--- trunk/tengDissertation/Introduction.tex	2006/04/15 01:08:18	2713
+++ trunk/tengDissertation/Introduction.tex	2006/04/17 20:39:26	2716
@@ -1247,8 +1247,14 @@ rigid body
 \end{equation}
 
 \section{\label{introSection:langevinDynamics}Langevin Dynamics}
-
-\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
+As an alternative to newtonian dynamics, Langevin dynamics, which
+mimics a simple heat bath with stochastic and dissipative forces,
+has been applied in a variety of studies. This section will review
+the theory of Langevin dynamics simulation. A brief derivation of
+generalized Langevin Dynamics will be given first. Follow that, we
+will discuss the physical meaning of the terms appearing in the
+equation as well as the calculation of friction tensor from
+hydrodynamics theory.
 
 \subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
 
@@ -1442,16 +1448,224 @@ And since the $q$ coordinates are harmonic oscillators
 \label{introEquation:secondFluctuationDissipation}
 \end{equation}
 
-\section{\label{introSection:hydroynamics}Hydrodynamics}
-
 \subsection{\label{introSection:frictionTensor} Friction Tensor}
-\subsection{\label{introSection:analyticalApproach}Analytical
-Approach}
-
-\subsection{\label{introSection:approximationApproach}Approximation
-Approach}
+Theoretically, the friction kernel can be determined using velocity
+autocorrelation function. However, this approach become impractical
+when the system become more and more complicate. Instead, various
+approaches based on hydrodynamics have been developed to calculate
+the friction coefficients. The friction effect is isotropic in
+Equation, \zeta can be taken as a scalar. In general, friction
+tensor \Xi is a $6\times 6$ matrix given by
+\[
+\Xi  = \left( {\begin{array}{*{20}c}
+   {\Xi _{}^{tt} } & {\Xi _{}^{rt} }  \\
+   {\Xi _{}^{tr} } & {\Xi _{}^{rr} }  \\
+\end{array}} \right).
+\]
+Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction
+tensor and rotational friction tensor respectively, while ${\Xi^{tr}
+}$ is translation-rotation coupling tensor and $ {\Xi^{rt} }$ is
+rotation-translation coupling tensor.
 
-\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-Body}
+\[
+\left( \begin{array}{l}
+ F_t  \\
+ \tau  \\
+ \end{array} \right) =  - \left( {\begin{array}{*{20}c}
+   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
+   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
+\end{array}} \right)\left( \begin{array}{l}
+ v \\
+ w \\
+ \end{array} \right)
+\]
+
+\subsubsection{\label{introSection:analyticalApproach}The Friction Tensor for Regular Shape}
+For a spherical particle, the translational and rotational friction
+constant can be calculated from Stoke's law,
+\[
+\Xi ^{tt}  = \left( {\begin{array}{*{20}c}
+   {6\pi \eta R} & 0 & 0  \\
+   0 & {6\pi \eta R} & 0  \\
+   0 & 0 & {6\pi \eta R}  \\
+\end{array}} \right)
+\]
+and
+\[
+\Xi ^{rr}  = \left( {\begin{array}{*{20}c}
+   {8\pi \eta R^3 } & 0 & 0  \\
+   0 & {8\pi \eta R^3 } & 0  \\
+   0 & 0 & {8\pi \eta R^3 }  \\
+\end{array}} \right)
+\]
+where $\eta$ is the viscosity of the solvent and $R$ is the
+hydrodynamics radius.
 
-\section{\label{introSection:correlationFunctions}Correlation Functions}
+Other non-spherical particles have more complex properties.
+
+\[
+S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
+} }}{b}
+\]
+
+
+\[
+S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
+}}{a}
+\]
+
+\[
+\begin{array}{l}
+ \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
+ \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
+ \end{array}
+\]
+
+\[
+\begin{array}{l}
+ \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
+ \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
+ \end{array}
+\]
+
+
+\subsubsection{\label{introSection:approximationApproach}The Friction Tensor for Arbitrary Shape}
+Unlike spherical and other regular shaped molecules, there is not
+analytical solution for friction tensor of any arbitrary shaped
+rigid molecules. The ellipsoid of revolution model and general
+triaxial ellipsoid model have been used to approximate the
+hydrodynamic properties of rigid bodies. However, since the mapping
+from all possible ellipsoidal space, $r$-space, to all possible
+combination of rotational diffusion coefficients, $D$-space is not
+unique\cite{Wegener79} as well as the intrinsic coupling between
+translational and rotational motion of rigid body\cite{}, general
+ellipsoid is not always suitable for modeling arbitrarily shaped
+rigid molecule. A number of studies have been devoted to determine
+the friction tensor for irregularly shaped rigid bodies using more
+advanced method\cite{} where the molecule of interest was modeled by
+combinations of spheres(beads)\cite{} and the hydrodynamics
+properties of the molecule can be calculated using the hydrodynamic
+interaction tensor. Let us consider a rigid assembly of $N$ beads
+immersed in a continuous medium. Due to hydrodynamics interaction,
+the ``net'' velocity of $i$th bead, $v'_i$ is different than its
+unperturbed velocity $v_i$,
+\[
+v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
+\]
+where $F_i$ is the frictional force, and $T_{ij}$ is the
+hydrodynamic interaction tensor. The friction force of $i$th bead is
+proportional to its ``net'' velocity
+\begin{equation}
+F_i  = \zeta _i v_i  - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }.
+\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}
+\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}
+\end{equation}
+Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
+A second order expression for element of different size was
+introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de
+la Torre and Bloomfield,
+\begin{equation}
+T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
+\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
+_i^2  + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} -
+\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
+overlapping beads with the same radius, $\sigma$, is given by
+\begin{equation}
+T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
+\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I +
+\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
+\label{introEquation:RPTensorOverlapped}
+\end{equation}
+
+%Bead Modeling
+
+\[
+B = \left( {\begin{array}{*{20}c}
+   {T_{11} } &  \ldots  & {T_{1N} }  \\
+    \vdots  &  \ddots  &  \vdots   \\
+   {T_{N1} } &  \cdots  & {T_{NN} }  \\
+\end{array}} \right)
+\]
+
+\[
+C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
+   {C_{11} } &  \ldots  & {C_{1N} }  \\
+    \vdots  &  \ddots  &  \vdots   \\
+   {C_{N1} } &  \cdots  & {C_{NN} }  \\
+\end{array}} \right)
+\]
+
+\begin{equation}
+\begin{array}{l}
+ \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\
+ \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}
+\end{equation}
+where
+\[
+U_i  = \left( {\begin{array}{*{20}c}
+   0 & { - z_i } & {y_i }  \\
+   {z_i } & 0 & { - x_i }  \\
+   { - y_i } & {x_i } & 0  \\
+\end{array}} \right)
+\]
+
+\[
+r_{OR}  = \left( \begin{array}{l}
+ x_{OR}  \\
+ y_{OR}  \\
+ z_{OR}  \\
+ \end{array} \right) = \left( {\begin{array}{*{20}c}
+   {\Xi _{yy}^{rr}  + \Xi _{zz}^{rr} } & { - \Xi _{xy}^{rr} } & { - \Xi _{xz}^{rr} }  \\
+   { - \Xi _{yx}^{rr} } & {\Xi _{zz}^{rr}  + \Xi _{xx}^{rr} } & { - \Xi _{yz}^{rr} }  \\
+   { - \Xi _{zx}^{rr} } & { - \Xi _{yz}^{rr} } & {\Xi _{xx}^{rr}  + \Xi _{yy}^{rr} }  \\
+\end{array}} \right)^{ - 1} \left( \begin{array}{l}
+ \Xi _{yz}^{tr}  - \Xi _{zy}^{tr}  \\
+ \Xi _{zx}^{tr}  - \Xi _{xz}^{tr}  \\
+ \Xi _{xy}^{tr}  - \Xi _{yx}^{tr}  \\
+ \end{array} \right)
+\]
+
+\[
+U_{OR}  = \left( {\begin{array}{*{20}c}
+   0 & { - z_{OR} } & {y_{OR} }  \\
+   {z_i } & 0 & { - x_{OR} }  \\
+   { - y_{OR} } & {x_{OR} } & 0  \\
+\end{array}} \right)
+\]
+
+\[
+\begin{array}{l}
+ \Xi _R^{tt}  = \Xi _{}^{tt}  \\
+ \Xi _R^{tr}  = \Xi _R^{rt}  = \Xi _{}^{tr}  - U_{OR} \Xi _{}^{tt}  \\
+ \Xi _R^{rr}  = \Xi _{}^{rr}  - U_{OR} \Xi _{}^{tt} U_{OR}  + \Xi _{}^{tr} U_{OR}  - U_{OR} \Xi _{}^{tr} ^{^T }  \\
+ \end{array}
+\]
+
+\[
+D_R  = \left( {\begin{array}{*{20}c}
+   {D_R^{tt} } & {D_R^{rt} }  \\
+   {D_R^{tr} } & {D_R^{rr} }  \\
+\end{array}} \right) = k_b T\left( {\begin{array}{*{20}c}
+   {\Xi _R^{tt} } & {\Xi _R^{rt} }  \\
+   {\Xi _R^{tr} } & {\Xi _R^{rr} }  \\
+\end{array}} \right)^{ - 1}
+\]
+
+
+%Approximation Methods
+
+%\section{\label{introSection:correlationFunctions}Correlation Functions}
