factorization will be |
| 736 |
> |
reversible in time. |
| 737 |
> |
|
| 738 |
> |
As an example, consider the following decomposition of $L$: |
| 739 |
> |
\begin{align} |
| 740 |
> |
iL_1 &= \dot{q}\frac{\partial}{\partial q}% |
| 741 |
> |
\label{introEq:Lp6a} \\% |
| 742 |
> |
% |
| 743 |
> |
iL_2 &= F(q)\frac{\partial}{\partial p}% |
| 744 |
> |
\label{introEq:Lp6b} |
| 745 |
> |
\end{align} |
| 746 |
> |
This leads to propagator $G( \Delta t )$ as, |
| 747 |
> |
\begin{equation} |
| 748 |
> |
G(\Delta t) = e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} \, |
| 749 |
> |
e^{\Delta t\,\dot{q}\frac{\partial}{\partial q}} \, |
| 750 |
> |
e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} |
| 751 |
> |
\label{introEq:Lp7} |
| 752 |
> |
\end{equation} |
| 753 |
> |
Operating $G(\Delta t)$ on $\Gamma(0)$, and utilizing the operator property |
| 754 |
> |
\begin{equation} |
| 755 |
> |
e^{c\frac{\partial}{\partial x}}\, f(x) = f(x+c) |
| 756 |
> |
\label{introEq:Lp8} |
| 757 |
> |
\end{equation} |
| 758 |
> |
Where $c$ is independent of $x$. One obtains the following: |
| 759 |
> |
\begin{align} |
| 760 |
> |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
| 761 |
> |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)] \label{introEq:Lp9a}\\% |
| 762 |
> |
% |
| 763 |
> |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr )% |
| 764 |
> |
\label{introEq:Lp9b}\\% |
| 765 |
> |
% |
| 766 |
> |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 767 |
> |
\frac{\Delta t}{2m}\, F[q(0)] \label{introEq:Lp9c} |
| 768 |
> |
\end{align} |
| 769 |
> |
Or written another way, |
| 770 |
> |
\begin{align} |
| 771 |
> |
q(t+\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 772 |
> |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2} % |
| 773 |
> |
\label{introEq:Lp10a} \\% |
| 774 |
> |
% |
| 775 |
> |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
| 776 |
> |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr] % |
| 777 |
> |
\label{introEq:Lp10b} |
| 778 |
> |
\end{align} |
| 779 |
> |
This is the velocity Verlet formulation presented in |
| 780 |
> |
Sec.~\ref{introSec:mdIntegrate}. Because this integration scheme is |
| 781 |
> |
comprised of unitary propagators, it is symplectic, and therefore area |
| 782 |
> |
preserving in phase space. From the preceding factorization, one can |
| 783 |
> |
see that the integration of the equations of motion would follow: |
| 784 |
> |
\begin{enumerate} |
| 785 |
> |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
| 786 |
> |
|
| 787 |
> |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 788 |
> |
|
| 789 |
> |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 790 |
> |
|
| 791 |
> |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 792 |
> |
\end{enumerate} |
| 793 |
> |
|
| 794 |
> |
\subsection{\label{introSec:MDsymplecticRot} Symplectic Propagation of the Rotation Matrix} |
| 795 |
> |
|
| 796 |
> |
Based on the factorization from the previous section, |
| 797 |
> |
Dullweber\emph{et al}.\cite{Dullweber1997}~ proposed a scheme for the |
| 798 |
> |
symplectic propagation of the rotation matrix, $\mathbf{A}$, as an |
| 799 |
> |
alternative method for the integration of orientational degrees of |
| 800 |
> |
freedom. The method starts with a straightforward splitting of the |
| 801 |
> |
Liouville operator: |
| 802 |
> |
\begin{align} |
| 803 |
> |
iL_{\text{pos}} &= \dot{q}\frac{\partial}{\partial q} + |
| 804 |
> |
\mathbf{\dot{A}}\frac{\partial}{\partial \mathbf{A}} |
| 805 |
> |
\label{introEq:SR1a} \\% |
| 806 |
> |
% |
| 807 |
> |
iL_F &= F(q)\frac{\partial}{\partial p} |
| 808 |
> |
\label{introEq:SR1b} \\% |
| 809 |
> |
iL_{\tau} &= \tau(\mathbf{A})\frac{\partial}{\partial \pi} |
| 810 |
> |
\label{introEq:SR1b} \\% |
| 811 |
> |
\end{align} |
| 812 |
> |
Where $\tau(\mathbf{A})$ is the torque of the system |
| 813 |
> |
due to the configuration, and $\pi$ is the conjugate |
| 814 |
> |
angular momenta of the system. The propagator, $G(\Delta t)$, becomes |
| 815 |
> |
\begin{equation} |
| 816 |
> |
G(\Delta t) = e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} \, |
| 817 |
> |
e^{\frac{\Delta t}{2} \tau(\mathbf{A})\frac{\partial}{\partial \pi}} \, |
| 818 |
> |
e^{\Delta t\,iL_{\text{pos}}} \, |
| 819 |
> |
e^{\frac{\Delta t}{2} \tau(\mathbf{A})\frac{\partial}{\partial \pi}} \, |
| 820 |
> |
e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} |
| 821 |
> |
\label{introEq:SR2} |
| 822 |
> |
\end{equation} |
| 823 |
> |
Propagation of the linear and angular momenta follows as in the Verlet |
| 824 |
> |
scheme. The propagation of positions also follows the Verlet scheme |
| 825 |
> |
with the addition of a further symplectic splitting of the rotation |
| 826 |
> |
matrix propagation, $\mathcal{U}_{\text{rot}}(\Delta t)$, within |
| 827 |
> |
$U_{\text{pos}}(\Delta t)$. |
| 828 |
> |
\begin{equation} |
| 829 |
> |
\mathcal{U}_{\text{rot}}(\Delta t) = |
| 830 |
> |
\mathcal{U}_x \biggl(\frac{\Delta t}{2}\biggr)\, |
| 831 |
> |
\mathcal{U}_y \biggl(\frac{\Delta t}{2}\biggr)\, |
| 832 |
> |
\mathcal{U}_z (\Delta t)\, |
| 833 |
> |
\mathcal{U}_y \biggl(\frac{\Delta t}{2}\biggr)\, |
| 834 |
> |
\mathcal{U}_x \biggl(\frac{\Delta t}{2}\biggr)\, |
| 835 |
> |
\label{introEq:SR3} |
| 836 |
> |
\end{equation} |
| 837 |
> |
Where $\mathcal{U}_j$ is a unitary rotation of $\mathbf{A}$ and |
| 838 |
> |
$\pi$ about each axis $j$. As all propagations are now |
| 839 |
> |
unitary and symplectic, the entire integration scheme is also |
| 840 |
> |
symplectic and time reversible. |
| 841 |
> |
|
| 842 |
> |
\section{\label{introSec:layout}Dissertation Layout} |
| 843 |
> |
|
| 844 |
> |
This dissertation is divided as follows:Ch.~\ref{chapt:RSA} |
| 845 |
> |
presents the random sequential adsorption simulations of related |
| 846 |
> |
pthalocyanines on a gold (111) surface. Ch.~\ref{chapt:OOPSE} |
| 847 |
> |
is about the writing of the molecular dynamics simulation package |
| 848 |
> |
{\sc oopse}. Ch.~\ref{chapt:lipid} regards the simulations of |
| 849 |
> |
phospholipid bilayers using a mesoscale model. And lastly, |
| 850 |
> |
Ch.~\ref{chapt:conclusion} concludes this dissertation with a |
| 851 |
> |
summary of all results. The chapters are arranged in chronological |
| 852 |
> |
order, and reflect the progression of techniques I employed during my |
| 853 |
> |
research. |
| 854 |
> |
|
| 855 |
> |
The chapter concerning random sequential adsorption simulations is a |
| 856 |
> |
study in applying Statistical Mechanics simulation techniques in order |
| 857 |
> |
to obtain a simple model capable of explaining the results. My |
| 858 |
> |
advisor, Dr. Gezelter, and I were approached by a colleague, |
| 859 |
> |
Dr. Lieberman, about possible explanations for the partial coverage of a |
| 860 |
> |
gold surface by a particular compound of hers. We suggested it might |
| 861 |
> |
be due to the statistical packing fraction of disks on a plane, and |
| 862 |
> |
set about to simulate this system. As the events in our model were |
| 863 |
> |
not dynamic in nature, a Monte Carlo method was employed. Here, if a |
| 864 |
> |
molecule landed on the surface without overlapping another, then its |
| 865 |
> |
landing was accepted. However, if there was overlap, the landing we |
| 866 |
> |
rejected and a new random landing location was chosen. This defined |
| 867 |
> |
our acceptance rules and allowed us to construct a Markov chain whose |
| 868 |
> |
limiting distribution was the surface coverage in which we were |
| 869 |
> |
interested. |
| 870 |
> |
|
| 871 |
> |
The following chapter, about the simulation package {\sc oopse}, |
| 872 |
> |
describes in detail the large body of scientific code that had to be |
| 873 |
> |
written in order to study phospholipid bilayers. Although there are |
| 874 |
> |
pre-existing molecular dynamic simulation packages available, none |
| 875 |
> |
were capable of implementing the models we were developing.{\sc oopse} |
| 876 |
> |
is a unique package capable of not only integrating the equations of |
| 877 |
> |
motion in Cartesian space, but is also able to integrate the |
| 878 |
> |
rotational motion of rigid bodies and dipoles. Add to this the |
| 879 |
> |
ability to perform calculations across parallel processors and a |
| 880 |
> |
flexible script syntax for creating systems, and {\sc oopse} becomes a |
| 881 |
> |
very powerful scientific instrument for the exploration of our model. |
| 882 |
> |
|
| 883 |
> |
Bringing us to Ch.~\ref{chapt:lipid}. Using {\sc oopse}, I have been |
| 884 |
> |
able to parameterize a mesoscale model for phospholipid simulations. |
| 885 |
> |
This model retains information about solvent ordering around the |
| 886 |
> |
bilayer, as well as information regarding the interaction of the |
| 887 |
> |
phospholipid head groups' dipoles with each other and the surrounding |
| 888 |
> |
solvent. These simulations give us insight into the dynamic events |
| 889 |
> |
that lead to the formation of phospholipid bilayers, as well as |
| 890 |
> |
provide the foundation for future exploration of bilayer phase |
| 891 |
> |
behavior with this model. |
| 892 |
> |
|
| 893 |
> |
Which leads into the last chapter, where I discuss future directions |
| 894 |
> |
for both{\sc oopse} and this mesoscale model. Additionally, I will |
| 895 |
> |
give a summary of results for this dissertation. |
| 896 |
> |
|
| 897 |
> |
|
