| 285 |
|
\end{equation} |
| 286 |
|
The difficulty is selecting points $\mathbf{r}^N$ such that they are |
| 287 |
|
sampled from the distribution $\rho_{kT}(\mathbf{r}^N)$. A solution |
| 288 |
< |
was proposed by Metropolis et al.\cite{metropolis:1953} which involved |
| 288 |
> |
was proposed by Metropolis \emph{et al}.\cite{metropolis:1953} which involved |
| 289 |
|
the use of a Markov chain whose limiting distribution was |
| 290 |
|
$\rho_{kT}(\mathbf{r}^N)$. |
| 291 |
|
|
| 481 |
|
simulation box on an infinite lattice in Cartesian space. Any given |
| 482 |
|
particle leaving the simulation box on one side will have an image of |
| 483 |
|
itself enter on the opposite side (see Fig.~\ref{introFig:pbc}). In |
| 484 |
< |
addition, this sets that any given particle pair has an image, real or |
| 485 |
< |
periodic, within $fix$ of each other. A discussion of the method used |
| 486 |
< |
to calculate the periodic image can be found in |
| 484 |
> |
addition, this sets that any two particles have an image, real or |
| 485 |
> |
periodic, within $\text{box}/2$ of each other. A discussion of the |
| 486 |
> |
method used to calculate the periodic image can be found in |
| 487 |
|
Sec.\ref{oopseSec:pbc}. |
| 488 |
|
|
| 489 |
|
\begin{figure} |
| 557 |
|
\mathcal{O}(\Delta t^4) |
| 558 |
|
\label{introEq:verletBack} |
| 559 |
|
\end{equation} |
| 560 |
< |
Adding together Eq.~\ref{introEq:verletForward} and |
| 560 |
> |
Where $m$ is the mass of the particle, $q(t)$ is the position at time |
| 561 |
> |
$t$, $v(t)$ the velocity, and $F(t)$ the force acting on the |
| 562 |
> |
particle. Adding together Eq.~\ref{introEq:verletForward} and |
| 563 |
|
Eq.~\ref{introEq:verletBack} results in, |
| 564 |
|
\begin{equation} |
| 565 |
< |
eq here |
| 565 |
> |
q(t+\Delta t)+q(t-\Delta t) = |
| 566 |
> |
2q(t) + \frac{F(t)}{m}\Delta t^2 + \mathcal{O}(\Delta t^4) |
| 567 |
|
\label{introEq:verletSum} |
| 568 |
|
\end{equation} |
| 569 |
|
Or equivalently, |
| 570 |
|
\begin{equation} |
| 571 |
< |
eq here |
| 571 |
> |
q(t+\Delta t) \approx |
| 572 |
> |
2q(t) - q(t-\Delta t) + \frac{F(t)}{m}\Delta t^2 |
| 573 |
|
\label{introEq:verletFinal} |
| 574 |
|
\end{equation} |
| 575 |
|
Which contains an error in the estimate of the new positions on the |
| 577 |
|
|
| 578 |
|
In practice, however, the simulations in this research were integrated |
| 579 |
|
with a velocity reformulation of the Verlet method.\cite{allen87:csl} |
| 580 |
< |
\begin{equation} |
| 581 |
< |
eq here |
| 582 |
< |
\label{introEq:MDvelVerletPos} |
| 583 |
< |
\end{equation} |
| 584 |
< |
\begin{equation} |
| 581 |
< |
eq here |
| 580 |
> |
\begin{align} |
| 581 |
> |
q(t+\Delta t) &= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 % |
| 582 |
> |
\label{introEq:MDvelVerletPos} \\% |
| 583 |
> |
% |
| 584 |
> |
v(t+\Delta t) &= v(t) + \frac{\Delta t}{2m}[F(t) + F(t+\Delta t)] % |
| 585 |
|
\label{introEq:MDvelVerletVel} |
| 586 |
< |
\end{equation} |
| 586 |
> |
\end{align} |
| 587 |
|
The original Verlet algorithm can be regained by substituting the |
| 588 |
|
velocity back into Eq.~\ref{introEq:MDvelVerletPos}. The Verlet |
| 589 |
|
formulations are chosen in this research because the algorithms have |
| 605 |
|
reversible. The fact that it shadows the true Hamiltonian in phase |
| 606 |
|
space is acceptable in actual simulations as one is interested in the |
| 607 |
|
ensemble average of the observable being measured. From the ergodic |
| 608 |
< |
hypothesis (Sec.~\ref{introSec:StatThermo}), it is known that the time |
| 608 |
> |
hypothesis (Sec.~\ref{introSec:statThermo}), it is known that the time |
| 609 |
|
average will match the ensemble average, therefore two similar |
| 610 |
|
trajectories in phase space should give matching statistical averages. |
| 611 |
|
|
| 612 |
|
\subsection{\label{introSec:MDfurther}Further Considerations} |
| 613 |
+ |
|
| 614 |
|
In the simulations presented in this research, a few additional |
| 615 |
|
parameters are needed to describe the motions. The simulations |
| 616 |
< |
involving water and phospholipids in Ch.~\ref{chaptLipids} are |
| 616 |
> |
involving water and phospholipids in Ch.~\ref{chapt:lipid} are |
| 617 |
|
required to integrate the equations of motions for dipoles on atoms. |
| 618 |
|
This involves an additional three parameters be specified for each |
| 619 |
|
dipole atom: $\phi$, $\theta$, and $\psi$. These three angles are |
| 620 |
|
taken to be the Euler angles, where $\phi$ is a rotation about the |
| 621 |
|
$z$-axis, and $\theta$ is a rotation about the new $x$-axis, and |
| 622 |
|
$\psi$ is a final rotation about the new $z$-axis (see |
| 623 |
< |
Fig.~\ref{introFig:euleerAngles}). This sequence of rotations can be |
| 624 |
< |
accumulated into a single $3 \times 3$ matrix $\mathbf{A}$ |
| 623 |
> |
Fig.~\ref{introFig:eulerAngles}). This sequence of rotations can be |
| 624 |
> |
accumulated into a single $3 \times 3$ matrix, $\mathbf{A}$, |
| 625 |
|
defined as follows: |
| 626 |
|
\begin{equation} |
| 627 |
< |
eq here |
| 627 |
> |
\mathbf{A} = |
| 628 |
> |
\begin{bmatrix} |
| 629 |
> |
\cos\phi\cos\psi-\sin\phi\cos\theta\sin\psi &% |
| 630 |
> |
\sin\phi\cos\psi+\cos\phi\cos\theta\sin\psi &% |
| 631 |
> |
\sin\theta\sin\psi \\% |
| 632 |
> |
% |
| 633 |
> |
-\cos\phi\sin\psi-\sin\phi\cos\theta\cos\psi &% |
| 634 |
> |
-\sin\phi\sin\psi+\cos\phi\cos\theta\cos\psi &% |
| 635 |
> |
\sin\theta\cos\psi \\% |
| 636 |
> |
% |
| 637 |
> |
\sin\phi\sin\theta &% |
| 638 |
> |
-\cos\phi\sin\theta &% |
| 639 |
> |
\cos\theta |
| 640 |
> |
\end{bmatrix} |
| 641 |
|
\label{introEq:EulerRotMat} |
| 642 |
|
\end{equation} |
| 643 |
|
|
| 644 |
< |
The equations of motion for Euler angles can be written down as |
| 645 |
< |
\cite{allen87:csl} |
| 646 |
< |
\begin{equation} |
| 647 |
< |
eq here |
| 648 |
< |
\label{introEq:MDeuleeerPsi} |
| 649 |
< |
\end{equation} |
| 644 |
> |
\begin{figure} |
| 645 |
> |
\caentering |
| 646 |
> |
\includegraphics[width=\linewidth]{eulerRotFig.eps} |
| 647 |
> |
\caption[Euler rotation of Cartesian coordinates]{The rotation scheme for Euler angles. First is a rotation of $\phi$ about the $z$ axis (blue rotation). Next is a rotation of $\theta$ about the new $x\prime$ axis (green rotation). Lastly is a final rotation of $\psi$ about the new $z\prime$ axis (red rotation).} |
| 648 |
> |
\label{introFig:eulerAngles} |
| 649 |
> |
\end{figure} |
| 650 |
> |
|
| 651 |
> |
The equations of motion for Euler angles can be written down |
| 652 |
> |
as\cite{allen87:csl} |
| 653 |
> |
\begin{align} |
| 654 |
> |
\dot{\phi} &= -\omega^s_x \frac{\sin\phi\cos\theta}{\sin\theta} + |
| 655 |
> |
\omega^s_y \frac{\cos\phi\cos\theta}{\sin\theta} + |
| 656 |
> |
\omega^s_z |
| 657 |
> |
\label{introEq:MDeulerPhi} \\% |
| 658 |
> |
% |
| 659 |
> |
\dot{\theta} &= \omega^s_x \cos\phi + \omega^s_y \sin\phi |
| 660 |
> |
\label{introEq:MDeulerTheta} \\% |
| 661 |
> |
% |
| 662 |
> |
\dot{\psi} &= \omega^s_x \frac{\sin\phi}{\sin\theta} - |
| 663 |
> |
\omega^s_y \frac{\cos\phi}{\sin\theta} |
| 664 |
> |
\label{introEq:MDeulerPsi} |
| 665 |
> |
\end{align} |
| 666 |
|
Where $\omega^s_i$ is the angular velocity in the lab space frame |
| 667 |
|
along Cartesian coordinate $i$. However, a difficulty arises when |
| 668 |
|
attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and |
| 669 |
|
Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in |
| 670 |
|
both equations means there is a non-physical instability present when |
| 671 |
< |
$\theta$ is 0 or $\pi$. |
| 672 |
< |
|
| 673 |
< |
To correct for this, the simulations integrate the rotation matrix, |
| 674 |
< |
$\mathbf{A}$, directly, thus avoiding the instability. |
| 642 |
< |
This method was proposed by Dullwebber |
| 643 |
< |
\emph{et. al.}\cite{Dullwebber:1997}, and is presented in |
| 671 |
> |
$\theta$ is 0 or $\pi$. To correct for this, the simulations integrate |
| 672 |
> |
the rotation matrix, $\mathbf{A}$, directly, thus avoiding the |
| 673 |
> |
instability. This method was proposed by Dullweber |
| 674 |
> |
\emph{et. al.}\cite{Dullweber1997}, and is presented in |
| 675 |
|
Sec.~\ref{introSec:MDsymplecticRot}. |
| 676 |
|
|
| 677 |
< |
\subsubsection{\label{introSec:MDliouville}Liouville Propagator} |
| 677 |
> |
\subsection{\label{introSec:MDliouville}Liouville Propagator} |
| 678 |
|
|
| 679 |
|
Before discussing the integration of the rotation matrix, it is |
| 680 |
|
necessary to understand the construction of a ``good'' integration |
| 681 |
|
scheme. It has been previously |
| 682 |
< |
discussed(Sec.~\ref{introSec:MDintegrate}) how it is desirable for an |
| 682 |
> |
discussed(Sec.~\ref{introSec:mdIntegrate}) how it is desirable for an |
| 683 |
|
integrator to be symplectic, or time reversible. The following is an |
| 684 |
|
outline of the Trotter factorization of the Liouville Propagator as a |
| 685 |
< |
scheme for generating symplectic integrators. \cite{Tuckerman:1992} |
| 685 |
> |
scheme for generating symplectic integrators.\cite{Tuckerman92} |
| 686 |
|
|
| 687 |
|
For a system with $f$ degrees of freedom the Liouville operator can be |
| 688 |
|
defined as, |
| 689 |
|
\begin{equation} |
| 690 |
< |
eq here |
| 690 |
> |
iL=\sum^f_{j=1} \biggl [\dot{q}_j\frac{\partial}{\partial q_j} + |
| 691 |
> |
F_j\frac{\partial}{\partial p_j} \biggr ] |
| 692 |
|
\label{introEq:LiouvilleOperator} |
| 693 |
|
\end{equation} |
| 694 |
< |
Here, $r_j$ and $p_j$ are the position and conjugate momenta of a |
| 695 |
< |
degree of freedom, and $f_j$ is the force on that degree of freedom. |
| 694 |
> |
Here, $q_j$ and $p_j$ are the position and conjugate momenta of a |
| 695 |
> |
degree of freedom, and $F_j$ is the force on that degree of freedom. |
| 696 |
|
$\Gamma$ is defined as the set of all positions and conjugate momenta, |
| 697 |
< |
$\{r_j,p_j\}$, and the propagator, $U(t)$, is defined |
| 697 |
> |
$\{q_j,p_j\}$, and the propagator, $U(t)$, is defined |
| 698 |
|
\begin {equation} |
| 699 |
< |
eq here |
| 699 |
> |
U(t) = e^{iLt} |
| 700 |
|
\label{introEq:Lpropagator} |
| 701 |
|
\end{equation} |
| 702 |
|
This allows the specification of $\Gamma$ at any time $t$ as |
| 703 |
|
\begin{equation} |
| 704 |
< |
eq here |
| 704 |
> |
\Gamma(t) = U(t)\Gamma(0) |
| 705 |
|
\label{introEq:Lp2} |
| 706 |
|
\end{equation} |
| 707 |
|
It is important to note, $U(t)$ is a unitary operator meaning |
| 712 |
|
|
| 713 |
|
Decomposing $L$ into two parts, $iL_1$ and $iL_2$, one can use the |
| 714 |
|
Trotter theorem to yield |
| 715 |
< |
\begin{equation} |
| 716 |
< |
eq here |
| 717 |
< |
\label{introEq:Lp4} |
| 718 |
< |
\end{equation} |
| 719 |
< |
Where $\Delta t = \frac{t}{P}$. |
| 715 |
> |
\begin{align} |
| 716 |
> |
e^{iLt} &= e^{i(L_1 + L_2)t} \notag \\% |
| 717 |
> |
% |
| 718 |
> |
&= \biggl [ e^{i(L_1 +L_2)\frac{t}{P}} \biggr]^P \notag \\% |
| 719 |
> |
% |
| 720 |
> |
&= \biggl [ e^{iL_1\frac{\Delta t}{2}}\, e^{iL_2\Delta t}\, |
| 721 |
> |
e^{iL_1\frac{\Delta t}{2}} \biggr ]^P + |
| 722 |
> |
\mathcal{O}\biggl (\frac{t^3}{P^2} \biggr ) \label{introEq:Lp4} |
| 723 |
> |
\end{align} |
| 724 |
> |
Where $\Delta t = t/P$. |
| 725 |
|
With this, a discrete time operator $G(\Delta t)$ can be defined: |
| 726 |
< |
\begin{equation} |
| 727 |
< |
eq here |
| 726 |
> |
\begin{align} |
| 727 |
> |
G(\Delta t) &= e^{iL_1\frac{\Delta t}{2}}\, e^{iL_2\Delta t}\, |
| 728 |
> |
e^{iL_1\frac{\Delta t}{2}} \notag \\% |
| 729 |
> |
% |
| 730 |
> |
&= U_1 \biggl ( \frac{\Delta t}{2} \biggr )\, U_2 ( \Delta t )\, |
| 731 |
> |
U_1 \biggl ( \frac{\Delta t}{2} \biggr ) |
| 732 |
|
\label{introEq:Lp5} |
| 733 |
< |
\end{equation} |
| 734 |
< |
Because $U_1(t)$ and $U_2(t)$ are unitary, $G|\Delta t)$ is also |
| 733 |
> |
\end{align} |
| 734 |
> |
Because $U_1(t)$ and $U_2(t)$ are unitary, $G(\Delta t)$ is also |
| 735 |
|
unitary. Meaning an integrator based on this factorization will be |
| 736 |
|
reversible in time. |
| 737 |
|
|
| 738 |
|
As an example, consider the following decomposition of $L$: |
| 739 |
+ |
\begin{align} |
| 740 |
+ |
iL_1 &= \dot{q}\frac{\partial}{\partial q}% |
| 741 |
+ |
\label{introEq:Lp6a} \\% |
| 742 |
+ |
% |
| 743 |
+ |
iL_2 &= F(q)\frac{\partial}{\partial p}% |
| 744 |
+ |
\label{introEq:Lp6b} |
| 745 |
+ |
\end{align} |
| 746 |
+ |
This leads to propagator $G( \Delta t )$ as, |
| 747 |
|
\begin{equation} |
| 748 |
< |
eq here |
| 749 |
< |
\label{introEq:Lp6} |
| 748 |
> |
G(\Delta t) = e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} \, |
| 749 |
> |
e^{\Delta t\,\dot{q}\frac{\partial}{\partial q}} \, |
| 750 |
> |
e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} |
| 751 |
> |
\label{introEq:Lp7} |
| 752 |
|
\end{equation} |
| 753 |
< |
Operating $G(\Delta t)$ on $\Gamma)0)$, and utilizing the operator property |
| 753 |
> |
Operating $G(\Delta t)$ on $\Gamma(0)$, and utilizing the operator property |
| 754 |
|
\begin{equation} |
| 755 |
< |
eq here |
| 755 |
> |
e^{c\frac{\partial}{\partial x}}\, f(x) = f(x+c) |
| 756 |
|
\label{introEq:Lp8} |
| 757 |
|
\end{equation} |
| 758 |
< |
Where $c$ is independent of $q$. One obtains the following: |
| 759 |
< |
\begin{equation} |
| 760 |
< |
eq here |
| 761 |
< |
\label{introEq:Lp8} |
| 762 |
< |
\end{equation} |
| 758 |
> |
Where $c$ is independent of $x$. One obtains the following: |
| 759 |
> |
\begin{align} |
| 760 |
> |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
| 761 |
> |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)] \label{introEq:Lp9a}\\% |
| 762 |
> |
% |
| 763 |
> |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr )% |
| 764 |
> |
\label{introEq:Lp9b}\\% |
| 765 |
> |
% |
| 766 |
> |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 767 |
> |
\frac{\Delta t}{2m}\, F[q(0)] \label{introEq:Lp9c} |
| 768 |
> |
\end{align} |
| 769 |
|
Or written another way, |
| 770 |
< |
\begin{equation} |
| 771 |
< |
eq here |
| 772 |
< |
\label{intorEq:Lp9} |
| 773 |
< |
\end{equation} |
| 770 |
> |
\begin{align} |
| 771 |
> |
q(t+\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 772 |
> |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2} % |
| 773 |
> |
\label{introEq:Lp10a} \\% |
| 774 |
> |
% |
| 775 |
> |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
| 776 |
> |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr] % |
| 777 |
> |
\label{introEq:Lp10b} |
| 778 |
> |
\end{align} |
| 779 |
|
This is the velocity Verlet formulation presented in |
| 780 |
< |
Sec.~\ref{introSec:MDintegrate}. Because this integration scheme is |
| 780 |
> |
Sec.~\ref{introSec:mdIntegrate}. Because this integration scheme is |
| 781 |
|
comprised of unitary propagators, it is symplectic, and therefore area |
| 782 |
|
preserving in phase space. From the preceding factorization, one can |
| 783 |
|
see that the integration of the equations of motion would follow: |
| 791 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 792 |
|
\end{enumerate} |
| 793 |
|
|
| 794 |
< |
\subsubsection{\label{introSec:MDsymplecticRot} Symplectic Propagation of the Rotation Matrix} |
| 794 |
> |
\subsection{\label{introSec:MDsymplecticRot} Symplectic Propagation of the Rotation Matrix} |
| 795 |
|
|
| 796 |
|
Based on the factorization from the previous section, |
| 797 |
< |
Dullweber\emph{et al.}\cite{Dullweber:1997}~ proposed a scheme for the |
| 797 |
> |
Dullweber\emph{et al}.\cite{Dullweber1997}~ proposed a scheme for the |
| 798 |
|
symplectic propagation of the rotation matrix, $\mathbf{A}$, as an |
| 799 |
|
alternative method for the integration of orientational degrees of |
| 800 |
|
freedom. The method starts with a straightforward splitting of the |
| 801 |
|
Liouville operator: |
| 802 |
< |
\begin{equation} |
| 803 |
< |
eq here |
| 804 |
< |
\label{introEq:SR1} |
| 805 |
< |
\end{equation} |
| 806 |
< |
Where $\boldsymbol{\tau}(\mathbf{A})$ are the torques of the system |
| 807 |
< |
due to the configuration, and $\boldsymbol{/pi}$ are the conjugate |
| 802 |
> |
\begin{align} |
| 803 |
> |
iL_{\text{pos}} &= \dot{q}\frac{\partial}{\partial q} + |
| 804 |
> |
\mathbf{\dot{A}}\frac{\partial}{\partial \mathbf{A}} |
| 805 |
> |
\label{introEq:SR1a} \\% |
| 806 |
> |
% |
| 807 |
> |
iL_F &= F(q)\frac{\partial}{\partial p} |
| 808 |
> |
\label{introEq:SR1b} \\% |
| 809 |
> |
iL_{\tau} &= \tau(\mathbf{A})\frac{\partial}{\partial \pi} |
| 810 |
> |
\label{introEq:SR1b} \\% |
| 811 |
> |
\end{align} |
| 812 |
> |
Where $\tau(\mathbf{A})$ is the torque of the system |
| 813 |
> |
due to the configuration, and $\pi$ is the conjugate |
| 814 |
|
angular momenta of the system. The propagator, $G(\Delta t)$, becomes |
| 815 |
|
\begin{equation} |
| 816 |
< |
eq here |
| 816 |
> |
G(\Delta t) = e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} \, |
| 817 |
> |
e^{\frac{\Delta t}{2} \tau(\mathbf{A})\frac{\partial}{\partial \pi}} \, |
| 818 |
> |
e^{\Delta t\,iL_{\text{pos}}} \, |
| 819 |
> |
e^{\frac{\Delta t}{2} \tau(\mathbf{A})\frac{\partial}{\partial \pi}} \, |
| 820 |
> |
e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} |
| 821 |
|
\label{introEq:SR2} |
| 822 |
|
\end{equation} |
| 823 |
|
Propagation of the linear and angular momenta follows as in the Verlet |
| 824 |
|
scheme. The propagation of positions also follows the Verlet scheme |
| 825 |
|
with the addition of a further symplectic splitting of the rotation |
| 826 |
< |
matrix propagation, $\mathcal{G}_{\text{rot}}(\Delta t)$. |
| 826 |
> |
matrix propagation, $\mathcal{U}_{\text{rot}}(\Delta t)$, within |
| 827 |
> |
$U_{\text{pos}}(\Delta t)$. |
| 828 |
|
\begin{equation} |
| 829 |
< |
eq here |
| 829 |
> |
\mathcal{U}_{\text{rot}}(\Delta t) = |
| 830 |
> |
\mathcal{U}_x \biggl(\frac{\Delta t}{2}\biggr)\, |
| 831 |
> |
\mathcal{U}_y \biggl(\frac{\Delta t}{2}\biggr)\, |
| 832 |
> |
\mathcal{U}_z (\Delta t)\, |
| 833 |
> |
\mathcal{U}_y \biggl(\frac{\Delta t}{2}\biggr)\, |
| 834 |
> |
\mathcal{U}_x \biggl(\frac{\Delta t}{2}\biggr)\, |
| 835 |
|
\label{introEq:SR3} |
| 836 |
|
\end{equation} |
| 837 |
< |
Where $\mathcal{G}_j$ is a unitary rotation of $\mathbf{A}$ and |
| 838 |
< |
$\boldsymbol{\pi}$ about each axis $j$. As all propagations are now |
| 837 |
> |
Where $\mathcal{U}_j$ is a unitary rotation of $\mathbf{A}$ and |
| 838 |
> |
$\pi$ about each axis $j$. As all propagations are now |
| 839 |
|
unitary and symplectic, the entire integration scheme is also |
| 840 |
|
symplectic and time reversible. |
| 841 |
|
|
| 842 |
|
\section{\label{introSec:layout}Dissertation Layout} |
| 843 |
|
|
| 844 |
< |
This dissertation is divided as follows:Chapt.~\ref{chapt:RSA} |
| 844 |
> |
This dissertation is divided as follows:Ch.~\ref{chapt:RSA} |
| 845 |
|
presents the random sequential adsorption simulations of related |
| 846 |
|
pthalocyanines on a gold (111) surface. Ch.~\ref{chapt:OOPSE} |
| 847 |
|
is about the writing of the molecular dynamics simulation package |
| 848 |
< |
{\sc oopse}, Ch.~\ref{chapt:lipid} regards the simulations of |
| 849 |
< |
phospholipid bilayers using a mesoscale model, and lastly, |
| 848 |
> |
{\sc oopse}. Ch.~\ref{chapt:lipid} regards the simulations of |
| 849 |
> |
phospholipid bilayers using a mesoscale model. And lastly, |
| 850 |
|
Ch.~\ref{chapt:conclusion} concludes this dissertation with a |
| 851 |
|
summary of all results. The chapters are arranged in chronological |
| 852 |
|
order, and reflect the progression of techniques I employed during my |
| 853 |
|
research. |
| 854 |
|
|
| 855 |
< |
The chapter concerning random sequential adsorption |
| 856 |
< |
simulations is a study in applying the principles of theoretical |
| 857 |
< |
research in order to obtain a simple model capable of explaining the |
| 858 |
< |
results. My advisor, Dr. Gezelter, and I were approached by a |
| 859 |
< |
colleague, Dr. Lieberman, about possible explanations for partial |
| 860 |
< |
coverage of a gold surface by a particular compound of hers. We |
| 861 |
< |
suggested it might be due to the statistical packing fraction of disks |
| 862 |
< |
on a plane, and set about to simulate this system. As the events in |
| 863 |
< |
our model were not dynamic in nature, a Monte Carlo method was |
| 864 |
< |
employed. Here, if a molecule landed on the surface without |
| 865 |
< |
overlapping another, then its landing was accepted. However, if there |
| 866 |
< |
was overlap, the landing we rejected and a new random landing location |
| 867 |
< |
was chosen. This defined our acceptance rules and allowed us to |
| 868 |
< |
construct a Markov chain whose limiting distribution was the surface |
| 869 |
< |
coverage in which we were interested. |
| 855 |
> |
The chapter concerning random sequential adsorption simulations is a |
| 856 |
> |
study in applying Statistical Mechanics simulation techniques in order |
| 857 |
> |
to obtain a simple model capable of explaining the results. My |
| 858 |
> |
advisor, Dr. Gezelter, and I were approached by a colleague, |
| 859 |
> |
Dr. Lieberman, about possible explanations for the partial coverage of a |
| 860 |
> |
gold surface by a particular compound of hers. We suggested it might |
| 861 |
> |
be due to the statistical packing fraction of disks on a plane, and |
| 862 |
> |
set about to simulate this system. As the events in our model were |
| 863 |
> |
not dynamic in nature, a Monte Carlo method was employed. Here, if a |
| 864 |
> |
molecule landed on the surface without overlapping another, then its |
| 865 |
> |
landing was accepted. However, if there was overlap, the landing we |
| 866 |
> |
rejected and a new random landing location was chosen. This defined |
| 867 |
> |
our acceptance rules and allowed us to construct a Markov chain whose |
| 868 |
> |
limiting distribution was the surface coverage in which we were |
| 869 |
> |
interested. |
| 870 |
|
|
| 871 |
|
The following chapter, about the simulation package {\sc oopse}, |
| 872 |
|
describes in detail the large body of scientific code that had to be |
| 873 |
< |
written in order to study phospholipid bilayer. Although there are |
| 873 |
> |
written in order to study phospholipid bilayers. Although there are |
| 874 |
|
pre-existing molecular dynamic simulation packages available, none |
| 875 |
|
were capable of implementing the models we were developing.{\sc oopse} |
| 876 |
|
is a unique package capable of not only integrating the equations of |
| 882 |
|
|
| 883 |
|
Bringing us to Ch.~\ref{chapt:lipid}. Using {\sc oopse}, I have been |
| 884 |
|
able to parameterize a mesoscale model for phospholipid simulations. |
| 885 |
< |
This model retains information about solvent ordering about the |
| 885 |
> |
This model retains information about solvent ordering around the |
| 886 |
|
bilayer, as well as information regarding the interaction of the |
| 887 |
< |
phospholipid head groups' dipole with each other and the surrounding |
| 887 |
> |
phospholipid head groups' dipoles with each other and the surrounding |
| 888 |
|
solvent. These simulations give us insight into the dynamic events |
| 889 |
|
that lead to the formation of phospholipid bilayers, as well as |
| 890 |
|
provide the foundation for future exploration of bilayer phase |
