| 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 |
| 153 |
> |
\pi_{mn} = \alpha_{mn} \times \accMe(m \rightarrow n) |
| 154 |
> |
\label{introEq:MCpi} |
| 155 |
|
\end{equation} |
| 156 |
< |
Where $fix$ is the probability of attempting the move $fix$, and $fix$ |
| 157 |
< |
is the probability of accepting the move $fix$. Defining a |
| 158 |
< |
probability vector, $fix$, such that |
| 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 |
| 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}). |
| 339 |
> |
With the use of an $fix$, however, comes a discontinuity in the |
| 340 |
> |
potential energy curve (Fig.~\ref{fix}). To fix this discontinuity, |
| 341 |
> |
one calculates the potential energy at the $r_{\text{cut}}$, and add |
| 342 |
> |
that value to the potential. This causes the function to go smoothly |
| 343 |
> |
to zero at the cutoff radius. This ensures conservation of energy |
| 344 |
> |
when integrating the Newtonian equations of motion. |
| 345 |
> |
|
| 346 |
> |
The second main simplification used in this research is the Verlet |
| 347 |
> |
neighbor list. \cite{allen87:csl} In the Verlet method, one generates |
| 348 |
> |
a list of all neighbor atoms, $j$, surrounding atom $i$ within some |
| 349 |
> |
cutoff $r_{\text{list}}$, where $r_{\text{list}}>r_{\text{cut}}$. |
| 350 |
> |
This list is created the first time forces are evaluated, then on |
| 351 |
> |
subsequent force evaluations, pair calculations are only calculated |
| 352 |
> |
from the neighbor lists. The lists are updated if any given particle |
| 353 |
> |
in the system moves farther than $r_{\text{list}}-r_{\text{cut}}$, |
| 354 |
> |
giving rise to the possibility that a particle has left or joined a |
| 355 |
> |
neighbor list. |
| 356 |
> |
|
| 357 |
> |
\subsection{\label{introSec:MDintegrate} Integration of the equations of motion} |
| 358 |
> |
|
| 359 |
> |
A starting point for the discussion of molecular dynamics integrators |
| 360 |
> |
is the Verlet algorithm. \cite{Frenkel1996} It begins with a Taylor |
| 361 |
> |
expansion of position in time: |
| 362 |
> |
\begin{equation} |
| 363 |
> |
eq here |
| 364 |
> |
\label{introEq:verletForward} |
| 365 |
> |
\end{equation} |
| 366 |
> |
As well as, |
| 367 |
> |
\begin{equation} |
| 368 |
> |
eq here |
| 369 |
> |
\label{introEq:verletBack} |
| 370 |
> |
\end{equation} |
| 371 |
> |
Adding together Eq.~\ref{introEq:verletForward} and |
| 372 |
> |
Eq.~\ref{introEq:verletBack} results in, |
| 373 |
> |
\begin{equation} |
| 374 |
> |
eq here |
| 375 |
> |
\label{introEq:verletSum} |
| 376 |
> |
\end{equation} |
| 377 |
> |
Or equivalently, |
| 378 |
> |
\begin{equation} |
| 379 |
> |
eq here |
| 380 |
> |
\label{introEq:verletFinal} |
| 381 |
> |
\end{equation} |
| 382 |
> |
Which contains an error in the estimate of the new positions on the |
| 383 |
> |
order of $\Delta t^4$. |
| 384 |
|
|
| 385 |
+ |
In practice, however, the simulations in this research were integrated |
| 386 |
+ |
with a velocity reformulation of teh Verlet method. \cite{allen87:csl} |
| 387 |
+ |
\begin{equation} |
| 388 |
+ |
eq here |
| 389 |
+ |
\label{introEq:MDvelVerletPos} |
| 390 |
+ |
\end{equation} |
| 391 |
+ |
\begin{equation} |
| 392 |
+ |
eq here |
| 393 |
+ |
\label{introEq:MDvelVerletVel} |
| 394 |
+ |
\end{equation} |
| 395 |
+ |
The original Verlet algorithm can be regained by substituting the |
| 396 |
+ |
velocity back into Eq.~\ref{introEq:MDvelVerletPos}. The Verlet |
| 397 |
+ |
formulations are chosen in this research because the algorithms have |
| 398 |
+ |
very little long term drift in energy conservation. Energy |
| 399 |
+ |
conservation in a molecular dynamics simulation is of extreme |
| 400 |
+ |
importance, as it is a measure of how closely one is following the |
| 401 |
+ |
``true'' trajectory wtih the finite integration scheme. An exact |
| 402 |
+ |
solution to the integration will conserve area in phase space, as well |
| 403 |
+ |
as be reversible in time, that is, the trajectory integrated forward |
| 404 |
+ |
or backwards will exactly match itself. Having a finite algorithm |
| 405 |
+ |
that both conserves area in phase space and is time reversible, |
| 406 |
+ |
therefore increases, but does not guarantee the ``correctness'' or the |
| 407 |
+ |
integrated trajectory. |
| 408 |
|
|
| 409 |
+ |
It can be shown, \cite{Frenkel1996} that although the Verlet algorithm |
| 410 |
+ |
does not rigorously preserve the actual Hamiltonian, it does preserve |
| 411 |
+ |
a pseudo-Hamiltonian which shadows the real one in phase space. This |
| 412 |
+ |
pseudo-Hamiltonian is proveably area-conserving as well as time |
| 413 |
+ |
reversible. The fact that it shadows the true Hamiltonian in phase |
| 414 |
+ |
space is acceptable in actual simulations as one is interested in the |
| 415 |
+ |
ensemble average of the observable being measured. From the ergodic |
| 416 |
+ |
hypothesis (Sec.~\ref{introSec:StatThermo}), it is known that the time |
| 417 |
+ |
average will match the ensemble average, therefore two similar |
| 418 |
+ |
trajectories in phase space should give matching statistical averages. |
| 419 |
+ |
|
| 420 |
+ |
\subsection{\label{introSec:MDfurther}Further Considerations} |
| 421 |
+ |
In the simulations presented in this research, a few additional |
| 422 |
+ |
parameters are needed to describe the motions. The simulations |
| 423 |
+ |
involving water and phospholipids in Chapt.~\ref{chaptLipids} are |
| 424 |
+ |
required to integrate the equations of motions for dipoles on atoms. |
| 425 |
+ |
This involves an additional three parameters be specified for each |
| 426 |
+ |
dipole atom: $\phi$, $\theta$, and $\psi$. These three angles are |
| 427 |
+ |
taken to be the Euler angles, where $\phi$ is a rotation about the |
| 428 |
+ |
$z$-axis, and $\theta$ is a rotation about the new $x$-axis, and |
| 429 |
+ |
$\psi$ is a final rotation about the new $z$-axis (see |
| 430 |
+ |
Fig.~\ref{introFig:euleerAngles}). This sequence of rotations can be |
| 431 |
+ |
accumulated into a single $3 \times 3$ matrix $\mathbf{A}$ |
| 432 |
+ |
defined as follows: |
| 433 |
+ |
\begin{equation} |
| 434 |
+ |
eq here |
| 435 |
+ |
\label{introEq:EulerRotMat} |
| 436 |
+ |
\end{equation} |
| 437 |
+ |
|
| 438 |
+ |
The equations of motion for Euler angles can be written down as |
| 439 |
+ |
\cite{allen87:csl} |
| 440 |
+ |
\begin{equation} |
| 441 |
+ |
eq here |
| 442 |
+ |
\label{introEq:MDeuleeerPsi} |
| 443 |
+ |
\end{equation} |
| 444 |
+ |
Where $\omega^s_i$ is the angular velocity in the lab space frame |
| 445 |
+ |
along cartesian coordinate $i$. However, a difficulty arises when |
| 446 |
+ |
attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and |
| 447 |
+ |
Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in |
| 448 |
+ |
both equations means there is a non-physical instability present when |
| 449 |
+ |
$\theta$ is 0 or $\pi$. |
| 450 |
+ |
|
| 451 |
+ |
To correct for this, the simulations integrate the rotation matrix, |
| 452 |
+ |
$\mathbf{A}$, directly, thus avoiding the instability. |
| 453 |
+ |
This method was proposed by Dullwebber |
| 454 |
+ |
\emph{et. al.}\cite{Dullwebber:1997}, and is presented in |
| 455 |
+ |
Sec.~\ref{introSec:MDsymplecticRot}. |
| 456 |
+ |
|
| 457 |
+ |
\subsubsection{\label{introSec:MDliouville}Liouville Propagator} |
| 458 |
+ |
|
| 459 |
+ |
Before discussing the integration of the rotation matrix, it is |
| 460 |
+ |
necessary to understand the construction of a ``good'' integration |
| 461 |
+ |
scheme. It has been previously |
| 462 |
+ |
discussed(Sec.~\ref{introSec:MDintegrate}) how it is desirable for an |
| 463 |
+ |
integrator to be symplectic, or time reversible. The following is an |
| 464 |
+ |
outline of the Trotter factorization of the Liouville Propagator as a |
| 465 |
+ |
scheme for generating symplectic integrators. \cite{Tuckerman:1992} |
| 466 |
+ |
|
| 467 |
+ |
For a system with $f$ degrees of freedom the Liouville operator can be |
| 468 |
+ |
defined as, |
| 469 |
+ |
\begin{equation} |
| 470 |
+ |
eq here |
| 471 |
+ |
\label{introEq:LiouvilleOperator} |
| 472 |
+ |
\end{equation} |
| 473 |
+ |
Here, $r_j$ and $p_j$ are the position and conjugate momenta of a |
| 474 |
+ |
degree of freedom, and $f_j$ is the force on that degree of freedom. |
| 475 |
+ |
$\Gamma$ is defined as the set of all positions nad conjugate momenta, |
| 476 |
+ |
$\{r_j,p_j\}$, and the propagator, $U(t)$, is defined |
| 477 |
+ |
\begin {equation} |
| 478 |
+ |
eq here |
| 479 |
+ |
\label{introEq:Lpropagator} |
| 480 |
+ |
\end{equation} |
| 481 |
+ |
This allows the specification of $\Gamma$ at any time $t$ as |
| 482 |
+ |
\begin{equation} |
| 483 |
+ |
eq here |
| 484 |
+ |
\label{introEq:Lp2} |
| 485 |
+ |
\end{equation} |
| 486 |
+ |
It is important to note, $U(t)$ is a unitary operator meaning |
| 487 |
+ |
\begin{equation} |
| 488 |
+ |
U(-t)=U^{-1}(t) |
| 489 |
+ |
\label{introEq:Lp3} |
| 490 |
+ |
\end{equation} |
| 491 |
+ |
|
| 492 |
+ |
Decomposing $L$ into two parts, $iL_1$ and $iL_2$, one can use the |
| 493 |
+ |
Trotter theorem to yield |
| 494 |
+ |
\begin{equation} |
| 495 |
+ |
eq here |
| 496 |
+ |
\label{introEq:Lp4} |
| 497 |
+ |
\end{equation} |
| 498 |
+ |
Where $\Delta t = \frac{t}{P}$. |
| 499 |
+ |
With this, a discrete time operator $G(\Delta t)$ can be defined: |
| 500 |
+ |
\begin{equation} |
| 501 |
+ |
eq here |
| 502 |
+ |
\label{introEq:Lp5} |
| 503 |
+ |
\end{equation} |
| 504 |
+ |
Because $U_1(t)$ and $U_2(t)$ are unitary, $G|\Delta t)$ is also |
| 505 |
+ |
unitary. Meaning an integrator based on this factorization will be |
| 506 |
+ |
reversible in time. |
| 507 |
+ |
|
| 508 |
+ |
As an example, consider the following decomposition of $L$: |
| 509 |
+ |
\begin{equation} |
| 510 |
+ |
eq here |
| 511 |
+ |
\label{introEq:Lp6} |
| 512 |
+ |
\end{equation} |
| 513 |
+ |
Operating $G(\Delta t)$ on $\Gamma)0)$, and utilizing the operator property |
| 514 |
+ |
\begin{equation} |
| 515 |
+ |
eq here |
| 516 |
+ |
\label{introEq:Lp8} |
| 517 |
+ |
\end{equation} |
| 518 |
+ |
Where $c$ is independent of $q$. One obtains the following: |
| 519 |
+ |
\begin{equation} |
| 520 |
+ |
eq here |
| 521 |
+ |
\label{introEq:Lp8} |
| 522 |
+ |
\end{equation} |
| 523 |
+ |
Or written another way, |
| 524 |
+ |
\begin{equation} |
| 525 |
+ |
eq here |
| 526 |
+ |
\label{intorEq:Lp9} |
| 527 |
+ |
\end{equation} |
| 528 |
+ |
This is the velocity Verlet formulation presented in |
| 529 |
+ |
Sec.~\ref{introSec:MDintegrate}. Because this integration scheme is |
| 530 |
+ |
comprised of unitary propagators, it is symplectic, and therefore area |
| 531 |
+ |
preserving in phase space. From the preceeding fatorization, one can |
| 532 |
+ |
see that the integration of the equations of motion would follow: |
| 533 |
+ |
\begin{enumerate} |
| 534 |
+ |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
| 535 |
+ |
|
| 536 |
+ |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 537 |
+ |
|
| 538 |
+ |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 539 |
+ |
|
| 540 |
+ |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 541 |
+ |
\end{enumerate} |
| 542 |
+ |
|
| 543 |
+ |
\subsubsection{\label{introSec:MDsymplecticRot} Symplectic Propagation of the Rotation Matrix} |
| 544 |
+ |
|
| 545 |
+ |
Based on the factorization from the previous section, |
| 546 |
+ |
Dullweber\emph{et al.}\cite{Dullweber:1997}~ proposed a scheme for the |
| 547 |
+ |
symplectic propagation of the rotation matrix, $\mathbf{A}$, as an |
| 548 |
+ |
alternative method for the integration of orientational degrees of |
| 549 |
+ |
freedom. The method starts with a straightforward splitting of the |
| 550 |
+ |
Liouville operator: |
| 551 |
+ |
\begin{equation} |
| 552 |
+ |
eq here |
| 553 |
+ |
\label{introEq:SR1} |
| 554 |
+ |
\end{equation} |
| 555 |
+ |
Where $\boldsymbol{\tau}(\mathbf{A})$ are the tourques of the system |
| 556 |
+ |
due to the configuration, and $\boldsymbol{/pi}$ are the conjugate |
| 557 |
+ |
angular momenta of the system. The propagator, $G(\Delta t)$, becomes |
| 558 |
+ |
\begin{equation} |
| 559 |
+ |
eq here |
| 560 |
+ |
\label{introEq:SR2} |
| 561 |
+ |
\end{equation} |
| 562 |
+ |
Propagation fo the linear and angular momenta follows as in the Verlet |
| 563 |
+ |
scheme. The propagation of positions also follows the verlet scheme |
| 564 |
+ |
with the addition of a further symplectic splitting of the rotation |
| 565 |
+ |
matrix propagation, $\mathcal{G}_{\text{rot}}(\Delta t)$. |
| 566 |
+ |
\begin{equation} |
| 567 |
+ |
eq here |
| 568 |
+ |
\label{introEq:SR3} |
| 569 |
+ |
\end{equation} |
| 570 |
+ |
Where $\mathcal{G}_j$ is a unitary rotation of $\mathbf{A}$ and |
| 571 |
+ |
$\boldsymbol{\pi}$ about each axis $j$. As all propagations are now |
| 572 |
+ |
unitary and symplectic, the entire integration scheme is also |
| 573 |
+ |
symplectic and time reversible. |
| 574 |
+ |
|
| 575 |
|
\section{\label{introSec:chapterLayout}Chapter Layout} |
| 576 |
|
|
| 577 |
|
\subsection{\label{introSec:RSA}Random Sequential Adsorption} |
| 578 |
|
|
| 579 |
|
\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package} |
| 580 |
|
|
| 581 |
< |
\subsection{\label{introSec:bilayers}A Mesoscale Model for Phospholipid Bilayers} |
| 581 |
> |
\subsection{\label{introSec:bilayers}A Mesoscale Model for |
| 582 |
> |
Phospholipid Bilayers} |
