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Comparing trunk/mattDisertation/Introduction.tex (file contents):
Revision 978 by mmeineke, Fri Jan 23 02:14:04 2004 UTC vs.
Revision 980 by mmeineke, Sat Jan 24 03:09:15 2004 UTC

+average will match the ensemble average, therefore two similar
+trajectories in phase space should give matching statistical averages.
-<
+\subsection{\label{introSec:MDfurtheeeeer}Further Considerations}
->
+\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
+$z$-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 $3\time3$ matrix $\underline{\mathbf{A}}$
->
+accumulated into a single $3 \times 3$ matrix $\mathbf{A}$
+defined as follows:
+\begin{equation}
+eq here
+\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:MDeuleerPhi} and
->
+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,
-<
+$\underline{\mathbf{A}}$, directly, thus avoiding the instability.
->
+$\mathbf{A}$, 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}

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Comparing trunk/mattDisertation/Introduction.tex (file contents): Revision 978 by mmeineke, Fri Jan 23 02:14:04 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 978 by mmeineke, Fri Jan 23 02:14:04 2004 UTC vs.
Revision 980 by mmeineke, Sat Jan 24 03:09:15 2004 UTC