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average will match the ensemble average, therefore two similar |
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trajectories in phase space should give matching statistical averages. |
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|
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< |
\subsection{\label{introSec:MDfurtheeeeer}Further Considerations} |
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> |
\subsection{\label{introSec:MDfurther}Further Considerations} |
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In the simulations presented in this research, a few additional |
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parameters are needed to describe the motions. The simulations |
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involving water and phospholipids in Chapt.~\ref{chaptLipids} are |
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$z$-axis, and $\theta$ is a rotation about the new $x$-axis, and |
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$\psi$ is a final rotation about the new $z$-axis (see |
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|
Fig.~\ref{introFig:euleerAngles}). This sequence of rotations can be |
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< |
accumulated into a single $3\time3$ matrix $\underline{\mathbf{A}}$ |
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> |
accumulated into a single $3 \times 3$ matrix $\mathbf{A}$ |
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defined as follows: |
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|
\begin{equation} |
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eq here |
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|
\end{equation} |
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Where $\omega^s_i$ is the angular velocity in the lab space frame |
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|
along cartesian coordinate $i$. However, a difficulty arises when |
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< |
attempting to integrate Eq.~\ref{introEq:MDeuleerPhi} and |
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> |
attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and |
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Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in |
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both equations means there is a non-physical instability present when |
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$\theta$ is 0 or $\pi$. |
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|
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To correct for this, the simulations integrate the rotation matrix, |
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< |
$\underline{\mathbf{A}}$, directly, thus avoiding the instability. |
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> |
$\mathbf{A}$, directly, thus avoiding the instability. |
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This method was proposed by Dullwebber |
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\emph{et. al.}\cite{Dullwebber:1997}, and is presented in |
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Sec.~\ref{introSec:MDsymplecticRot}. |
