--- trunk/tengDissertation/Langevin.tex	2006/06/26 13:42:53	2889
+++ trunk/tengDissertation/Langevin.tex	2006/06/26 13:43:22	2890
@@ -579,12 +579,11 @@ temperature fluctuation versus time.} \label{langevin:
 temperature fluctuation versus time.} \label{langevin:temperature}
 \end{figure}
 
-\subsection{Langevin Dynamics of Banana Shaped Molecule}
+\subsection{Langevin Dynamics of Banana Shaped Molecules}
 
 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.
-
 Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation
 made of 256 pentane molecules and two banana shaped molecules at
 273~K. It has an equivalent implicit solvent system containing only
@@ -592,7 +591,7 @@ made of 2266 small identical beads with size of 0.3 $\
 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
+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
@@ -607,8 +606,13 @@ -6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-1
 0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\
 \end{array}} \right).
 \]
-
-
+Curves of velocity auto-correlation functions in
+Fig.~\ref{langevin:vacf} were shown to match each other very well.
+However, because of the stochastic nature, simulation using Langevin
+dynamics was shown to decay slightly fast. In order to study the
+rotational motion of the molecules, we also calculated the auto-
+correlation function of the principle axis of the second GB
+particle, $u$.
 
 \begin{figure}
 \centering
@@ -628,17 +632,28 @@ molecules and 256 pentane molecules.} \label{langevin:
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{vacf.eps}
-\caption[Plots of Velocity Auto-correlation functions]{Velocity
-Auto-correlation function of NVE (blue) and Langevin dynamics
-(red).} \label{langevin:twoBanana}
+\caption[Plots of Velocity Auto-correlation Functions]{Velocity
+auto-correlation functions in NVE (blue) and Langevin dynamics
+(red).} \label{langevin:vacf}
 \end{figure}
 
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{uacf.eps}
-\caption[Snapshot from Simulation of Two Banana Shaped Molecules and
-256 Pentane Molecules]{Snapshot from simulation of two Banana shaped
-molecules and 256 pentane molecules.} \label{langevin:twoBanana}
+\caption[Auto-correlation functions of the principle axis of the
+middle GB particle]{Auto-correlation functions of the principle axis
+of the middle GB particle in NVE (blue) and Langevin dynamics
+(red).} \label{langevin:twoBanana}
 \end{figure}
 
 \section{Conclusions}
+
+We have presented a new Langevin algorithm by incorporating the
+hydrodynamics properties of arbitrary shaped molecules into an
+advanced symplectic integration scheme. The temperature control
+ability of this algorithm was demonstrated by a set of simulations
+with different viscosities. It was also shown to have significant
+advantage of producing rapid thermal equilibration over
+Nos\'{e}-Hoover method. Further studies in systems involving banana
+shaped molecules illustrated that the dynamic properties could be
+preserved by using this new algorithm as an implicit solvent model.