--- trunk/tengDissertation/Langevin.tex	2006/06/28 17:36:32	2904
+++ trunk/tengDissertation/Langevin.tex	2006/06/28 19:09:25	2905
@@ -180,8 +180,8 @@ S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt 
 and oblate :
 \[
 S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
-}}{a}
-\],
+}}{a}.
+\]
 one can write down the translational and rotational resistance
 tensors
 \[
@@ -231,8 +231,8 @@ This equation is the basis for deriving the hydrodynam
 \label{introEquation:tensorExpression}
 \end{equation}
 This equation is the basis for deriving the hydrodynamic tensor. In
-1930, Oseen and Burgers gave a simple solution to Equation
-\ref{introEquation:tensorExpression}
+1930, Oseen and Burgers gave a simple solution to
+Eq.~\ref{introEquation:tensorExpression}
 \begin{equation}
 T_{ij}  = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij}
 R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor}
@@ -248,9 +248,9 @@ Both of the Equation \ref{introEquation:oseenTensor} a
 \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
 \label{introEquation:RPTensorNonOverlapped}
 \end{equation}
-Both of the Equation \ref{introEquation:oseenTensor} and Equation
-\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij}
-\ge \sigma _i  + \sigma _j$. An alternative expression for
+Both of the Eq.~\ref{introEquation:oseenTensor} and
+Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption
+$R_{ij} \ge \sigma _i  + \sigma _j$. An alternative expression for
 overlapping beads with the same radius, $\sigma$, is given by
 \begin{equation}
 T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
@@ -258,7 +258,6 @@ T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( 
 \frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
 \label{introEquation:RPTensorOverlapped}
 \end{equation}
-
 To calculate the resistance tensor at an arbitrary origin $O$, we
 construct a $3N \times 3N$ matrix consisting of $N \times N$
 $B_{ij}$ blocks
@@ -276,15 +275,14 @@ matrix, we obtain
 \]
 where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$
 matrix, we obtain
-
 \[
 C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
    {C_{11} } &  \ldots  & {C_{1N} }  \\
     \vdots  &  \ddots  &  \vdots   \\
    {C_{N1} } &  \cdots  & {C_{NN} }  \\
-\end{array}} \right)
+\end{array}} \right),
 \]
-, which can be partitioned into $N \times N$ $3 \times 3$ block
+which can be partitioned into $N \times N$ $3 \times 3$ block
 $C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$
 \[
 U_i  = \left( {\begin{array}{*{20}c}
@@ -296,14 +294,12 @@ arbitrary origin $O$ can be written as
 where $x_i$, $y_i$, $z_i$ are the components of the vector joining
 bead $i$ and origin $O$. Hence, the elements of resistance tensor at
 arbitrary origin $O$ can be written as
-\begin{equation}
-\begin{array}{l}
- \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\
+\begin{eqnarray}
+ \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\
  \Xi _{}^{tr}  = \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
- \Xi _{}^{rr}  =  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j  \\
- \end{array}
+ \Xi _{}^{rr}  =  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\
 \label{introEquation:ResistanceTensorArbitraryOrigin}
-\end{equation}
+\end{eqnarray}
 
 The resistance tensor depends on the origin to which they refer. The
 proper location for applying friction force is the center of
@@ -339,9 +335,9 @@ Using Equations \ref{introEquation:definitionCR} and
    { - y_{OP} } & {x_{OP} } & 0  \\
 \end{array}} \right)
 \]
-Using Equations \ref{introEquation:definitionCR} and
-\ref{introEquation:resistanceTensorTransformation}, one can locate
-the position of center of resistance,
+Using Eq.~\ref{introEquation:definitionCR} and
+Eq.~\ref{introEquation:resistanceTensorTransformation}, one can
+locate the position of center of resistance,
 \begin{eqnarray*}
  \left( \begin{array}{l}
  x_{OR}  \\
@@ -358,7 +354,6 @@ the position of center of resistance,
  (\Xi _O^{tr} )_{xy}  - (\Xi _O^{tr} )_{yx}  \\
  \end{array} \right) \\
 \end{eqnarray*}
-
 where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
 joining center of resistance $R$ and origin $O$.
 
@@ -371,8 +366,8 @@ $V_i = V_i(v_i,\omega _i)$. The right side of Eq.~
 \end{equation}
 where $M_i$ is a $6\times6$ generalized diagonal mass (include mass
 and moment of inertial) matrix and $V_i$ is a generalized velocity,
-$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~
-(\ref{LDGeneralizedForm}) consists of three generalized forces in
+$V_i = V_i(v_i,\omega _i)$. The right side of
+Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in
 lab-fixed frame, systematic force $F_{s,i}$, dissipative force
 $F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the
 system in Newtownian mechanics typically refers to lab-fixed frame,
@@ -382,12 +377,12 @@ in body-fixed frame and converted back to lab-fixed fr
 \[
 \begin{array}{l}
  F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\
- F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\
- \end{array}.
+ F_{r,i}^l (t) = A^T F_{r,i}^b (t). \\
+ \end{array}
 \]
 Here, the body-fixed friction force $F_{r,i}^b$ is proportional to
 the body-fixed velocity at center of resistance $v_{R,i}^b$ and
-angular velocity $\omega _i$,
+angular velocity $\omega _i$
 \begin{equation}
 F_{r,i}^b (t) = \left( \begin{array}{l}
  f_{r,i}^b (t) \\
@@ -407,7 +402,6 @@ with zero mean and variance
 \left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle  =
 2k_B T\Xi _R \delta (t - t'). \label{randomForce}
 \end{equation}
-
 The equation of motion for $v_i$ can be written as
 \begin{equation}
 m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) +
@@ -429,7 +423,6 @@ + \tau _{r,i}^b(t)
 \dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t)
 + \tau _{r,i}^b(t)
 \end{equation}
-
 Embedding the friction terms into force and torque, one can
 integrate the langevin equations of motion for rigid body of
 arbitrary shape in a velocity-Verlet style 2-part algorithm, where
@@ -449,7 +442,6 @@ $h= \delta t$:
 \mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j}
     (t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right).
 \end{align*}
-
 In this context, the $\mathrm{rotate}$ function is the reversible
 product of the three body-fixed rotations,
 \begin{equation}
@@ -484,11 +476,10 @@ All other rotations follow in a straightforward manner
 \end{array}
 \right).
 \end{equation}
-All other rotations follow in a straightforward manner.
+All other rotations follow in a straightforward manner. After the
+first part of the propagation, the forces and body-fixed torques are
+calculated at the new positions and orientations
 
-After the first part of the propagation, the forces and body-fixed
-torques are calculated at the new positions and orientations
-
 {\tt doForces:}
 \begin{align*}
 {\bf f}(t + h) &\leftarrow
@@ -500,12 +491,9 @@ torques are calculated at the new positions and orient
 {\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h)
     \cdot {\bf \tau}^s(t + h).
 \end{align*}
+Once the forces and torques have been obtained at the new time step,
+the velocities can be advanced to the same time value.
 
-{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix
-$\mathsf{A}$ is calculated in {\tt moveA}.  Once the forces and
-torques have been obtained at the new time step, the velocities can
-be advanced to the same time value.
-
 {\tt moveB:}
 \begin{align*}
 {\bf v}\left(t + h \right)  &\leftarrow  {\bf v}\left(t + h / 2
