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Revision 977 by mmeineke, Thu Jan 22 21:13:55 2004 UTC vs.
Revision 980 by mmeineke, Sat Jan 24 03:09:15 2004 UTC

+the lack of particle images in the $x$, $y$, or $z$ directions past a
+disance of $fix$.
-<
+With the use of an $fix$, however, comes a discontinuity in the potential energy curve (Fig.~\ref{fix}).
->
+With the use of an $fix$, however, comes a discontinuity in the
->
+potential energy curve (Fig.~\ref{fix}). To fix this discontinuity,
->
+one calculates the potential energy at the $r_{\text{cut}}$, and add
->
+that value to the potential.  This causes the function to go smoothly
->
+to zero at the cutoff radius.  This ensures conservation of energy
->
+when integrating the Newtonian equations of motion.
->
->
+The second main simplification used in this research is the Verlet
->
+neighbor list. \cite{allen87:csl} In the Verlet method, one generates
->
+a list of all neighbor atoms, $j$, surrounding atom $i$ within some
->
+cutoff $r_{\text{list}}$, where $r_{\text{list}}>r_{\text{cut}}$.
->
+This list is created the first time forces are evaluated, then on
->
+subsequent force evaluations, pair calculations are only calculated
->
+from the neighbor lists.  The lists are updated if any given particle
->
+in the system moves farther than $r_{\text{list}}-r_{\text{cut}}$,
->
+giving rise to the possibility that a particle has left or joined a
->
+neighbor list.
->
->
+\subsection{\label{introSec:MDintegrate} Integration of the equations of motion}
->
->
+A starting point for the discussion of molecular dynamics integrators
->
+is the Verlet algorithm. \cite{Frenkel1996} It begins with a Taylor
->
+expansion of position in time:
->
+\begin{equation}
->
+eq here
->
+\label{introEq:verletForward}
->
+\end{equation}
->
+As well as,
->
+\begin{equation}
->
+eq here
->
+\label{introEq:verletBack}
->
+\end{equation}
->
+Adding together Eq.~\ref{introEq:verletForward} and
->
+Eq.~\ref{introEq:verletBack} results in,
->
+\begin{equation}
->
+eq here
->
+\label{introEq:verletSum}
->
+\end{equation}
->
+Or equivalently,
->
+\begin{equation}
->
+eq here
->
+\label{introEq:verletFinal}
->
+\end{equation}
->
+Which contains an error in the estimate of the new positions on the
->
+order of $\Delta t^4$.
->
->
+In practice, however, the simulations in this research were integrated
->
+with a velocity reformulation of teh Verlet method. \cite{allen87:csl}
->
+\begin{equation}
->
+eq here
->
+\label{introEq:MDvelVerletPos}
->
+\end{equation}
->
+\begin{equation}
->
+eq here
->
+\label{introEq:MDvelVerletVel}
->
+\end{equation}
->
+The original Verlet algorithm can be regained by substituting the
->
+velocity back into Eq.~\ref{introEq:MDvelVerletPos}.  The Verlet
->
+formulations are chosen in this research because the algorithms have
->
+very little long term drift in energy conservation.  Energy
->
+conservation in a molecular dynamics simulation is of extreme
->
+importance, as it is a measure of how closely one is following the
->
+``true'' trajectory wtih the finite integration scheme.  An exact
->
+solution to the integration will conserve area in phase space, as well
->
+as be reversible in time, that is, the trajectory integrated forward
->
+or backwards will exactly match itself.  Having a finite algorithm
->
+that both conserves area in phase space and is time reversible,
->
+therefore increases, but does not guarantee the ``correctness'' or the
->
+integrated trajectory.
->
->
+It can be shown, \cite{Frenkel1996} that although the Verlet algorithm
->
+does not rigorously preserve the actual Hamiltonian, it does preserve
->
+a pseudo-Hamiltonian which shadows the real one in phase space.  This
->
+pseudo-Hamiltonian is proveably area-conserving as well as time
->
+reversible.  The fact that it shadows the true Hamiltonian in phase
->
+space is acceptable in actual simulations as one is interested in the
->
+ensemble average of the observable being measured.  From the ergodic
->
+hypothesis (Sec.~\ref{introSec:StatThermo}), it is known that the time
->
+average will match the ensemble average, therefore two similar
->
+trajectories in phase space should give matching statistical averages.
->
->
+\subsection{\label{introSec:MDfurther}Further Considerations}
->
+In the simulations presented in this research, a few additional
->
+parameters are needed to describe the motions.  The simulations
->
+involving water and phospholipids in Chapt.~\ref{chaptLipids} are
->
+required to integrate the equations of motions for dipoles on atoms.
->
+This involves an additional three parameters be specified for each
->
+dipole atom: $\phi$, $\theta$, and $\psi$.  These three angles are
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+taken to be the Euler angles, where $\phi$ is a rotation about the
->
+$z$-axis, and $\theta$ is a rotation about the new $x$-axis, and
->
+$\psi$ is a final rotation about the new $z$-axis (see
->
+Fig.~\ref{introFig:euleerAngles}).  This sequence of rotations can be
->
+accumulated into a single $3 \times 3$ matrix $\mathbf{A}$
->
+defined as follows:
->
+\begin{equation}
->
+eq here
->
+\label{introEq:EulerRotMat}
->
+\end{equation}
->
->
+The equations of motion for Euler angles can be written down as
->
+\cite{allen87:csl}
->
+\begin{equation}
->
+eq here
->
+\label{introEq:MDeuleeerPsi}
->
+\end{equation}
->
+Where $\omega^s_i$ is the angular velocity in the lab space frame
->
+along cartesian coordinate $i$.  However, a difficulty arises when
->
+attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and
->
+Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in
->
+both equations means there is a non-physical instability present when
->
+$\theta$ is 0 or $\pi$.
->
->
+To correct for this, the simulations integrate the rotation matrix,
->
+$\mathbf{A}$, directly, thus avoiding the instability.
->
+This method was proposed by Dullwebber
->
+\emph{et. al.}\cite{Dullwebber:1997}, and is presented in
->
+Sec.~\ref{introSec:MDsymplecticRot}.
->
->
+\subsubsection{\label{introSec:MDliouville}Liouville Propagator}
-+
+Before discussing the integration of the rotation matrix, it is
-+
+necessary to understand the construction of a ``good'' integration
-+
+scheme.  It has been previously
-+
+discussed(Sec.~\ref{introSec:MDintegrate}) how it is desirable for an
-+
+integrator to be symplectic, or time reversible.  The following is an
-+
+outline of the Trotter factorization of the Liouville Propagator as a
-+
+scheme for generating symplectic integrators. \cite{Tuckerman:1992}
-+
+For a system with $f$ degrees of freedom the Liouville operator can be
-+
+defined as,
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:LiouvilleOperator}
-+
+\end{equation}
-+
+Here, $r_j$ and $p_j$ are the position and conjugate momenta of a
-+
+degree of freedom, and $f_j$ is the force on that degree of freedom.
-+
+$\Gamma$ is defined as the set of all positions nad conjugate momenta,
-+
+$\{r_j,p_j\}$, and the propagator, $U(t)$, is defined
-+
+\begin {equation}
-+
+eq here
-+
+\label{introEq:Lpropagator}
-+
+\end{equation}
-+
+This allows the specification of $\Gamma$ at any time $t$ as
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:Lp2}
-+
+\end{equation}
-+
+It is important to note, $U(t)$ is a unitary operator meaning
-+
+\begin{equation}
-+
+U(-t)=U^{-1}(t)
-+
+\label{introEq:Lp3}
-+
+\end{equation}
-+
-+
+Decomposing $L$ into two parts, $iL_1$ and $iL_2$, one can use the
-+
+Trotter theorem to yield
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:Lp4}
-+
+\end{equation}
-+
+Where $\Delta t = \frac{t}{P}$.
-+
+With this, a discrete time operator $G(\Delta t)$ can be defined:
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:Lp5}
-+
+\end{equation}
-+
+Because $U_1(t)$ and $U_2(t)$ are unitary, $G|\Delta t)$ is also
-+
+unitary.  Meaning an integrator based on this factorization will be
-+
+reversible in time.
-+
-+
+As an example, consider the following decomposition of $L$:
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:Lp6}
-+
+\end{equation}
-+
+Operating $G(\Delta t)$ on $\Gamma)0)$, and utilizing the operator property
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:Lp8}
-+
+\end{equation}
-+
+Where $c$ is independent of $q$.  One obtains the following:
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:Lp8}
-+
+\end{equation}
-+
+Or written another way,
-+
+\begin{equation}
-+
+eq here
-+
+\label{intorEq:Lp9}
-+
+\end{equation}
-+
+This is the velocity Verlet formulation presented in
-+
+Sec.~\ref{introSec:MDintegrate}.  Because this integration scheme is
-+
+comprised of unitary propagators, it is symplectic, and therefore area
-+
+preserving in phase space.  From the preceeding fatorization, one can
-+
+see that the integration of the equations of motion would follow:
-+
+\begin{enumerate}
-+
+\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
-+
-+
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
-+
-+
+\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move.
-+
-+
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
-+
+\end{enumerate}
-+
-+
+\subsubsection{\label{introSec:MDsymplecticRot} Symplectic Propagation of the Rotation Matrix}
-+
-+
+Based on the factorization from the previous section,
-+
+Dullweber\emph{et al.}\cite{Dullweber:1997}~ proposed a scheme for the
-+
+symplectic propagation of the rotation matrix, $\mathbf{A}$, as an
-+
+alternative method for the integration of orientational degrees of
-+
+freedom. The method starts with a straightforward splitting of the
-+
+Liouville operator:
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:SR1}
-+
+\end{equation}
-+
+Where $\boldsymbol{\tau}(\mathbf{A})$ are the tourques of the system
-+
+due to the configuration, and $\boldsymbol{/pi}$ are the conjugate
-+
+angular momenta of the system. The propagator, $G(\Delta t)$, becomes
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:SR2}
-+
+\end{equation}
-+
+Propagation fo the linear and angular momenta follows as in the Verlet
-+
+scheme.  The propagation of positions also follows the verlet scheme
-+
+with the addition of a further symplectic splitting of the rotation
-+
+matrix propagation, $\mathcal{G}_{\text{rot}}(\Delta t)$.
-+
+\begin{equation}
-+
+eq here
-+
+\label{introEq:SR3}
-+
+\end{equation}
-+
+Where $\mathcal{G}_j$ is a unitary rotation of $\mathbf{A}$ and
-+
+$\boldsymbol{\pi}$ about each axis $j$.  As all propagations are now
-+
+unitary and symplectic, the entire integration scheme is also
-+
+symplectic and time reversible.
-+
+\section{\label{introSec:chapterLayout}Chapter Layout}
+\subsection{\label{introSec:RSA}Random Sequential Adsorption}
+\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package}
-<
+\subsection{\label{introSec:bilayers}A Mesoscale Model for Phospholipid Bilayers}
->
+\subsection{\label{introSec:bilayers}A Mesoscale Model for
->
+Phospholipid Bilayers}

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Comparing trunk/mattDisertation/Introduction.tex (file contents): Revision 977 by mmeineke, Thu Jan 22 21:13:55 2004 UTC vs. Revision 980 by mmeineke, Sat Jan 24 03:09:15 2004 UTC

Diff Legend

Comparing trunk/mattDisertation/Introduction.tex (file contents):
Revision 977 by mmeineke, Thu Jan 22 21:13:55 2004 UTC vs.
Revision 980 by mmeineke, Sat Jan 24 03:09:15 2004 UTC