--- trunk/tengDissertation/Langevin.tex	2006/06/29 23:00:35	2909
+++ trunk/tengDissertation/Langevin.tex	2006/07/03 20:22:28	2922
@@ -172,15 +172,15 @@ tensors
 one can write down the translational and rotational resistance
 tensors
 \begin{eqnarray*}
- \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}}. \\
- \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S +
+ \Xi _a^{tt}  & = & 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}}. \\
+ \Xi _b^{tt}  & = & \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S +
  2a}},
 \end{eqnarray*}
 and
 \begin{eqnarray*}
- \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}}, \\
- \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}}.
-\end{eqnarray}
+ \Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}}, \\
+ \Xi _b^{rr} & = & \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}}.
+\end{eqnarray*}
 
 \subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}}
 
@@ -565,14 +565,14 @@ identified the center of resistance at $(0\AA, 0.7482\
 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 $(0\AA, 0.7482\AA,
--0.1988\AA)$, as well as the resistance tensor,
+identified the center of resistance at $(0 \AA, 0.7482 \AA,
+-0.1988 \AA)$, as well as the resistance tensor,
 \[
 \left( {\begin{array}{*{20}c}
 0.9261 & 0 & 0&0&0.08585&0.2057\\
 0& 0.9270&-0.007063& 0.08585&0&0\\
-0&0.007063&0.7494&0.2057&0&0\\
-0&0.0858&0.2057& 58.64& 0&-8.5736\\
+0&-0.007063&0.7494&0.2057&0&0\\
+0&0.0858&0.2057& 58.64& 0&0\\
 0.08585&0&0&0&48.30&3.219&\\
 0.2057&0&0&0&3.219&10.7373\\
 \end{array}} \right).
@@ -593,7 +593,7 @@ auto- correlation function of the principle axis of th
 However, because of the stochastic nature, simulation using Langevin
 dynamics was shown to decay slightly faster than MD. 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
+auto-correlation function of the principle axis of the second GB
 particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was
 probably due to the reason that the viscosity using in the
 simulations only partially preserved the dynamics of the system.
@@ -617,7 +617,7 @@ auto-correlation functions of NVE (explicit solvent) i
 \centering
 \includegraphics[width=\linewidth]{vacf.eps}
 \caption[Plots of Velocity Auto-correlation Functions]{Velocity
-auto-correlation functions of NVE (explicit solvent) in blue) and
+auto-correlation functions of NVE (explicit solvent) in blue and
 Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf}
 \end{figure}
 
