| 40 |
|
|
| 41 |
|
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 |
< |
named, because it heavily uses random numbers in the solution of the |
| 44 |
< |
problem. |
| 43 |
> |
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)\langle f(x) \rangle |
| 57 |
> |
\label{eq:MCex2} |
| 58 |
> |
\end{equation} |
| 59 |
> |
Where $\langle f(x) \rangle$ 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 |
|
|
| 66 |
+ |
However, in Statistical Mechanics, one is typically interested in |
| 67 |
+ |
integrals of the form: |
| 68 |
+ |
\begin{equation} |
| 69 |
+ |
\langle A \rangle = \frac{\int d^N \mathbf{r}~A(\mathbf{r}^N)% |
| 70 |
+ |
e^{-\beta V(\mathbf{r}^N)}}% |
| 71 |
+ |
{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}} |
| 72 |
+ |
\label{eq:mcEnsAvg} |
| 73 |
+ |
\end{equation} |
| 74 |
+ |
Where $\mathbf{r}^N$ stands for the coordinates of all $N$ particles |
| 75 |
+ |
and $A$ is some observable that is only dependent on |
| 76 |
+ |
position. $\langle A \rangle$ is the ensemble average of $A$ as |
| 77 |
+ |
presented in Sec.~\ref{introSec:statThermo}. Because $A$ is |
| 78 |
+ |
independent of momentum, the momenta contribution of the integral can |
| 79 |
+ |
be factored out, leaving the configurational integral. Application of |
| 80 |
+ |
the brute force method to this system would yield highly inefficient |
| 81 |
+ |
results. Due to the Boltzman weighting of this integral, most random |
| 82 |
+ |
configurations will have a near zero contribution to the ensemble |
| 83 |
+ |
average. This is where a importance sampling comes into |
| 84 |
+ |
play.\cite{allen87:csl} |
| 85 |
|
|
| 86 |
< |
\subsection{\label{introSec:md}Molecular Dynamics Simulations} |
| 86 |
> |
Importance Sampling is a method where one selects a distribution from |
| 87 |
> |
which the random configurations are chosen in order to more |
| 88 |
> |
efficiently calculate the integral.\cite{Frenkel1996} Consider again |
| 89 |
> |
Eq.~\ref{eq:MCex1} rewritten to be: |
| 90 |
> |
\begin{equation} |
| 91 |
> |
I = \int^b_a \frac{f(x)}{\rho(x)} \rho(x) dx |
| 92 |
> |
\label{introEq:Importance1} |
| 93 |
> |
\end{equation} |
| 94 |
> |
Where $\rho(x)$ is an arbitrary probability distribution in $x$. If |
| 95 |
> |
one conducts $\tau$ trials selecting a random number, $\zeta_\tau$, |
| 96 |
> |
from the distribution $\rho(x)$ on the interval $[a,b]$, then |
| 97 |
> |
Eq.~\ref{introEq:Importance1} becomes |
| 98 |
> |
\begin{equation} |
| 99 |
> |
I= \biggl \langle \frac{f(x)}{\rho(x)} \biggr \rangle_{\text{trials}} |
| 100 |
> |
\label{introEq:Importance2} |
| 101 |
> |
\end{equation} |
| 102 |
> |
Looking at Eq.~\ref{eq:mcEnsAvg}, and realizing |
| 103 |
> |
\begin {equation} |
| 104 |
> |
\rho_{kT}(\mathbf{r}^N) = |
| 105 |
> |
\frac{e^{-\beta V(\mathbf{r}^N)}} |
| 106 |
> |
{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}} |
| 107 |
> |
\label{introEq:MCboltzman} |
| 108 |
> |
\end{equation} |
| 109 |
> |
Where $\rho_{kT}$ is the boltzman distribution. The ensemble average |
| 110 |
> |
can be rewritten as |
| 111 |
> |
\begin{equation} |
| 112 |
> |
\langle A \rangle = \int d^N \mathbf{r}~A(\mathbf{r}^N) |
| 113 |
> |
\rho_{kT}(\mathbf{r}^N) |
| 114 |
> |
\label{introEq:Importance3} |
| 115 |
> |
\end{equation} |
| 116 |
> |
Applying Eq.~\ref{introEq:Importance1} one obtains |
| 117 |
> |
\begin{equation} |
| 118 |
> |
\langle A \rangle = \biggl \langle |
| 119 |
> |
\frac{ A \rho_{kT}(\mathbf{r}^N) } |
| 120 |
> |
{\rho(\mathbf{r}^N)} \biggr \rangle_{\text{trials}} |
| 121 |
> |
\label{introEq:Importance4} |
| 122 |
> |
\end{equation} |
| 123 |
> |
By selecting $\rho(\mathbf{r}^N)$ to be $\rho_{kT}(\mathbf{r}^N)$ |
| 124 |
> |
Eq.~\ref{introEq:Importance4} becomes |
| 125 |
> |
\begin{equation} |
| 126 |
> |
\langle A \rangle = \langle A(\mathbf{r}^N) \rangle_{\text{trials}} |
| 127 |
> |
\label{introEq:Importance5} |
| 128 |
> |
\end{equation} |
| 129 |
> |
The difficulty is selecting points $\mathbf{r}^N$ such that they are |
| 130 |
> |
sampled from the distribution $\rho_{kT}(\mathbf{r}^N)$. A solution |
| 131 |
> |
was proposed by Metropolis et al.\cite{metropolis:1953} which involved |
| 132 |
> |
the use of a Markov chain whose limiting distribution was |
| 133 |
> |
$\rho_{kT}(\mathbf{r}^N)$. |
| 134 |
|
|
| 135 |
< |
time averages |
| 135 |
> |
\subsubsection{\label{introSec:markovChains}Markov Chains} |
| 136 |
|
|
| 137 |
< |
time integrating schemes |
| 137 |
> |
A Markov chain is a chain of states satisfying the following |
| 138 |
> |
conditions:\cite{leach01:mm} |
| 139 |
> |
\begin{enumerate} |
| 140 |
> |
\item The outcome of each trial depends only on the outcome of the previous trial. |
| 141 |
> |
\item Each trial belongs to a finite set of outcomes called the state space. |
| 142 |
> |
\end{enumerate} |
| 143 |
> |
If given two configuartions, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$, |
| 144 |
> |
$\rho_m$ and $\rho_n$ are the probablilities of being in state |
| 145 |
> |
$\mathbf{r}^N_m$ and $\mathbf{r}^N_n$ respectively. Further, the two |
| 146 |
> |
states are linked by a transition probability, $\pi_{mn}$, which is the |
| 147 |
> |
probability of going from state $m$ to state $n$. |
| 148 |
|
|
| 149 |
< |
time reversible |
| 149 |
> |
\newcommand{\accMe}{\operatorname{acc}} |
| 150 |
|
|
| 151 |
< |
symplectic methods |
| 151 |
> |
The transition probability is given by the following: |
| 152 |
> |
\begin{equation} |
| 153 |
> |
\pi_{mn} = \alpha_{mn} \times \accMe(m \rightarrow n) |
| 154 |
> |
\label{introEq:MCpi} |
| 155 |
> |
\end{equation} |
| 156 |
> |
Where $\alpha_{mn}$ is the probability of attempting the move $m |
| 157 |
> |
\rightarrow n$, and $\accMe$ is the probability of accepting the move |
| 158 |
> |
$m \rightarrow n$. Defining a probability vector, |
| 159 |
> |
$\boldsymbol{\rho}$, such that |
| 160 |
> |
\begin{equation} |
| 161 |
> |
\boldsymbol{\rho} = \{\rho_1, \rho_2, \ldots \rho_m, \rho_n, |
| 162 |
> |
\ldots \rho_N \} |
| 163 |
> |
\label{introEq:MCrhoVector} |
| 164 |
> |
\end{equation} |
| 165 |
> |
a transition matrix $\boldsymbol{\Pi}$ can be defined, |
| 166 |
> |
whose elements are $\pi_{mn}$, for each given transition. The |
| 167 |
> |
limiting distribution of the Markov chain can then be found by |
| 168 |
> |
applying the transition matrix an infinite number of times to the |
| 169 |
> |
distribution vector. |
| 170 |
> |
\begin{equation} |
| 171 |
> |
\boldsymbol{\rho}_{\text{limit}} = |
| 172 |
> |
\lim_{N \rightarrow \infty} \boldsymbol{\rho}_{\text{initial}} |
| 173 |
> |
\boldsymbol{\Pi}^N |
| 174 |
> |
\label{introEq:MCmarkovLimit} |
| 175 |
> |
\end{equation} |
| 176 |
> |
The limiting distribution of the chain is independent of the starting |
| 177 |
> |
distribution, and successive applications of the transition matrix |
| 178 |
> |
will only yield the limiting distribution again. |
| 179 |
> |
\begin{equation} |
| 180 |
> |
\boldsymbol{\rho}_{\text{limit}} = \boldsymbol{\rho}_{\text{initial}} |
| 181 |
> |
\boldsymbol{\Pi} |
| 182 |
> |
\label{introEq:MCmarkovEquil} |
| 183 |
> |
\end{equation} |
| 184 |
|
|
| 185 |
< |
Extended ensembles (NVT NPT) |
| 185 |
> |
\subsubsection{\label{introSec:metropolisMethod}The Metropolis Method} |
| 186 |
|
|
| 187 |
< |
constrained dynamics |
| 187 |
> |
In the Metropolis method\cite{metropolis:1953} |
| 188 |
> |
Eq.~\ref{introEq:MCmarkovEquil} is solved such that |
| 189 |
> |
$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzman distribution |
| 190 |
> |
of states. The method accomplishes this by imposing the strong |
| 191 |
> |
condition of microscopic reversibility on the equilibrium |
| 192 |
> |
distribution. Meaning, that at equilibrium the probability of going |
| 193 |
> |
from $m$ to $n$ is the same as going from $n$ to $m$. |
| 194 |
> |
\begin{equation} |
| 195 |
> |
\rho_m\pi_{mn} = \rho_n\pi_{nm} |
| 196 |
> |
\label{introEq:MCmicroReverse} |
| 197 |
> |
\end{equation} |
| 198 |
> |
Further, $\boldsymbol{\alpha}$ is chosen to be a symetric matrix in |
| 199 |
> |
the Metropolis method. Using Eq.~\ref{introEq:MCpi}, |
| 200 |
> |
Eq.~\ref{introEq:MCmicroReverse} becomes |
| 201 |
> |
\begin{equation} |
| 202 |
> |
\frac{\accMe(m \rightarrow n)}{\accMe(n \rightarrow m)} = |
| 203 |
> |
\frac{\rho_n}{\rho_m} |
| 204 |
> |
\label{introEq:MCmicro2} |
| 205 |
> |
\end{equation} |
| 206 |
> |
For a Boltxman limiting distribution, |
| 207 |
> |
\begin{equation} |
| 208 |
> |
\frac{\rho_n}{\rho_m} = e^{-\beta[\mathcal{U}(n) - \mathcal{U}(m)]} |
| 209 |
> |
= e^{-\beta \Delta \mathcal{U}} |
| 210 |
> |
\label{introEq:MCmicro3} |
| 211 |
> |
\end{equation} |
| 212 |
> |
This allows for the following set of acceptance rules be defined: |
| 213 |
> |
\begin{equation} |
| 214 |
> |
EQ Here |
| 215 |
> |
\end{equation} |
| 216 |
|
|
| 217 |
+ |
Using the acceptance criteria from Eq.~\ref{fix} the Metropolis method |
| 218 |
+ |
proceeds as follows |
| 219 |
+ |
\begin{itemize} |
| 220 |
+ |
\item Generate an initial configuration $fix$ which has some finite probability in $fix$. |
| 221 |
+ |
\item Modify $fix$, to generate configuratioon $fix$. |
| 222 |
+ |
\item If configuration $n$ lowers the energy of the system, accept the move with unity ($fix$ becomes $fix$). Otherwise accept with probability $fix$. |
| 223 |
+ |
\item Accumulate the average for the configurational observable of intereest. |
| 224 |
+ |
\item Repeat from step 2 until average converges. |
| 225 |
+ |
\end{itemize} |
| 226 |
+ |
One important note is that the average is accumulated whether the move |
| 227 |
+ |
is accepted or not, this ensures proper weighting of the average. |
| 228 |
+ |
Using Eq.~\ref{fix} it becomes clear that the accumulated averages are |
| 229 |
+ |
the ensemble averages, as this method ensures that the limiting |
| 230 |
+ |
distribution is the Boltzman distribution. |
| 231 |
+ |
|
| 232 |
+ |
\subsection{\label{introSec:MD}Molecular Dynamics Simulations} |
| 233 |
+ |
|
| 234 |
+ |
The main simulation tool used in this research is Molecular Dynamics. |
| 235 |
+ |
Molecular Dynamics is when the equations of motion for a system are |
| 236 |
+ |
integrated in order to obtain information about both the positions and |
| 237 |
+ |
momentum of a system, allowing the calculation of not only |
| 238 |
+ |
configurational observables, but momenta dependent ones as well: |
| 239 |
+ |
diffusion constants, velocity auto correlations, folding/unfolding |
| 240 |
+ |
events, etc. Due to the principle of ergodicity, Eq.~\ref{fix}, the |
| 241 |
+ |
average of these observables over the time period of the simulation |
| 242 |
+ |
are taken to be the ensemble averages for the system. |
| 243 |
+ |
|
| 244 |
+ |
The choice of when to use molecular dynamics over Monte Carlo |
| 245 |
+ |
techniques, is normally decided by the observables in which the |
| 246 |
+ |
researcher is interested. If the observabvles depend on momenta in |
| 247 |
+ |
any fashion, then the only choice is molecular dynamics in some form. |
| 248 |
+ |
However, when the observable is dependent only on the configuration, |
| 249 |
+ |
then most of the time Monte Carlo techniques will be more efficent. |
| 250 |
+ |
|
| 251 |
+ |
The focus of research in the second half of this dissertation is |
| 252 |
+ |
centered around the dynamic properties of phospholipid bilayers, |
| 253 |
+ |
making molecular dynamics key in the simulation of those properties. |
| 254 |
+ |
|
| 255 |
+ |
\subsubsection{Molecular dynamics Algorithm} |
| 256 |
+ |
|
| 257 |
+ |
To illustrate how the molecular dynamics technique is applied, the |
| 258 |
+ |
following sections will describe the sequence involved in a |
| 259 |
+ |
simulation. Sec.~\ref{fix} deals with the initialization of a |
| 260 |
+ |
simulation. Sec.~\ref{fix} discusses issues involved with the |
| 261 |
+ |
calculation of the forces. Sec.~\ref{fix} concludes the algorithm |
| 262 |
+ |
discussion with the integration of the equations of motion. \cite{fix} |
| 263 |
+ |
|
| 264 |
+ |
\subsubsection{initialization} |
| 265 |
+ |
|
| 266 |
+ |
When selecting the initial configuration for the simulation it is |
| 267 |
+ |
important to consider what dynamics one is hoping to observe. |
| 268 |
+ |
Ch.~\ref{fix} deals with the formation and equilibrium dynamics of |
| 269 |
+ |
phospholipid membranes. Therefore in these simulations initial |
| 270 |
+ |
positions were selected that in some cases dispersed the lipids in |
| 271 |
+ |
water, and in other cases structured the lipids into preformed |
| 272 |
+ |
bilayers. Important considerations at this stage of the simulation are: |
| 273 |
+ |
\begin{itemize} |
| 274 |
+ |
\item There are no major overlaps of molecular or atomic orbitals |
| 275 |
+ |
\item Velocities are chosen in such a way as to not gie the system a non=zero total momentum or angular momentum. |
| 276 |
+ |
\item It is also sometimes desireable to select the velocities to correctly sample the target temperature. |
| 277 |
+ |
\end{itemize} |
| 278 |
+ |
|
| 279 |
+ |
The first point is important due to the amount of potential energy |
| 280 |
+ |
generated by having two particles too close together. If overlap |
| 281 |
+ |
occurs, the first evaluation of forces will return numbers so large as |
| 282 |
+ |
to render the numerical integration of teh motion meaningless. The |
| 283 |
+ |
second consideration keeps the system from drifting or rotating as a |
| 284 |
+ |
whole. This arises from the fact that most simulations are of systems |
| 285 |
+ |
in equilibrium in the absence of outside forces. Therefore any net |
| 286 |
+ |
movement would be unphysical and an artifact of the simulation method |
| 287 |
+ |
used. The final point addresses teh selection of the magnitude of the |
| 288 |
+ |
initial velocities. For many simulations it is convienient to use |
| 289 |
+ |
this opportunity to scale the amount of kinetic energy to reflect the |
| 290 |
+ |
desired thermal distribution of the system. However, it must be noted |
| 291 |
+ |
that most systems will require further velocity rescaling after the |
| 292 |
+ |
first few initial simulation steps due to either loss or gain of |
| 293 |
+ |
kinetic energy from energy stored in potential degrees of freedom. |
| 294 |
+ |
|
| 295 |
+ |
\subsubsection{Force Evaluation} |
| 296 |
+ |
|
| 297 |
+ |
The evaluation of forces is the most computationally expensive portion |
| 298 |
+ |
of a given molecular dynamics simulation. This is due entirely to the |
| 299 |
+ |
evaluation of long range forces in a simulation, typically pair-wise. |
| 300 |
+ |
These forces are most commonly the Van der Waals force, and sometimes |
| 301 |
+ |
Coulombic forces as well. For a pair-wise force, there are $fix$ |
| 302 |
+ |
pairs to be evaluated, where $n$ is the number of particles in the |
| 303 |
+ |
system. This leads to the calculations scaling as $fix$, making large |
| 304 |
+ |
simulations prohibitive in the absence of any computation saving |
| 305 |
+ |
techniques. |
| 306 |
+ |
|
| 307 |
+ |
Another consideration one must resolve, is that in a given simulation |
| 308 |
+ |
a disproportionate number of the particles will feel the effects of |
| 309 |
+ |
the surface. \cite{fix} For a cubic system of 1000 particles arranged |
| 310 |
+ |
in a $10x10x10$ cube, 488 particles will be exposed to the surface. |
| 311 |
+ |
Unless one is simulating an isolated particle group in a vacuum, the |
| 312 |
+ |
behavior of the system will be far from the desired bulk |
| 313 |
+ |
charecteristics. To offset this, simulations employ the use of |
| 314 |
+ |
periodic boundary images. \cite{fix} |
| 315 |
+ |
|
| 316 |
+ |
The technique involves the use of an algorithm that replicates the |
| 317 |
+ |
simulation box on an infinite lattice in cartesian space. Any given |
| 318 |
+ |
particle leaving the simulation box on one side will have an image of |
| 319 |
+ |
itself enter on the opposite side (see Fig.~\ref{fix}). |
| 320 |
+ |
\begin{equation} |
| 321 |
+ |
EQ Here |
| 322 |
+ |
\end{equation} |
| 323 |
+ |
In addition, this sets that any given particle pair has an image, real |
| 324 |
+ |
or periodic, within $fix$ of each other. A discussion of the method |
| 325 |
+ |
used to calculate the periodic image can be found in Sec.\ref{fix}. |
| 326 |
+ |
|
| 327 |
+ |
Returning to the topic of the computational scale of the force |
| 328 |
+ |
evaluation, the use of periodic boundary conditions requires that a |
| 329 |
+ |
cutoff radius be employed. Using a cutoff radius improves the |
| 330 |
+ |
efficiency of the force evaluation, as particles farther than a |
| 331 |
+ |
predetermined distance, $fix$, are not included in the |
| 332 |
+ |
calculation. \cite{fix} In a simultation with periodic images, $fix$ |
| 333 |
+ |
has a maximum value of $fix$. Fig.~\ref{fix} illustrates how using an |
| 334 |
+ |
$fix$ larger than this value, or in the extreme limit of no $fix$ at |
| 335 |
+ |
all, the corners of the simulation box are unequally weighted due to |
| 336 |
+ |
the lack of particle images in the $x$, $y$, or $z$ directions past a |
| 337 |
+ |
disance of $fix$. |
| 338 |
+ |
|
| 339 |
+ |
With the use of an $fix$, however, comes a discontinuity in the potential energy curve (Fig.~\ref{fix}). |
| 340 |
+ |
|
| 341 |
+ |
|
| 342 |
|
\section{\label{introSec:chapterLayout}Chapter Layout} |
| 343 |
|
|
| 344 |
|
\subsection{\label{introSec:RSA}Random Sequential Adsorption} |
