| 3 |
|
\section{Background on the Problem\label{In:sec:pro}} |
| 4 |
|
Phospholipid molecules are the primary topic of this dissertation |
| 5 |
|
because of their critical role as the foundation of biological |
| 6 |
< |
membranes. Lipids, when dispersed in water, self assemble into a |
| 7 |
< |
mumber of topologically distinct bilayer structures. The phase |
| 8 |
< |
behavior of lipid bilayers has been explored |
| 9 |
< |
experimentally~\cite{Cevc87}, however, a complete understanding of the |
| 10 |
< |
mechanism and driving forces behind the various phases has not been |
| 11 |
< |
achieved. |
| 6 |
> |
membranes. The chemical structure of phospholipids includes a head |
| 7 |
> |
group with a large dipole moment which is due to the large charge |
| 8 |
> |
separation between phosphate and amino alcohol, and a nonpolar tail |
| 9 |
> |
that contains fatty acid chains. Depending on the specific alcohol |
| 10 |
> |
which the phosphate and fatty acid chains are esterified to, the |
| 11 |
> |
phospholipids are divided into glycerophospholipids and |
| 12 |
> |
sphingophospholipids.~\cite{Cevc80} The chemical structures are shown |
| 13 |
> |
in figure~\ref{Infig:lipid}. |
| 14 |
> |
\begin{figure} |
| 15 |
> |
\centering |
| 16 |
> |
\includegraphics[width=\linewidth]{./figures/inLipid.pdf} |
| 17 |
> |
\caption{The chemical structure of glycerophospholipids (left) and |
| 18 |
> |
sphingophospholipids (right).\cite{Cevc80}} |
| 19 |
> |
\label{Infig:lipid} |
| 20 |
> |
\end{figure} |
| 21 |
> |
Glycerophospholipids are the dominant phospholipids in biological |
| 22 |
> |
membranes. The type of glycerophospholipid depends on the identity of |
| 23 |
> |
the X group, and the chains. For example, if X is choline |
| 24 |
> |
[(CH$_3$)$_3$N$^+$CH$_2$CH$_2$OH], the lipids are known as |
| 25 |
> |
phosphatidylcholine (PC), or if X is ethanolamine |
| 26 |
> |
[H$_3$N$^+$CH$_2$CH$_2$OH], the lipids are known as |
| 27 |
> |
phosphatidyethanolamine (PE). Table~\ref{Intab:pc} listed a number |
| 28 |
> |
types of phosphatidycholine with different fatty acids as the lipid |
| 29 |
> |
chains. |
| 30 |
> |
\begin{table*} |
| 31 |
> |
\begin{minipage}{\linewidth} |
| 32 |
> |
\begin{center} |
| 33 |
> |
\caption{A number types of phosphatidycholine.} |
| 34 |
> |
\begin{tabular}{lll} |
| 35 |
> |
\hline |
| 36 |
> |
& Fatty acid & Generic Name \\ |
| 37 |
> |
\hline |
| 38 |
> |
\textcolor{red}{DMPC} & Myristic: CH$_3$(CH$_2$)$_{12}$COOH & |
| 39 |
> |
\textcolor{red}{D}i\textcolor{red}{M}yristoyl\textcolor{red}{P}hosphatidyl\textcolor{red}{C}holine \\ |
| 40 |
> |
\textcolor{red}{DPPC} & Palmitic: CH$_3$(CH$_2$)$_{14}$COOH & \textcolor{red}{D}i\textcolor{red}{P}almtoyl\textcolor{red}{P}hosphatidyl\textcolor{red}{C}holine |
| 41 |
> |
\\ |
| 42 |
> |
\textcolor{red}{DSPC} & Stearic: CH$_3$(CH$_2$)$_{16}$COOH & \textcolor{red}{D}i\textcolor{red}{S}tearoyl\textcolor{red}{P}hosphatidyl\textcolor{red}{C}holine \\ |
| 43 |
> |
\end{tabular} |
| 44 |
> |
\label{Intab:pc} |
| 45 |
> |
\end{center} |
| 46 |
> |
\end{minipage} |
| 47 |
> |
\end{table*} |
| 48 |
> |
When dispersed in water, lipids self assemble into a mumber of |
| 49 |
> |
topologically distinct bilayer structures. The phase behavior of lipid |
| 50 |
> |
bilayers has been explored experimentally~\cite{Cevc80}, however, a |
| 51 |
> |
complete understanding of the mechanism and driving forces behind the |
| 52 |
> |
various phases has not been achieved. |
| 53 |
|
|
| 54 |
< |
\subsection{Ripple Phase\label{In:ssec:ripple}} |
| 54 |
> |
\subsection{The Ripple Phase\label{In:ssec:ripple}} |
| 55 |
|
The $P_{\beta'}$ {\it ripple phase} of lipid bilayers, named from the |
| 56 |
|
periodic buckling of the membrane, is an intermediate phase which is |
| 57 |
|
developed either from heating the gel phase $L_{\beta'}$ or cooling |
| 58 |
< |
the fluid phase $L_\alpha$. A Sketch is shown in |
| 59 |
< |
figure~\ref{Infig:phaseDiagram}.~\cite{Cevc87} |
| 58 |
> |
the fluid phase $L_\alpha$. A sketch of the phases is shown in |
| 59 |
> |
figure~\ref{Infig:phaseDiagram}.~\cite{Cevc80} |
| 60 |
|
\begin{figure} |
| 61 |
|
\centering |
| 62 |
|
\includegraphics[width=\linewidth]{./figures/inPhaseDiagram.pdf} |
| 63 |
< |
\caption{A phase diagram of lipid bilayer. With increasing the |
| 64 |
< |
temperature, the bilayer can go through a gel ($L_{\beta'}$), ripple |
| 65 |
< |
($P_{\beta'}$) to fluid ($L_\alpha$) phase transition.} |
| 63 |
> |
\caption{Phases of PC lipid bilayers. With increasing |
| 64 |
> |
temperature, phosphotidylcholine (PC) bilayers can go through |
| 65 |
> |
$L_{\beta'} \rightarrow P_{\beta'}$ (gel $\rightarrow$ ripple) and |
| 66 |
> |
$P_{\beta'} \rightarrow L_\alpha$ (ripple $\rightarrow$ fluid) phase |
| 67 |
> |
transitions.~\cite{Cevc80}} |
| 68 |
|
\label{Infig:phaseDiagram} |
| 69 |
|
\end{figure} |
| 70 |
< |
Most structural information of the ripple phase has been obtained by |
| 71 |
< |
the X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture |
| 70 |
> |
Most structural information about the ripple phase has been obtained |
| 71 |
> |
by X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture |
| 72 |
|
electron microscopy (FFEM).~\cite{Copeland80,Meyer96} The X-ray |
| 73 |
|
diffraction work by Katsaras {\it et al.} showed that a rich phase |
| 74 |
|
diagram exhibiting both {\it asymmetric} and {\it symmetric} ripples |
| 79 |
|
\begin{figure} |
| 80 |
|
\centering |
| 81 |
|
\includegraphics[width=\linewidth]{./figures/inRipple.pdf} |
| 82 |
< |
\caption{The experimental observed ripple phase. The top image is |
| 83 |
< |
obtained by X-ray diffraction~\cite{Sun96}, and the bottom one is |
| 84 |
< |
observed by AFM.~\cite{Kaasgaard03}} |
| 82 |
> |
\caption{Experimental observations of the riple phase. The top image |
| 83 |
> |
is an electrostatic density map obtained by Sun {\it et al.} using |
| 84 |
> |
X-ray diffraction~\cite{Sun96}. The lower figures are the surface |
| 85 |
> |
topology of various ripple domains in bilayers supported in mica. The |
| 86 |
> |
AFM images were observed by Kaasgaard {\it et |
| 87 |
> |
al.}.~\cite{Kaasgaard03}} |
| 88 |
|
\label{Infig:ripple} |
| 89 |
|
\end{figure} |
| 90 |
< |
Figure~\ref{Infig:ripple} shows the ripple phase oberved by X-ray |
| 90 |
> |
Figure~\ref{Infig:ripple} shows the ripple phase oberved by both X-ray |
| 91 |
|
diffraction and AFM. The experimental results provide strong support |
| 92 |
|
for a 2-dimensional triangular packing lattice of the lipid molecules |
| 93 |
|
within the ripple phase. This is a notable change from the observed |
| 94 |
< |
lipid packing within the gel phase,~\cite{Cevc87} although Tenchov |
| 95 |
< |
{\it et al.} have recently observed near-hexagonal packing in some |
| 94 |
> |
lipid packing within the gel phase,~\cite{Cevc80} although Tenchov |
| 95 |
> |
{\it et al.} have recently observed near-triangular packing in some |
| 96 |
|
phosphatidylcholine (PC) gel phases.~\cite{Tenchov2001} However, the |
| 97 |
|
physical mechanism for the formation of the ripple phase has never |
| 98 |
|
been explained and the microscopic structure of the ripple phase has |
| 99 |
|
never been elucidated by experiments. Computational simulation is a |
| 100 |
|
perfect tool to study the microscopic properties for a |
| 101 |
< |
system. However, the large length scale the ripple structure and the |
| 102 |
< |
long time scale of the formation of the ripples are crucial obstacles |
| 103 |
< |
to performing the actual work. The principal ideas explored in this |
| 104 |
< |
dissertation are attempts to break the computational task up by |
| 101 |
> |
system. However, the large length scale of the ripple structures and |
| 102 |
> |
the long time required for the formation of the ripples are crucial |
| 103 |
> |
obstacles to performing the actual work. The principal ideas explored |
| 104 |
> |
in this dissertation are attempts to break the computational task up |
| 105 |
> |
by |
| 106 |
|
\begin{itemize} |
| 107 |
|
\item Simplifying the lipid model. |
| 108 |
< |
\item Improving algorithm for integrating the equations of motion. |
| 108 |
> |
\item Improving the algorithm for integrating the equations of motion. |
| 109 |
|
\end{itemize} |
| 110 |
|
In chapters~\ref{chap:mc} and~\ref{chap:md}, we use a simple point |
| 111 |
< |
dipole spin model and a coarse-grained molecualr scale model to |
| 111 |
> |
dipole spin model and a coarse-grained molecular scale model to |
| 112 |
|
perform the Monte Carlo and Molecular Dynamics simulations |
| 113 |
|
respectively, and in chapter~\ref{chap:ld}, we develop a Langevin |
| 114 |
|
Dynamics algorithm which excludes the explicit solvent to improve the |
| 115 |
|
efficiency of the simulations. |
| 116 |
|
|
| 117 |
< |
\subsection{Lattice Model\label{In:ssec:model}} |
| 118 |
< |
The gel-like characteristic (relatively immobile molecules) exhibited |
| 117 |
> |
\subsection{Lattice Models\label{In:ssec:model}} |
| 118 |
> |
The gel-like characteristic (laterally immobile molecules) exhibited |
| 119 |
|
by the ripple phase makes it feasible to apply a lattice model to |
| 120 |
|
study the system. The popular $2$ dimensional lattice models, {\it |
| 121 |
|
i.e.}, the Ising, $X-Y$, and Heisenberg models, show {\it frustration} |
| 138 |
|
\includegraphics[width=3in]{./figures/inFrustration.pdf} |
| 139 |
|
\caption{Frustration on triangular lattice, the spins and dipoles are |
| 140 |
|
represented by arrows. The multiple local minima of energy states |
| 141 |
< |
induce the frustration for spins and dipoles picking the directions.} |
| 141 |
> |
induce frustration for spins and dipoles resulting in disordered |
| 142 |
> |
low-temperature phases.} |
| 143 |
|
\label{Infig:frustration} |
| 144 |
|
\end{figure} |
| 145 |
< |
The spins in figure~\ref{Infig:frustration} shows an illustration of |
| 146 |
< |
the frustration for $J < 0$ on a triangular lattice. There are |
| 147 |
< |
multiple local minima energy states which are independent of the |
| 148 |
< |
direction of the spin on top of the triangle, therefore infinite |
| 149 |
< |
possibilities for the packing of spins which induces what is known as |
| 150 |
< |
``complete regular frustration'' which leads to disordered low |
| 151 |
< |
temperature phases. The similarity goes to the dipoles on a hexagonal |
| 152 |
< |
lattice, which are shown by the dipoles in |
| 153 |
< |
figure~\ref{Infig:frustration}. In this circumstance, the dipoles want |
| 154 |
< |
to be aligned, however, due to the long wave fluctuation, at low |
| 155 |
< |
temperature, the aligned state becomes unstable, vortex is formed and |
| 156 |
< |
results in multiple local minima of energy states. The dipole on the |
| 109 |
< |
center of the hexagonal lattice is frustrated. |
| 145 |
> |
The spins in figure~\ref{Infig:frustration} illustrate frustration for |
| 146 |
> |
$J < 0$ on a triangular lattice. There are multiple local minima |
| 147 |
> |
energy states which are independent of the direction of the spin on |
| 148 |
> |
top of the triangle, therefore infinite possibilities for orienting |
| 149 |
> |
large numbers spins. This induces what is known as ``complete regular |
| 150 |
> |
frustration'' which leads to disordered low temperature phases. This |
| 151 |
> |
behavior extends to dipoles on a triangular lattices, which are shown |
| 152 |
> |
in the lower portion of figure~\ref{Infig:frustration}. In this case, |
| 153 |
> |
dipole-aligned structures are energetically favorable, however, at low |
| 154 |
> |
temperature, vortices are easily formed, and, this results in multiple |
| 155 |
> |
local minima of energy states for a central dipole. The dipole on the |
| 156 |
> |
center of the hexagonal lattice is therefore frustrated. |
| 157 |
|
|
| 158 |
|
The lack of translational degrees of freedom in lattice models |
| 159 |
|
prevents their utilization in models for surface buckling. In |
| 167 |
|
to key concepts of classical statistical mechanics that we used in |
| 168 |
|
this dissertation. Tolman gives an excellent introduction to the |
| 169 |
|
principles of statistical mechanics.~\cite{Tolman1979} A large part of |
| 170 |
< |
section~\ref{In:sec:SM} will follow Tolman's notation. |
| 170 |
> |
section~\ref{In:sec:SM} follows Tolman's notation. |
| 171 |
|
|
| 172 |
|
\subsection{Ensembles\label{In:ssec:ensemble}} |
| 173 |
|
In classical mechanics, the state of the system is completely |
| 277 |
|
and the rate of density change is zero in the neighborhood of any |
| 278 |
|
selected moving representative points in the phase space. |
| 279 |
|
|
| 280 |
< |
The condition of the ensemble is determined by the density |
| 280 |
> |
The type of thermodynamic ensemble is determined by the density |
| 281 |
|
distribution. If we consider the density distribution as only a |
| 282 |
|
function of $q$ and $p$, which means the rate of change of the phase |
| 283 |
|
space density in the neighborhood of all representative points in the |
| 306 |
|
\rho = \mathrm{const.} |
| 307 |
|
\label{Ineq:uniformEnsemble} |
| 308 |
|
\end{equation} |
| 309 |
< |
the ensemble is called {\it uniform ensemble}. |
| 309 |
> |
the ensemble is called {\it uniform ensemble}, but this ensemble has |
| 310 |
> |
little relevance for physical chemistry. It is an ensemble with |
| 311 |
> |
essentially infinite temperature. |
| 312 |
|
|
| 313 |
|
\subsubsection{The Microcanonical Ensemble\label{In:sssec:microcanonical}} |
| 314 |
< |
Another useful ensemble is the {\it microcanonical ensemble}, for |
| 315 |
< |
which: |
| 314 |
> |
The most useful ensemble for Molecular Dynamics is the {\it |
| 315 |
> |
microcanonical ensemble}, for which: |
| 316 |
|
\begin{equation} |
| 317 |
|
\rho = \delta \left( H(q^N, p^N) - E \right) \frac{1}{\Sigma (N, V, E)} |
| 318 |
|
\label{Ineq:microcanonicalEnsemble} |
| 329 |
|
\end{equation} |
| 330 |
|
where $k_B$ is the Boltzmann constant and $C^N$ is a number which |
| 331 |
|
makes the argument of $\ln$ dimensionless. In this case, $C^N$ is the |
| 332 |
< |
total phase space volume of one state. The entropy of a microcanonical |
| 333 |
< |
ensemble is given by |
| 332 |
> |
total phase space volume of one state which has the same units as |
| 333 |
> |
$\Sigma(N, V, E)$. The entropy of a microcanonical ensemble is given |
| 334 |
> |
by |
| 335 |
|
\begin{equation} |
| 336 |
|
S = k_B \ln \left(\frac{\Sigma(N, V, E)}{C^N}\right). |
| 337 |
|
\label{Ineq:entropy} |
| 348 |
|
Z_N = \int d \vec q~^N \int_\Gamma d \vec p~^N e^{-H(q^N, p^N) / k_B T}, |
| 349 |
|
\label{Ineq:partitionFunction} |
| 350 |
|
\end{equation} |
| 351 |
< |
which is also known as the canonical{\it partition function}. $\Gamma$ |
| 352 |
< |
indicates that the integral is over all phase space. In the canonical |
| 353 |
< |
ensemble, $N$, the total number of particles, $V$, total volume, and |
| 354 |
< |
$T$, the temperature, are constants. The systems with the lowest |
| 355 |
< |
energies hold the largest population. According to maximum principle, |
| 356 |
< |
thermodynamics maximizes the entropy $S$, implying that |
| 351 |
> |
which is also known as the canonical {\it partition |
| 352 |
> |
function}. $\Gamma$ indicates that the integral is over all phase |
| 353 |
> |
space. In the canonical ensemble, $N$, the total number of particles, |
| 354 |
> |
$V$, total volume, and $T$, the temperature, are constants. The |
| 355 |
> |
systems with the lowest energies hold the largest |
| 356 |
> |
population. Thermodynamics maximizes the entropy, $S$, implying that |
| 357 |
|
\begin{equation} |
| 358 |
|
\begin{array}{ccc} |
| 359 |
|
\delta S = 0 & \mathrm{and} & \delta^2 S < 0. |
| 374 |
|
system and the distribution of microscopic states. |
| 375 |
|
|
| 376 |
|
There is an implicit assumption that our arguments are based on so |
| 377 |
< |
far. A representative point in the phase space is equally likely to be |
| 378 |
< |
found in any energetically allowed region. In other words, all |
| 379 |
< |
energetically accessible states are represented equally, the |
| 380 |
< |
probabilities to find the system in any of the accessible states is |
| 381 |
< |
equal. This is called the principle of equal a {\it priori} |
| 377 |
> |
far. Tow representative points in phase space are equally likely to be |
| 378 |
> |
found if they have the same energy. In other words, all energetically |
| 379 |
> |
accessible states are represented , and the probabilities to find the |
| 380 |
> |
system in any of the accessible states is equal to that states |
| 381 |
> |
Boltzmann weight. This is called the principle of equal a {\it priori} |
| 382 |
|
probabilities. |
| 383 |
|
|
| 384 |
|
\subsection{Statistical Averages\label{In:ssec:average}} |
| 413 |
|
\frac{1}{T} \int_{0}^{T} F[q^N(t), p^N(t)] dt |
| 414 |
|
\label{Ineq:timeAverage2} |
| 415 |
|
\end{equation} |
| 416 |
< |
for an infinite time interval. |
| 416 |
> |
for an finite time interval, $T$. |
| 417 |
|
|
| 418 |
|
\subsubsection{Ergodicity\label{In:sssec:ergodicity}} |
| 419 |
|
The {\it ergodic hypothesis}, an important hypothesis governing modern |
| 456 |
|
\frac{1}{T} \int_{0}^{T} dt A(t) B(t + \tau), |
| 457 |
|
\label{Ineq:crosscorrelationFunction} |
| 458 |
|
\end{equation} |
| 459 |
< |
and called {\it cross correlation function}. |
| 459 |
> |
and is called a {\it cross correlation function}. |
| 460 |
|
|
| 461 |
|
We know from the ergodic hypothesis that there is a relationship |
| 462 |
|
between time average and ensemble average. We can put the correlation |
| 463 |
< |
function in a classical mechanics form, |
| 463 |
> |
function in a classical mechanical form, |
| 464 |
|
\begin{equation} |
| 465 |
< |
C_{AA}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N(t), p^N(t)] |
| 466 |
< |
A[q^N(t+\tau), q^N(t+\tau)] \rho(q, p) |
| 465 |
> |
C_{AA}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N, p^N] |
| 466 |
> |
A[q^N(\tau), p^N(\tau)] \rho(q^N, p^N) |
| 467 |
|
\label{Ineq:autocorrelationFunctionCM} |
| 468 |
|
\end{equation} |
| 469 |
< |
and |
| 469 |
> |
where $q^N(\tau)$, $p^N(\tau)$ is the phase space point that follows |
| 470 |
> |
classical evolution of the point $q^N$, $p^N$ after a tme $\tau$ has |
| 471 |
> |
elapsed, and |
| 472 |
|
\begin{equation} |
| 473 |
< |
C_{AB}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N(t), p^N(t)] |
| 474 |
< |
B[q^N(t+\tau), q^N(t+\tau)] \rho(q, p) |
| 473 |
> |
C_{AB}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N, p^N] |
| 474 |
> |
B[q^N(\tau), p^N(\tau)] \rho(q^N, p^N) |
| 475 |
|
\label{Ineq:crosscorrelationFunctionCM} |
| 476 |
|
\end{equation} |
| 477 |
< |
as autocorrelation function and cross correlation function |
| 477 |
> |
as the autocorrelation function and cross correlation functions |
| 478 |
|
respectively. $\rho(q, p)$ is the density distribution at equillibrium |
| 479 |
< |
in phase space. In many cases, the correlation function decay is a |
| 480 |
< |
single exponential |
| 479 |
> |
in phase space. In many cases, correlation functions decay as a |
| 480 |
> |
single exponential in time |
| 481 |
|
\begin{equation} |
| 482 |
|
C(t) \sim e^{-t / \tau_r}, |
| 483 |
|
\label{Ineq:relaxation} |
| 484 |
|
\end{equation} |
| 485 |
< |
where $\tau_r$ is known as relaxation time which discribes the rate of |
| 485 |
> |
where $\tau_r$ is known as relaxation time which describes the rate of |
| 486 |
|
the decay. |
| 487 |
|
|
| 488 |
< |
\section{Methodolody\label{In:sec:method}} |
| 489 |
< |
The simulations performed in this dissertation are branched into two |
| 490 |
< |
main catalog, Monte Carlo and Molecular Dynamics. There are two main |
| 491 |
< |
difference between Monte Carlo and Molecular Dynamics simulations. One |
| 492 |
< |
is that the Monte Carlo simulation is time independent, and Molecular |
| 493 |
< |
Dynamics simulation is time involved. Another dissimilar is that the |
| 494 |
< |
Monte Carlo is a stochastic process, the configuration of the system |
| 495 |
< |
is not determinated by its past, however, using Moleuclar Dynamics, |
| 496 |
< |
the system is propagated by Newton's equation of motion, the |
| 497 |
< |
trajectory of the system evolved in the phase space is determined. A |
| 498 |
< |
brief introduction of the two algorithms are given in |
| 499 |
< |
section~\ref{In:ssec:mc} and~\ref{In:ssec:md}. An extension of the |
| 500 |
< |
Molecular Dynamics, Langevin Dynamics, is introduced by |
| 488 |
> |
\section{Methodology\label{In:sec:method}} |
| 489 |
> |
The simulations performed in this dissertation branch into two main |
| 490 |
> |
categories, Monte Carlo and Molecular Dynamics. There are two main |
| 491 |
> |
differences between Monte Carlo and Molecular Dynamics |
| 492 |
> |
simulations. One is that the Monte Carlo simulations are time |
| 493 |
> |
independent methods of sampling structural features of an ensemble, |
| 494 |
> |
while Molecular Dynamics simulations provide dynamic |
| 495 |
> |
information. Additionally, Monte Carlo methods are stochastic |
| 496 |
> |
processes; the future configurations of the system are not determined |
| 497 |
> |
by its past. However, in Molecular Dynamics, the system is propagated |
| 498 |
> |
by Hamilton's equations of motion, and the trajectory of the system |
| 499 |
> |
evolving in phase space is deterministic. Brief introductions of the |
| 500 |
> |
two algorithms are given in section~\ref{In:ssec:mc} |
| 501 |
> |
and~\ref{In:ssec:md}. Langevin Dynamics, an extension of the Molecular |
| 502 |
> |
Dynamics that includes implicit solvent effects, is introduced by |
| 503 |
|
section~\ref{In:ssec:ld}. |
| 504 |
|
|
| 505 |
|
\subsection{Monte Carlo\label{In:ssec:mc}} |
| 506 |
< |
Monte Carlo algorithm was first introduced by Metropolis {\it et |
| 507 |
< |
al.}.~\cite{Metropolis53} Basic Monte Carlo algorithm is usually |
| 508 |
< |
applied to the canonical ensemble, a Boltzmann-weighted ensemble, in |
| 509 |
< |
which the $N$, the total number of particles, $V$, total volume, $T$, |
| 510 |
< |
temperature are constants. The average energy is given by substituding |
| 511 |
< |
Eq.~\ref{Ineq:canonicalEnsemble} into Eq.~\ref{Ineq:statAverage2}, |
| 506 |
> |
A Monte Carlo integration algorithm was first introduced by Metropolis |
| 507 |
> |
{\it et al.}~\cite{Metropolis53} The basic Metropolis Monte Carlo |
| 508 |
> |
algorithm is usually applied to the canonical ensemble, a |
| 509 |
> |
Boltzmann-weighted ensemble, in which $N$, the total number of |
| 510 |
> |
particles, $V$, the total volume, and $T$, the temperature are |
| 511 |
> |
constants. An average in this ensemble is given |
| 512 |
|
\begin{equation} |
| 513 |
< |
\langle E \rangle = \frac{1}{Z_N} \int d \vec q~^N \int d \vec p~^N E e^{-H(q^N, p^N) / k_B T}. |
| 513 |
> |
\langle A \rangle = \frac{1}{Z_N} \int d \vec q~^N \int d \vec p~^N |
| 514 |
> |
A(q^N, p^N) e^{-H(q^N, p^N) / k_B T}. |
| 515 |
|
\label{Ineq:energyofCanonicalEnsemble} |
| 516 |
|
\end{equation} |
| 517 |
< |
So are the other properties of the system. The Hamiltonian is the |
| 518 |
< |
summation of Kinetic energy $K(p^N)$ as a function of momenta and |
| 519 |
< |
Potential energy $U(q^N)$ as a function of positions, |
| 517 |
> |
The Hamiltonian is the sum of the kinetic energy, $K(p^N)$, a function |
| 518 |
> |
of momenta and the potential energy, $U(q^N)$, a function of |
| 519 |
> |
positions, |
| 520 |
|
\begin{equation} |
| 521 |
|
H(q^N, p^N) = K(p^N) + U(q^N). |
| 522 |
|
\label{Ineq:hamiltonian} |
| 523 |
|
\end{equation} |
| 524 |
< |
If the property $A$ is only a function of position ($ A = A(q^N)$), |
| 525 |
< |
the mean value of $A$ is reduced to |
| 524 |
> |
If the property $A$ is a function only of position ($ A = A(q^N)$), |
| 525 |
> |
the mean value of $A$ can be reduced to |
| 526 |
|
\begin{equation} |
| 527 |
< |
\langle A \rangle = \frac{\int d \vec q~^N \int d \vec p~^N A e^{-U(q^N) / k_B T}}{\int d \vec q~^N \int d \vec p~^N e^{-U(q^N) / k_B T}}, |
| 527 |
> |
\langle A \rangle = \frac{\int d \vec q~^N A e^{-U(q^N) / k_B T}}{\int d \vec q~^N e^{-U(q^N) / k_B T}}, |
| 528 |
|
\label{Ineq:configurationIntegral} |
| 529 |
|
\end{equation} |
| 530 |
|
The kinetic energy $K(p^N)$ is factored out in |
| 531 |
|
Eq.~\ref{Ineq:configurationIntegral}. $\langle A |
| 532 |
< |
\rangle$ is a configuration integral now, and the |
| 532 |
> |
\rangle$ is now a configuration integral, and |
| 533 |
|
Eq.~\ref{Ineq:configurationIntegral} is equivalent to |
| 534 |
|
\begin{equation} |
| 535 |
< |
\langle A \rangle = \int d \vec q~^N A \rho(q^N). |
| 535 |
> |
\langle A \rangle = \int d \vec q~^N A \rho(q^N), |
| 536 |
|
\label{Ineq:configurationAve} |
| 537 |
|
\end{equation} |
| 538 |
+ |
where $\rho(q^N)$ is a configurational probability |
| 539 |
+ |
\begin{equation} |
| 540 |
+ |
\rho(q^N) = \frac{e^{-U(q^N) / k_B T}}{\int d \vec q~^N e^{-U(q^N) / k_B T}}. |
| 541 |
+ |
\label{Ineq:configurationProb} |
| 542 |
+ |
\end{equation} |
| 543 |
|
|
| 544 |
< |
In a Monte Carlo simulation of canonical ensemble, the probability of |
| 545 |
< |
the system being in a state $s$ is $\rho_s$, the change of this |
| 546 |
< |
probability with time is given by |
| 544 |
> |
In a Monte Carlo simulation of a system in the canonical ensemble, the |
| 545 |
> |
probability of the system being in a state $s$ is $\rho_s$, the change |
| 546 |
> |
of this probability with time is given by |
| 547 |
|
\begin{equation} |
| 548 |
|
\frac{d \rho_s}{dt} = \sum_{s'} [ -w_{ss'}\rho_s + w_{s's}\rho_{s'} ], |
| 549 |
|
\label{Ineq:timeChangeofProb} |
| 554 |
|
\frac{d \rho_{s}^{equilibrium}}{dt} = 0, |
| 555 |
|
\label{Ineq:equiProb} |
| 556 |
|
\end{equation} |
| 557 |
< |
which means $\sum_{s'} [ -w_{ss'}\rho_s + w_{s's}\rho_{s'} ]$ is $0$ |
| 558 |
< |
for all $s'$. So |
| 557 |
> |
the sum of transition probabilities $\sum_{s'} [ -w_{ss'}\rho_s + |
| 558 |
> |
w_{s's}\rho_{s'} ]$ is $0$ for all $s'$. So |
| 559 |
|
\begin{equation} |
| 560 |
|
\frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = \frac{w_{s's}}{w_{ss'}}. |
| 561 |
|
\label{Ineq:relationshipofRhoandW} |
| 562 |
|
\end{equation} |
| 563 |
< |
If |
| 563 |
> |
If the ratio of state populations |
| 564 |
|
\begin{equation} |
| 505 |
– |
\frac{w_{s's}}{w_{ss'}} = e^{-(U_s - U_{s'}) / k_B T}, |
| 506 |
– |
\label{Ineq:conditionforBoltzmannStatistics} |
| 507 |
– |
\end{equation} |
| 508 |
– |
then |
| 509 |
– |
\begin{equation} |
| 565 |
|
\frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = e^{-(U_s - U_{s'}) / k_B T}. |
| 566 |
|
\label{Ineq:satisfyofBoltzmannStatistics} |
| 567 |
|
\end{equation} |
| 568 |
< |
Eq.~\ref{Ineq:satisfyofBoltzmannStatistics} implies that |
| 569 |
< |
$\rho^{equilibrium}$ satisfies Boltzmann statistics. An algorithm, |
| 570 |
< |
shows how Monte Carlo simulation generates a transition probability |
| 571 |
< |
governed by \ref{Ineq:conditionforBoltzmannStatistics}, is schemed as |
| 568 |
> |
then the ratio of transition probabilities, |
| 569 |
> |
\begin{equation} |
| 570 |
> |
\frac{w_{s's}}{w_{ss'}} = e^{-(U_s - U_{s'}) / k_B T}, |
| 571 |
> |
\label{Ineq:conditionforBoltzmannStatistics} |
| 572 |
> |
\end{equation} |
| 573 |
> |
An algorithm that indicates how a Monte Carlo simulation generates a |
| 574 |
> |
transition probability governed by |
| 575 |
> |
\ref{Ineq:conditionforBoltzmannStatistics}, is given schematically as, |
| 576 |
|
\begin{enumerate} |
| 577 |
< |
\item\label{Initm:oldEnergy} Choose an particle randomly, calculate the energy. |
| 578 |
< |
\item\label{Initm:newEnergy} Make a random displacement for particle, |
| 579 |
< |
calculate the new energy. |
| 577 |
> |
\item\label{Initm:oldEnergy} Choose a particle randomly, and calculate |
| 578 |
> |
the energy of the rest of the system due to the current location of |
| 579 |
> |
the particle. |
| 580 |
> |
\item\label{Initm:newEnergy} Make a random displacement of the particle, |
| 581 |
> |
calculate the new energy taking the movement of the particle into account. |
| 582 |
|
\begin{itemize} |
| 583 |
< |
\item Keep the new configuration and return to step~\ref{Initm:oldEnergy} if energy |
| 584 |
< |
goes down. |
| 524 |
< |
\item Pick a random number between $[0,1]$ if energy goes up. |
| 583 |
> |
\item If the energy goes down, keep the new configuration. |
| 584 |
> |
\item If the energy goes up, pick a random number between $[0,1]$. |
| 585 |
|
\begin{itemize} |
| 586 |
< |
\item Keep the new configuration and return to |
| 587 |
< |
step~\ref{Initm:oldEnergy} if the random number smaller than |
| 588 |
< |
$e^{-(U_{new} - U_{old})} / k_B T$. |
| 589 |
< |
\item Keep the old configuration and return to |
| 530 |
< |
step~\ref{Initm:oldEnergy} if the random number larger than |
| 531 |
< |
$e^{-(U_{new} - U_{old})} / k_B T$. |
| 586 |
> |
\item If the random number smaller than |
| 587 |
> |
$e^{-(U_{new} - U_{old})} / k_B T$, keep the new configuration. |
| 588 |
> |
\item If the random number is larger than |
| 589 |
> |
$e^{-(U_{new} - U_{old})} / k_B T$, keep the old configuration. |
| 590 |
|
\end{itemize} |
| 591 |
|
\end{itemize} |
| 592 |
< |
\item\label{Initm:accumulateAvg} Accumulate the average after it converges. |
| 592 |
> |
\item\label{Initm:accumulateAvg} Accumulate the averages based on the |
| 593 |
> |
current configuration. |
| 594 |
> |
\item Go to step~\ref{Initm:oldEnergy}. |
| 595 |
|
\end{enumerate} |
| 596 |
< |
It is important to notice that the old configuration has to be sampled |
| 597 |
< |
again if it is kept. |
| 596 |
> |
It is important for sampling accuracy that the old configuration is |
| 597 |
> |
sampled again if it is kept. |
| 598 |
|
|
| 599 |
|
\subsection{Molecular Dynamics\label{In:ssec:md}} |
| 600 |
|
Although some of properites of the system can be calculated from the |
| 601 |
< |
ensemble average in Monte Carlo simulations, due to the nature of |
| 602 |
< |
lacking in the time dependence, it is impossible to gain information |
| 603 |
< |
of those dynamic properties from Monte Carlo simulations. Molecular |
| 604 |
< |
Dynamics is a measurement of the evolution of the positions and |
| 605 |
< |
momenta of the particles in the system. The evolution of the system |
| 606 |
< |
obeys laws of classical mechanics, in most situations, there is no |
| 607 |
< |
need for the count of the quantum effects. For a real experiment, the |
| 608 |
< |
instantaneous positions and momenta of the particles in the system are |
| 609 |
< |
neither important nor measurable, the observable quantities are |
| 610 |
< |
usually a average value for a finite time interval. These quantities |
| 611 |
< |
are expressed as a function of positions and momenta in Melecular |
| 612 |
< |
Dynamics simulations. Like the thermal temperature of the system is |
| 553 |
< |
defined by |
| 601 |
> |
ensemble average in Monte Carlo simulations, due to the absence of the |
| 602 |
> |
time dependence, it is impossible to gain information on dynamic |
| 603 |
> |
properties from Monte Carlo simulations. Molecular Dynamics evolves |
| 604 |
> |
the positions and momenta of the particles in the system. The |
| 605 |
> |
evolution of the system obeys the laws of classical mechanics, and in |
| 606 |
> |
most situations, there is no need to account for quantum effects. In a |
| 607 |
> |
real experiment, the instantaneous positions and momenta of the |
| 608 |
> |
particles in the system are ofter neither important nor measurable, |
| 609 |
> |
the observable quantities are usually an average value for a finite |
| 610 |
> |
time interval. These quantities are expressed as a function of |
| 611 |
> |
positions and momenta in Molecular Dynamics simulations. For example, |
| 612 |
> |
temperature of the system is defined by |
| 613 |
|
\begin{equation} |
| 614 |
< |
\frac{1}{2} k_B T = \langle \frac{1}{2} m v_\alpha \rangle, |
| 614 |
> |
\frac{3}{2} N k_B T = \langle \sum_{i=1}^N \frac{1}{2} m_i v_i \rangle, |
| 615 |
|
\label{Ineq:temperature} |
| 616 |
|
\end{equation} |
| 617 |
< |
here $m$ is the mass of the particle and $v_\alpha$ is the $\alpha$ |
| 618 |
< |
component of the velocity of the particle. The right side of |
| 619 |
< |
Eq.~\ref{Ineq:temperature} is the average kinetic energy of the |
| 561 |
< |
system. A simple Molecular Dynamics simulation scheme |
| 562 |
< |
is:~\cite{Frenkel1996} |
| 563 |
< |
\begin{enumerate} |
| 564 |
< |
\item\label{Initm:initialize} Assign the initial positions and momenta |
| 565 |
< |
for the particles in the system. |
| 566 |
< |
\item\label{Initm:calcForce} Calculate the forces. |
| 567 |
< |
\item\label{Initm:equationofMotion} Integrate the equation of motion. |
| 568 |
< |
\begin{itemize} |
| 569 |
< |
\item Return to step~\ref{Initm:calcForce} if the equillibrium is |
| 570 |
< |
not achieved. |
| 571 |
< |
\item Go to step~\ref{Initm:calcAvg} if the equillibrium is |
| 572 |
< |
achieved. |
| 573 |
< |
\end{itemize} |
| 574 |
< |
\item\label{Initm:calcAvg} Compute the quantities we are interested in. |
| 575 |
< |
\end{enumerate} |
| 576 |
< |
The initial positions of the particles are chosen as that there is no |
| 577 |
< |
overlap for the particles. The initial velocities at first are |
| 578 |
< |
distributed randomly to the particles, and then shifted to make the |
| 579 |
< |
momentum of the system $0$, at last scaled to the desired temperature |
| 580 |
< |
of the simulation according Eq.~\ref{Ineq:temperature}. |
| 617 |
> |
here $m_i$ is the mass of particle $i$ and $v_i$ is the velocity of |
| 618 |
> |
particle $i$. The right side of Eq.~\ref{Ineq:temperature} is the |
| 619 |
> |
average kinetic energy of the system. |
| 620 |
|
|
| 621 |
< |
The core of Molecular Dynamics simulations is step~\ref{Initm:calcForce} |
| 622 |
< |
and~\ref{Initm:equationofMotion}. The calculation of the forces are |
| 623 |
< |
often involved numerous effort, this is the most time consuming step |
| 624 |
< |
in the Molecular Dynamics scheme. The evaluation of the forces is |
| 625 |
< |
followed by |
| 621 |
> |
The initial positions of the particles are chosen so that there is no |
| 622 |
> |
overlap of the particles. The initial velocities at first are |
| 623 |
> |
distributed randomly to the particles using a Maxwell-Boltzmann |
| 624 |
> |
distribution, and then shifted to make the total linear momentum of |
| 625 |
> |
the system $0$. |
| 626 |
> |
|
| 627 |
> |
The core of Molecular Dynamics simulations is the calculation of |
| 628 |
> |
forces and the integration algorithm. Calculation of the forces often |
| 629 |
> |
involves enormous effort. This is the most time consuming step in the |
| 630 |
> |
Molecular Dynamics scheme. Evaluation of the forces is mathematically |
| 631 |
> |
simple, |
| 632 |
|
\begin{equation} |
| 633 |
|
f(q) = - \frac{\partial U(q)}{\partial q}, |
| 634 |
|
\label{Ineq:force} |
| 635 |
|
\end{equation} |
| 636 |
< |
$U(q)$ is the potential of the system. Once the forces computed, are |
| 637 |
< |
the positions and velocities updated by integrating Newton's equation |
| 638 |
< |
of motion, |
| 639 |
< |
\begin{equation} |
| 640 |
< |
f(q) = \frac{dp}{dt} = \frac{m dv}{dt}. |
| 636 |
> |
where $U(q)$ is the potential of the system. However, the numerical |
| 637 |
> |
details of this computation are often quite complex. Once the forces |
| 638 |
> |
computed, the positions and velocities are updated by integrating |
| 639 |
> |
Hamilton's equations of motion, |
| 640 |
> |
\begin{eqnarray} |
| 641 |
> |
\dot p_i & = & -\frac{\partial H}{\partial q_i} = -\frac{\partial |
| 642 |
> |
U(q_i)}{\partial q_i} = f(q_i) \\ |
| 643 |
> |
\dot q_i & = & p_i |
| 644 |
|
\label{Ineq:newton} |
| 645 |
< |
\end{equation} |
| 646 |
< |
Here is an example of integrating algorithms, Verlet algorithm, which |
| 647 |
< |
is one of the best algorithms to integrate Newton's equation of |
| 648 |
< |
motion. The Taylor expension of the position at time $t$ is |
| 645 |
> |
\end{eqnarray} |
| 646 |
> |
The classic example of an integrating algorithm is the Verlet |
| 647 |
> |
algorithm, which is one of the simplest algorithms for integrating the |
| 648 |
> |
equations of motion. The Taylor expansion of the position at time $t$ |
| 649 |
> |
is |
| 650 |
|
\begin{equation} |
| 651 |
< |
q(t+\Delta t)= q(t) + v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 + |
| 651 |
> |
q(t+\Delta t)= q(t) + \frac{p(t)}{m} \Delta t + \frac{f(t)}{2m}\Delta t^2 + |
| 652 |
|
\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} + |
| 653 |
|
\mathcal{O}(\Delta t^4) |
| 654 |
|
\label{Ineq:verletFuture} |
| 655 |
|
\end{equation} |
| 656 |
|
for a later time $t+\Delta t$, and |
| 657 |
|
\begin{equation} |
| 658 |
< |
q(t-\Delta t)= q(t) - v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 - |
| 658 |
> |
q(t-\Delta t)= q(t) - \frac{p(t)}{m} \Delta t + \frac{f(t)}{2m}\Delta t^2 - |
| 659 |
|
\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} + |
| 660 |
|
\mathcal{O}(\Delta t^4) , |
| 661 |
|
\label{Ineq:verletPrevious} |
| 662 |
|
\end{equation} |
| 663 |
< |
for a previous time $t-\Delta t$. The summation of the |
| 664 |
< |
Eq.~\ref{Ineq:verletFuture} and~\ref{Ineq:verletPrevious} gives |
| 663 |
> |
for a previous time $t-\Delta t$. Adding Eq.~\ref{Ineq:verletFuture} |
| 664 |
> |
and~\ref{Ineq:verletPrevious} gives |
| 665 |
|
\begin{equation} |
| 666 |
|
q(t+\Delta t)+q(t-\Delta t) = |
| 667 |
|
2q(t) + \frac{f(t)}{m}\Delta t^2 + \mathcal{O}(\Delta t^4), |
| 673 |
|
2q(t) - q(t-\Delta t) + \frac{f(t)}{m}\Delta t^2. |
| 674 |
|
\label{Ineq:newPosition} |
| 675 |
|
\end{equation} |
| 676 |
< |
The higher order of the $\Delta t$ is omitted. |
| 676 |
> |
The higher order terms in $\Delta t$ are omitted. |
| 677 |
|
|
| 678 |
< |
Numerous technics and tricks are applied to Molecular Dynamics |
| 679 |
< |
simulation to gain more efficiency or more accuracy. The simulation |
| 680 |
< |
engine used in this dissertation for the Molecular Dynamics |
| 681 |
< |
simulations is {\sc oopse}, more details of the algorithms and |
| 682 |
< |
technics can be found in~\cite{Meineke2005}. |
| 678 |
> |
Numerous techniques and tricks have been applied to Molecular Dynamics |
| 679 |
> |
simulations to gain greater efficiency or accuracy. The engine used in |
| 680 |
> |
this dissertation for the Molecular Dynamics simulations is {\sc |
| 681 |
> |
oopse}. More details of the algorithms and techniques used in this |
| 682 |
> |
code can be found in Ref.~\cite{Meineke2005}. |
| 683 |
|
|
| 684 |
|
\subsection{Langevin Dynamics\label{In:ssec:ld}} |
| 685 |
< |
In many cases, the properites of the solvent in a system, like the |
| 686 |
< |
lipid-water system studied in this dissertation, are less important to |
| 687 |
< |
the researchers. However, the major computational expense is spent on |
| 688 |
< |
the solvent in the Molecular Dynamics simulations because of the large |
| 689 |
< |
number of the solvent molecules compared to that of solute molecules, |
| 690 |
< |
the ratio of the number of lipid molecules to the number of water |
| 691 |
< |
molecules is $1:25$ in our lipid-water system. The efficiency of the |
| 692 |
< |
Molecular Dynamics simulations is greatly reduced. |
| 685 |
> |
In many cases, the properites of the solvent (like the water in the |
| 686 |
> |
lipid-water system studied in this dissertation) are less interesting |
| 687 |
> |
to the researchers than the behavior of the solute. However, the major |
| 688 |
> |
computational expense is ofter the solvent-solvent interactions, this |
| 689 |
> |
is due to the large number of the solvent molecules when compared to |
| 690 |
> |
the number of solute molecules. The ratio of the number of lipid |
| 691 |
> |
molecules to the number of water molecules is $1:25$ in our |
| 692 |
> |
lipid-water system. The efficiency of the Molecular Dynamics |
| 693 |
> |
simulations is greatly reduced, with up to 85\% of CPU time spent |
| 694 |
> |
calculating only water-water interactions. |
| 695 |
|
|
| 696 |
< |
As an extension of the Molecular Dynamics simulations, the Langevin |
| 697 |
< |
Dynamics seeks a way to avoid integrating equation of motion for |
| 698 |
< |
solvent particles without losing the Brownian properites of solute |
| 699 |
< |
particles. A common approximation is that the coupling of the solute |
| 700 |
< |
and solvent is expressed as a set of harmonic oscillators. So the |
| 701 |
< |
Hamiltonian of such a system is written as |
| 696 |
> |
As an extension of the Molecular Dynamics methodologies, Langevin |
| 697 |
> |
Dynamics seeks a way to avoid integrating the equations of motion for |
| 698 |
> |
solvent particles without losing the solvent effects on the solute |
| 699 |
> |
particles. One common approximation is to express the coupling of the |
| 700 |
> |
solute and solvent as a set of harmonic oscillators. The Hamiltonian |
| 701 |
> |
of such a system is written as |
| 702 |
|
\begin{equation} |
| 703 |
|
H = \frac{p^2}{2m} + U(q) + H_B + \Delta U(q), |
| 704 |
|
\label{Ineq:hamiltonianofCoupling} |
| 705 |
|
\end{equation} |
| 706 |
< |
where $H_B$ is the Hamiltonian of the bath which equals to |
| 706 |
> |
where $H_B$ is the Hamiltonian of the bath which is a set of N |
| 707 |
> |
harmonic oscillators |
| 708 |
|
\begin{equation} |
| 709 |
|
H_B = \sum_{\alpha = 1}^{N} \left\{ \frac{p_\alpha^2}{2m_\alpha} + |
| 710 |
|
\frac{1}{2} m_\alpha \omega_\alpha^2 q_\alpha^2\right\}, |
| 711 |
|
\label{Ineq:hamiltonianofBath} |
| 712 |
|
\end{equation} |
| 713 |
< |
$\alpha$ is all the degree of freedoms of the bath, $\omega$ is the |
| 714 |
< |
bath frequency, and $\Delta U(q)$ is the bilinear coupling given by |
| 713 |
> |
$\alpha$ runs over all the degree of freedoms of the bath, |
| 714 |
> |
$\omega_\alpha$ is the bath frequency of oscillator $\alpha$, and |
| 715 |
> |
$\Delta U(q)$ is the bilinear coupling given by |
| 716 |
|
\begin{equation} |
| 717 |
|
\Delta U = -\sum_{\alpha = 1}^{N} g_\alpha q_\alpha q, |
| 718 |
|
\label{Ineq:systemBathCoupling} |
| 719 |
|
\end{equation} |
| 720 |
< |
where $g$ is the coupling constant. By solving the Hamilton's equation |
| 721 |
< |
of motion, the {\it Generalized Langevin Equation} for this system is |
| 722 |
< |
derived to |
| 720 |
> |
where $g_\alpha$ is the coupling constant for oscillator $\alpha$. By |
| 721 |
> |
solving the Hamilton's equations of motion, the {\it Generalized |
| 722 |
> |
Langevin Equation} for this system is derived as |
| 723 |
|
\begin{equation} |
| 724 |
|
m \ddot q = -\frac{\partial W(q)}{\partial q} - \int_0^t \xi(t) \dot q(t-t')dt' + R(t), |
| 725 |
|
\label{Ineq:gle} |
| 727 |
|
with mean force, |
| 728 |
|
\begin{equation} |
| 729 |
|
W(q) = U(q) - \sum_{\alpha = 1}^N \frac{g_\alpha^2}{2m_\alpha |
| 730 |
< |
\omega_\alpha^2}q^2, |
| 730 |
> |
\omega_\alpha^2}q^2. |
| 731 |
|
\label{Ineq:meanForce} |
| 732 |
|
\end{equation} |
| 733 |
< |
being only a dependence of coordinates of the solute particles, {\it |
| 734 |
< |
friction kernel}, |
| 733 |
> |
The {\it friction kernel}, $\xi(t)$, depends only on the coordinates |
| 734 |
> |
of the solute particles, |
| 735 |
|
\begin{equation} |
| 736 |
|
\xi(t) = \sum_{\alpha = 1}^N \frac{-g_\alpha}{m_\alpha |
| 737 |
|
\omega_\alpha} \cos(\omega_\alpha t), |
| 738 |
|
\label{Ineq:xiforGLE} |
| 739 |
|
\end{equation} |
| 740 |
< |
and the random force, |
| 740 |
> |
and a ``random'' force, |
| 741 |
|
\begin{equation} |
| 742 |
|
R(t) = \sum_{\alpha = 1}^N \left( g_\alpha q_\alpha(0)-\frac{g_\alpha}{m_\alpha |
| 743 |
|
\omega_\alpha^2}q(0)\right) \cos(\omega_\alpha t) + \frac{\dot |
| 744 |
|
q_\alpha(0)}{\omega_\alpha} \sin(\omega_\alpha t), |
| 745 |
|
\label{Ineq:randomForceforGLE} |
| 746 |
|
\end{equation} |
| 747 |
< |
as only a dependence of the initial conditions. The relationship of |
| 748 |
< |
friction kernel $\xi(t)$ and random force $R(t)$ is given by |
| 747 |
> |
that depends only on the initial conditions. The relationship of |
| 748 |
> |
friction kernel $\xi(t)$ and random force $R(t)$ is given by the |
| 749 |
> |
second fluctuation dissipation theorem, |
| 750 |
|
\begin{equation} |
| 751 |
< |
\xi(t) = \frac{1}{k_B T} \langle R(t)R(0) \rangle |
| 751 |
> |
\xi(t) = \frac{1}{k_B T} \langle R(t)R(0) \rangle. |
| 752 |
|
\label{Ineq:relationshipofXiandR} |
| 753 |
|
\end{equation} |
| 754 |
< |
from their definitions. In Langevin limit, the friction is treated |
| 755 |
< |
static, which means |
| 754 |
> |
In the harmonic bath this relation is exact and provable from the |
| 755 |
> |
definitions of these quantities. In the limit of static friction, |
| 756 |
|
\begin{equation} |
| 757 |
|
\xi(t) = 2 \xi_0 \delta(t). |
| 758 |
|
\label{Ineq:xiofStaticFriction} |
| 759 |
|
\end{equation} |
| 760 |
< |
After substitude $\xi(t)$ into Eq.~\ref{Ineq:gle} with |
| 761 |
< |
Eq.~\ref{Ineq:xiofStaticFriction}, {\it Langevin Equation} is extracted |
| 762 |
< |
to |
| 760 |
> |
After substituting $\xi(t)$ into Eq.~\ref{Ineq:gle} with |
| 761 |
> |
Eq.~\ref{Ineq:xiofStaticFriction}, the {\it Langevin Equation} is |
| 762 |
> |
extracted, |
| 763 |
|
\begin{equation} |
| 764 |
|
m \ddot q = -\frac{\partial U(q)}{\partial q} - \xi \dot q(t) + R(t). |
| 765 |
|
\label{Ineq:langevinEquation} |
| 766 |
|
\end{equation} |
| 767 |
< |
The applying of Langevin Equation to dynamic simulations is discussed |
| 768 |
< |
in Ch.~\ref{chap:ld}. |
| 767 |
> |
Application of the Langevin Equation to dynamic simulations is |
| 768 |
> |
discussed in Ch.~\ref{chap:ld}. |
