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+isolated and conserve energy, Microcanonical ensemble(NVE) has a
+partition function like,
+\begin{equation}
-<
+\Omega (N,V,E) = e^{\beta TS}
-<
+\label{introEqaution:NVEPartition}.
->
+\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
+\end{equation}
+A canonical ensemble(NVT)is an ensemble of systems, each of which
+can share its energy with a large heat reservoir. The distribution
+\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
+\end{equation}
+The most obvious change being that matrix $J$ now depends on $x$.
-–
+The free rigid body is an example of Poisson system (actually a
-–
+Lie-Poisson system) with Hamiltonian function of angular kinetic
-–
+energy.
-–
+\begin{equation}
-–
+J(\pi ) = \left( {\begin{array}{*{20}c}
-–
+& {\pi _3 } & { - \pi _2 }  \\
-–
+   { - \pi _3 } & 0 & {\pi _1 }  \\
-–
+   {\pi _2 } & { - \pi _1 } & 0  \\
-–
+\end{array}} \right)
-–
+\end{equation}
-–
+\begin{equation}
-–
+H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2
-–
+}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right)
-–
+\end{equation}
-–
+\subsection{\label{introSection:exactFlow}Exact Flow}
+Let $x(t)$ be the exact solution of the ODE system,
+splitting gives a second-order decomposition,
+\begin{equation}
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
-<
+_{1,h/2} ,
-<
+\label{introEqaution:secondOrderSplitting}
->
+_{1,h/2} , \label{introEquation:secondOrderSplitting}
+\end{equation}
+which has a local error proportional to $h^3$. Sprang splitting's
+popularity in molecular simulation community attribute to its
+method using quaternion representation was developed by Omelyan.
+However, both of these methods are iterative and inefficient. In
+this section, we will present a symplectic Lie-Poisson integrator
-<
+for rigid body developed by Dullweber and his coworkers\cite{}.
->
+for rigid body developed by Dullweber and his
->
+coworkers\cite{Dullweber1997} in depth.
-<
+\subsection{\label{introSection:lieAlgebra}Lie Algebra}
-<
-<
+\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
-<
-<
+\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
-<
-<
+\section{\label{introSection:correlationFunctions}Correlation Functions}
->
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body}
->
+The motion of the rigid body is Hamiltonian with the Hamiltonian
->
+function
->
+\begin{equation}
->
+H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
->
+V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
->
+\label{introEquation:RBHamiltonian}
->
+\end{equation}
->
+Here, $q$ and $Q$  are the position and rotation matrix for the
->
+rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ , and
->
+$J$, a diagonal matrix, is defined by
->
+\[
->
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
->
+\]
->
+where $I_{ii}$ is the diagonal element of the inertia tensor. This
->
+constrained Hamiltonian equation subjects to a holonomic constraint,
->
+\begin{equation}
->
+Q^T Q = 1$, \label{introEquation:orthogonalConstraint}
->
+\end{equation}
->
+which is used to ensure rotation matrix's orthogonality.
->
+Differentiating \ref{introEquation:orthogonalConstraint} and using
->
+Equation \ref{introEquation:RBMotionMomentum}, one may obtain,
->
+\begin{equation}
->
+Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
->
+\label{introEquation:RBFirstOrderConstraint}
->
+\end{equation}
-<
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
->
+Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
->
+\ref{introEquation:motionHamiltonianMomentum}), one can write down
->
+the equations of motion,
->
+\[
->
+\begin{array}{c}
->
+ \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\
->
+ \frac{{dp}}{{dt}} =  - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
->
+ \frac{{dQ}}{{dt}} = PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
->
+ \frac{{dP}}{{dt}} =  - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\
->
+ \end{array}
->
+\]
->
->
+In general, there are two ways to satisfy the holonomic constraints.
->
+We can use constraint force provided by lagrange multiplier on the
->
+normal manifold to keep the motion on constraint space. Or we can
->
+simply evolve the system in constraint manifold. The two method are
->
+proved to be equivalent. The holonomic constraint and equations of
->
+motions define a constraint manifold for rigid body
->
+\[
->
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
->
+\right\}.
->
+\]
-<
+\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
->
+Unfortunately, this constraint manifold is not the cotangent bundle
->
+$T_{\star}SO(3)$. However, it turns out that under symplectic
->
+transformation, the cotangent space and the phase space are
->
+diffeomorphic. Introducing
->
+\[
->
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
->
+\]
->
+the mechanical system subject to a holonomic constraint manifold $M$
->
+can be re-formulated as a Hamiltonian system on the cotangent space
->
+\[
->
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
->
+,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
->
+\]
-+
+For a body fixed vector $X_i$ with respect to the center of mass of
-+
+the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
-+
+given as
-+
+\begin{equation}
-+
+X_i^{lab} = Q X_i + q.
-+
+\end{equation}
-+
+Therefore, potential energy $V(q,Q)$ is defined by
-+
+\[
-+
+V(q,Q) = V(Q X_0 + q).
-+
+\]
-+
+Hence, the force and torque are given by
-+
+\[
-+
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
-+
+\]
-+
+and
-+
+\[
-+
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
-+
+\]
-+
+respectively.
-+
-+
+As a common choice to describe the rotation dynamics of the rigid
-+
+body, angular momentum on body frame $\Pi  = Q^t P$ is introduced to
-+
+rewrite the equations of motion,
-+
+\begin{equation}
-+
+ \begin{array}{l}
-+
+ \mathop \Pi \limits^ \bullet   = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda  \\
-+
+ \mathop Q\limits^{{\rm{   }} \bullet }  = Q\Pi {\rm{ }}J^{ - 1}  \\
-+
+ \end{array}
-+
+ \label{introEqaution:RBMotionPI}
-+
+\end{equation}
-+
+, as well as holonomic constraints,
-+
+\[
-+
+\begin{array}{l}
-+
+ \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
-+
+ Q^T Q = 1 \\
-+
+ \end{array}
-+
+\]
-+
-+
+For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
-+
+so(3)^ \star$, the hat-map isomorphism,
-+
+\begin{equation}
-+
+v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
-+
+{\begin{array}{*{20}c}
-+
+& { - v_3 } & {v_2 }  \\
-+
+   {v_3 } & 0 & { - v_1 }  \\
-+
+   { - v_2 } & {v_1 } & 0  \\
-+
+\end{array}} \right),
-+
+\label{introEquation:hatmapIsomorphism}
-+
+\end{equation}
-+
+will let us associate the matrix products with traditional vector
-+
+operations
-+
+\[
-+
+\hat vu = v \times u
-+
+\]
-+
-+
+Using \ref{introEqaution:RBMotionPI}, one can construct a skew
-+
+matrix,
-+
+\begin{equation}
-+
+(\mathop \Pi \limits^ \bullet   - \mathop \Pi \limits^ \bullet  ^T
-+
+){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi  + \Pi J^{
-+
+- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]} -
-+
+(\Lambda  - \Lambda ^T ) . \label{introEquation:skewMatrixPI}
-+
+\end{equation}
-+
+Since $\Lambda$ is symmetric, the last term of Equation
-+
+\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
-+
+multiplier $\Lambda$ is absent from the equations of motion. This
-+
+unique property eliminate the requirement of iterations which can
-+
+not be avoided in other methods\cite{}.
-+
-+
+Applying hat-map isomorphism, we obtain the equation of motion for
-+
+angular momentum on body frame
-+
+\begin{equation}
-+
+\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
-+
+F_i (r,Q)} \right) \times X_i }.
-+
+\label{introEquation:bodyAngularMotion}
-+
+\end{equation}
-+
+In the same manner, the equation of motion for rotation matrix is
-+
+given by
-+
+\[
-+
+\dot Q = Qskew(I^{ - 1} \pi )
-+
+\]
-+
-+
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
-+
+Lie-Poisson Integrator for Free Rigid Body}
-+
-+
+If there is not external forces exerted on the rigid body, the only
-+
+contribution to the rotational is from the kinetic potential (the
-+
+first term of \ref{ introEquation:bodyAngularMotion}). The free
-+
+rigid body is an example of Lie-Poisson system with Hamiltonian
-+
+function
-+
+\begin{equation}
-+
+T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
-+
+\label{introEquation:rotationalKineticRB}
-+
+\end{equation}
-+
+where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
-+
+Lie-Poisson structure matrix,
-+
+\begin{equation}
-+
+J(\pi ) = \left( {\begin{array}{*{20}c}
-+
+& {\pi _3 } & { - \pi _2 }  \\
-+
+   { - \pi _3 } & 0 & {\pi _1 }  \\
-+
+   {\pi _2 } & { - \pi _1 } & 0  \\
-+
+\end{array}} \right)
-+
+\end{equation}
-+
+Thus, the dynamics of free rigid body is governed by
-+
+\begin{equation}
-+
+\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
-+
+\end{equation}
-+
-+
+One may notice that each $T_i^r$ in Equation
-+
+\ref{introEquation:rotationalKineticRB} can be solved exactly. For
-+
+instance, the equations of motion due to $T_1^r$ are given by
-+
+\begin{equation}
-+
+\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
-+
+\label{introEqaution:RBMotionSingleTerm}
-+
+\end{equation}
-+
+where
-+
+\[ R_1  = \left( {\begin{array}{*{20}c}
-+
+& 0 & 0  \\
-+
+& 0 & {\pi _1 }  \\
-+
+& { - \pi _1 } & 0  \\
-+
+\end{array}} \right).
-+
+\]
-+
+The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is
-+
+\[
-+
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
-+
+Q(0)e^{\Delta tR_1 }
-+
+\]
-+
+with
-+
+\[
-+
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
-+
+& 0 & 0  \\
-+
+& {\cos \theta _1 } & {\sin \theta _1 }  \\
-+
+& { - \sin \theta _1 } & {\cos \theta _1 }  \\
-+
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
-+
+\]
-+
+To reduce the cost of computing expensive functions in e^{\Delta
-+
+tR_1 }, we can use Cayley transformation,
-+
+\[
-+
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
-+
+)
-+
+\]
-+
-+
+The flow maps for $T_2^r$ and $T_2^r$ can be found in the same
-+
+manner.
-+
-+
+In order to construct a second-order symplectic method, we split the
-+
+angular kinetic Hamiltonian function can into five terms
-+
+\[
-+
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
-+
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
-+
+(\pi _1 )
-+
+\].
-+
+Concatenating flows corresponding to these five terms, we can obtain
-+
+an symplectic integrator,
-+
+\[
-+
+\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
-+
+\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
-+
+\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
-+
+_1 }.
-+
+\]
-+
-+
+The non-canonical Lie-Poisson bracket ${F, G}$ of two function
-+
+$F(\pi )$ and $G(\pi )$ is defined by
-+
+\[
-+
+\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
-+
+)
-+
+\]
-+
+If the Poisson bracket of a function $F$ with an arbitrary smooth
-+
+function $G$ is zero, $F$ is a \emph{Casimir}, which is the
-+
+conserved quantity in Poisson system. We can easily verify that the
-+
+norm of the angular momentum, $\parallel \pi
-+
+\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel
-+
+\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
-+
+then by the chain rule
-+
+\[
-+
+\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
-+
+}}{2})\pi
-+
+\]
-+
+Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
-+
+\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
-+
+Lie-Poisson integrator is found to be extremely efficient and stable
-+
+which can be explained by the fact the small angle approximation is
-+
+used and the norm of the angular momentum is conserved.
-+
-+
+\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
-+
+Splitting for Rigid Body}
-+
-+
+The Hamiltonian of rigid body can be separated in terms of kinetic
-+
+energy and potential energy,
-+
+\[
-+
+H = T(p,\pi ) + V(q,Q)
-+
+\]
-+
+The equations of motion corresponding to potential energy and
-+
+kinetic energy are listed in the below table,
-+
+\begin{center}
-+
+\begin{tabular}{|l|l|}
-+
+  \hline
-+
+  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
-+
+  Potential & Kinetic \\
-+
+  $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
-+
+  $\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
-+
+  $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
-+
+  $ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
-+
+  \hline
-+
+\end{tabular}
-+
+\end{center}
-+
+A second-order symplectic method is now obtained by the composition
-+
+of the flow maps,
-+
+\[
-+
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
-+
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
-+
+\]
-+
+Moreover, \varphi _{\Delta t/2,V} can be divided into two sub-flows
-+
+which corresponding to force and torque respectively,
-+
+\[
-+
+\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
-+
+_{\Delta t/2,\tau }.
-+
+\]
-+
+Since the associated operators of $\varphi _{\Delta t/2,F} $ and
-+
+$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition
-+
+order inside \varphi _{\Delta t/2,V} does not matter.
-+
-+
+Furthermore, kinetic potential can be separated to translational
-+
+kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$,
-+
+\begin{equation}
-+
+T(p,\pi ) =T^t (p) + T^r (\pi ).
-+
+\end{equation}
-+
+where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
-+
+defined by \ref{introEquation:rotationalKineticRB}. Therefore, the
-+
+corresponding flow maps are given by
-+
+\[
-+
+\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
-+
+_{\Delta t,T^r }.
-+
+\]
-+
+Finally, we obtain the overall symplectic flow maps for free moving
-+
+rigid body
-+
+\begin{equation}
-+
+\begin{array}{c}
-+
+ \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
-+
+  \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \\
-+
+  \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
-+
+ \end{array}
-+
+\label{introEquation:overallRBFlowMaps}
-+
+\end{equation}
-+
-+
+\section{\label{introSection:langevinDynamics}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}
+\begin{equation}
+\label{introEquation:secondFluctuationDissipation}
+\end{equation}
-–
+\section{\label{introSection:hydroynamics}Hydrodynamics}
-–
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
-<
+\subsection{\label{introSection:analyticalApproach}Analytical
-<
+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:approximationApproach}Approximation
-<
+Approach}
->
+\[
->
+\left( \begin{array}{l}
->
+ F_t  \\
->
+ \tau  \\
->
+ \end{array} \right) =  - \left( {\begin{array}{*{20}c}
->
+   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
->
+   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
->
+\end{array}} \right)\left( \begin{array}{l}
->
+ v \\
->
+ w \\
->
+ \end{array} \right)
->
+\]
-<
+\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-<
+Body}
->
+\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,
->
+\[
->
+\Xi ^{tt}  = \left( {\begin{array}{*{20}c}
->
+   {6\pi \eta R} & 0 & 0  \\
->
+& {6\pi \eta R} & 0  \\
->
+& 0 & {6\pi \eta R}  \\
->
+\end{array}} \right)
->
+\]
->
+and
->
+\[
->
+\Xi ^{rr}  = \left( {\begin{array}{*{20}c}
->
+   {8\pi \eta R^3 } & 0 & 0  \\
->
+& {8\pi \eta R^3 } & 0  \\
->
+& 0 & {8\pi \eta R^3 }  \\
->
+\end{array}} \right)
->
+\]
->
+where $\eta$ is the viscosity of the solvent and $R$ is the
->
+hydrodynamics radius.
->
->
+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
->
+, 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
->
+\[
->
+U_i  = \left( {\begin{array}{*{20}c}
->
+& { - z_i } & {y_i }  \\
->
+   {z_i } & 0 & { - x_i }  \\
->
+   { - y_i } & {x_i } & 0  \\
->
+\end{array}} \right)
->
+\]
->
->
+\[
->
+r_{OR}  = \left( \begin{array}{l}
->
+ x_{OR}  \\
->
+ y_{OR}  \\
->
+ z_{OR}  \\
->
+ \end{array} \right) = \left( {\begin{array}{*{20}c}
->
+   {\Xi _{yy}^{rr}  + \Xi _{zz}^{rr} } & { - \Xi _{xy}^{rr} } & { - \Xi _{xz}^{rr} }  \\
->
+   { - \Xi _{yx}^{rr} } & {\Xi _{zz}^{rr}  + \Xi _{xx}^{rr} } & { - \Xi _{yz}^{rr} }  \\
->
+   { - \Xi _{zx}^{rr} } & { - \Xi _{yz}^{rr} } & {\Xi _{xx}^{rr}  + \Xi _{yy}^{rr} }  \\
->
+\end{array}} \right)^{ - 1} \left( \begin{array}{l}
->
+ \Xi _{yz}^{tr}  - \Xi _{zy}^{tr}  \\
->
+ \Xi _{zx}^{tr}  - \Xi _{xz}^{tr}  \\
->
+ \Xi _{xy}^{tr}  - \Xi _{yx}^{tr}  \\
->
+ \end{array} \right)
->
+\]
->
->
+\[
->
+U_{OR}  = \left( {\begin{array}{*{20}c}
->
+& { - z_{OR} } & {y_{OR} }  \\
->
+   {z_i } & 0 & { - x_{OR} }  \\
->
+   { - y_{OR} } & {x_{OR} } & 0  \\
->
+\end{array}} \right)
->
+\]
->
->
+\[
->
+\begin{array}{l}
->
+ \Xi _R^{tt}  = \Xi _{}^{tt}  \\
->
+ \Xi _R^{tr}  = \Xi _R^{rt}  = \Xi _{}^{tr}  - U_{OR} \Xi _{}^{tt}  \\
->
+ \Xi _R^{rr}  = \Xi _{}^{rr}  - U_{OR} \Xi _{}^{tt} U_{OR}  + \Xi _{}^{tr} U_{OR}  - U_{OR} \Xi _{}^{tr} ^{^T }  \\
->
+ \end{array}
->
+\]
->
->
+\[
->
+D_R  = \left( {\begin{array}{*{20}c}
->
+   {D_R^{tt} } & {D_R^{rt} }  \\
->
+   {D_R^{tr} } & {D_R^{rr} }  \\
->
+\end{array}} \right) = k_b T\left( {\begin{array}{*{20}c}
->
+   {\Xi _R^{tt} } & {\Xi _R^{rt} }  \\
->
+   {\Xi _R^{tr} } & {\Xi _R^{rr} }  \\
->
+\end{array}} \right)^{ - 1}
->
+\]
->
->
->
+%Approximation Methods
->
->
+%\section{\label{introSection:correlationFunctions}Correlation Functions}

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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2705 by tim, Wed Apr 12 21:20:13 2006 UTC vs. Revision 2716 by tim, Mon Apr 17 20:39:26 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2705 by tim, Wed Apr 12 21:20:13 2006 UTC vs.
Revision 2716 by tim, Mon Apr 17 20:39:26 2006 UTC