| 53 |
|
\end{equation} |
| 54 |
|
The equation can be recast as: |
| 55 |
|
\begin{equation} |
| 56 |
< |
I = (b-a)<f(x)> |
| 56 |
> |
I = (b-a)\langle f(x) \rangle |
| 57 |
|
\label{eq:MCex2} |
| 58 |
|
\end{equation} |
| 59 |
< |
Where $<f(x)>$ is the unweighted average over the interval |
| 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 |
| 66 |
|
However, in Statistical Mechanics, one is typically interested in |
| 67 |
|
integrals of the form: |
| 68 |
|
\begin{equation} |
| 69 |
< |
<A> = \frac{A}{exp^{-\beta}} |
| 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 $r^N$ stands for the coordinates of all $N$ particles and $A$ is |
| 75 |
< |
some observable that is only dependent on position. $<A>$ is the |
| 76 |
< |
ensemble average of $A$ as presented in |
| 77 |
< |
Sec.~\ref{introSec:statThermo}. Because $A$ is independent of |
| 78 |
< |
momentum, the momenta contribution of the integral can be factored |
| 79 |
< |
out, leaving the configurational integral. Application of the brute |
| 80 |
< |
force method to this system would yield highly inefficient |
| 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 |
| 88 |
|
efficiently calculate the integral.\cite{Frenkel1996} Consider again |
| 89 |
|
Eq.~\ref{eq:MCex1} rewritten to be: |
| 90 |
|
\begin{equation} |
| 91 |
< |
EQ Here |
| 91 |
> |
I = \int^b_a \frac{f(x)}{\rho(x)} \rho(x) dx |
| 92 |
> |
\label{introEq:Importance1} |
| 93 |
|
\end{equation} |
| 94 |
< |
Where $fix$ is an arbitrary probability distribution in $x$. If one |
| 95 |
< |
conducts $fix$ trials selecting a random number, $fix$, from the |
| 96 |
< |
distribution $fix$ on the interval $[a,b]$, then Eq.~\ref{fix} becomes |
| 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 |
< |
EQ Here |
| 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{fix}, and realizing |
| 102 |
> |
Looking at Eq.~\ref{eq:mcEnsAvg}, and realizing |
| 103 |
|
\begin {equation} |
| 104 |
< |
EQ Here |
| 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 |
< |
The ensemble average can be rewritten as |
| 109 |
> |
Where $\rho_{kT}$ is the boltzman distribution. The ensemble average |
| 110 |
> |
can be rewritten as |
| 111 |
|
\begin{equation} |
| 112 |
< |
EQ Here |
| 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 |
< |
Appllying Eq.~ref{fix} one obtains |
| 116 |
> |
Applying Eq.~\ref{introEq:Importance1} one obtains |
| 117 |
|
\begin{equation} |
| 118 |
< |
EQ Here |
| 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 $fix$ to be $fix$ Eq.~ref{fix} becomes |
| 123 |
> |
By selecting $\rho(\mathbf{r}^N)$ to be $\rho_{kT}(\mathbf{r}^N)$ |
| 124 |
> |
Eq.~\ref{introEq:Importance4} becomes |
| 125 |
|
\begin{equation} |
| 126 |
< |
EQ Here |
| 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 $fix$ such that they are sampled |
| 130 |
< |
from the distribution $fix$. A solution was proposed by Metropolis et |
| 131 |
< |
al.\cite{fix} which involved the use of a Markov chain whose limiting |
| 132 |
< |
distribution was $fix$. |
| 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 |
< |
\subsection{Markov Chains} |
| 135 |
> |
\subsubsection{\label{introSec:markovChains}Markov Chains} |
| 136 |
|
|
| 137 |
|
A Markov chain is a chain of states satisfying the following |
| 138 |
< |
conditions:\cite{fix} |
| 139 |
< |
\begin{itemize} |
| 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{itemize} |
| 143 |
< |
If given two configuartions, $fix$ and $fix$, $fix$ and $fix$ are the |
| 144 |
< |
probablilities of being in state $fix$ and $fix$ respectively. |
| 145 |
< |
Further, the two states are linked by a transition probability, $fix$, |
| 146 |
< |
which is the probability of going from state $m$ to state $n$. |
| 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 |
+ |
\newcommand{\accMe}{\operatorname{acc}} |
| 150 |
+ |
|
| 151 |
|
The transition probability is given by the following: |
| 152 |
|
\begin{equation} |
| 153 |
< |
EQ Here |
| 154 |
< |
\end{equation} |
| 155 |
< |
Where $fix$ is the probability of attempting the move $fix$, and $fix$ |
| 156 |
< |
is the probability of accepting the move $fix$. Defining a |
| 157 |
< |
probability vector, $fix$, such that |
| 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 |
< |
EQ Here |
| 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 $fix$ can be defined, whose elements are $fix$, |
| 166 |
< |
for each given transition. The limiting distribution of the Markov |
| 167 |
< |
chain can then be found by applying the transition matrix an infinite |
| 168 |
< |
number of times to the distribution vector. |
| 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 |
< |
EQ Here |
| 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} |
| 148 |
– |
|
| 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 |
< |
EQ Here |
| 180 |
> |
\boldsymbol{\rho}_{\text{limit}} = \boldsymbol{\rho}_{\text{initial}} |
| 181 |
> |
\boldsymbol{\Pi} |
| 182 |
> |
\label{introEq:MCmarkovEquil} |
| 183 |
|
\end{equation} |
| 184 |
|
|
| 185 |
< |
\subsection{fix} |
| 185 |
> |
\subsubsection{\label{introSec:metropolisMethod}The Metropolis Method} |
| 186 |
|
|
| 187 |
< |
In the Metropolis method \cite{fix} Eq.~ref{fix} is solved such that |
| 188 |
< |
$fix$ matches the Boltzman distribution of states. The method |
| 189 |
< |
accomplishes this by imposing the strong condition of microscopic |
| 190 |
< |
reversibility on the equilibrium distribution. Meaning, that at |
| 191 |
< |
equilibrium the probability of going from $m$ to $n$ is the same as |
| 192 |
< |
going from $n$ to $m$. |
| 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 |
< |
EQ Here |
| 195 |
> |
\rho_m\pi_{mn} = \rho_n\pi_{nm} |
| 196 |
> |
\label{introEq:MCmicroReverse} |
| 197 |
|
\end{equation} |
| 198 |
< |
Further, $fix$ is chosen to be a symetric matrix in the Metropolis |
| 199 |
< |
method. Using Eq.~\ref{fix}, Eq.~\ref{fix} becomes |
| 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 |
< |
EQ Here |
| 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 |
| 206 |
> |
For a Boltxman limiting distribution, |
| 207 |
|
\begin{equation} |
| 208 |
< |
EQ Here |
| 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} |
| 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} |
| 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 |
| 252 |
|
centered around the dynamic properties of phospholipid bilayers, |
| 253 |
|
making molecular dynamics key in the simulation of those properties. |
| 254 |
|
|
| 255 |
< |
\subsection{Molecular dynamics Algorithm} |
| 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 |
| 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 |
< |
\subsection{initialization} |
| 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. |
| 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 |
< |
\subsection{Force Evaluation} |
| 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 |
