--- trunk/tengDissertation/Langevin.tex	2006/06/21 16:37:09	2873
+++ trunk/tengDissertation/Langevin.tex	2006/06/21 19:46:01	2876
@@ -580,14 +580,35 @@ In order to compare the
 
 \subsection{Langevin Dynamics of Banana Shaped Molecule}
 
-In order to compare the
+In order to verify that Langevin dynamics can mimic the dynamics of
+the systems absent of explicit solvents, we carried out two sets of
+simulations and compare their dynamic properties.
 
-\begin{figure}
-\centering
-\includegraphics[width=\linewidth]{one_banana.eps}
-\caption[]{.} \label{langevin:banana}
-\end{figure}
+\subsubsection{Simulations Contain One Banana Shaped Molecule}
 
+Fig.~\ref{langevin:oneBanana} shows a snapshot of the simulation
+made of 256 pentane molecules and one banana shaped molecule at
+273~K. It has an equivalent implicit solvent model containing only
+one banana shaped molecule with viscosity of 0.289 center poise. To
+calculate the hydrodynamic properties of the banana shaped molecule,
+we create a rough shell model (see Fig.~\ref{langevin:roughShell}),
+in which the banana shaped molecule is represented as a ``shell''
+made of 2266 small identical beads with size of 0.3 $\AA$ on the
+surface. Applying the procedure described in
+Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
+identified the center of resistance at $(0, 0.7482, -0.1988)$, as
+well as the resistance tensor,
+\[
+\left( {\begin{array}{*{20}c}
+0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\
+3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\
+-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\
+5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\
+0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\
+0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\
+\end{array}} \right).
+\]
+
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{roughShell.eps}
@@ -597,6 +618,16 @@ model for banana shaped molecule.} \label{langevin:rou
 
 \begin{figure}
 \centering
+\includegraphics[width=\linewidth]{oneBanana.eps}
+\caption[Snapshot from Simulation of One Banana Shaped Molecules and
+256 Pentane Molecules]{Snapshot from simulation of one Banana shaped
+molecules and 256 pentane molecules.} \label{langevin:oneBanana}
+\end{figure}
+
+\subsubsection{Simulations Contain Two Banana Shaped Molecules}
+
+\begin{figure}
+\centering
 \includegraphics[width=\linewidth]{twoBanana.eps}
 \caption[Snapshot from Simulation of Two Banana Shaped Molecules and
 256 Pentane Molecules]{Snapshot from simulation of two Banana shaped