--- trunk/tengDissertation/Introduction.tex	2006/04/13 04:47:47	2706
+++ trunk/tengDissertation/Introduction.tex	2006/05/26 18:25:41	2779
@@ -570,21 +570,6 @@ The free rigid body is an example of Poisson system (a
 \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}
-   0 & {\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}
 
@@ -837,7 +822,7 @@ q(\Delta t)} \right]. %
 %
 q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
 q(\Delta t)} \right]. %
- \label{introEquation:positionVerlet1}
+ \label{introEquation:positionVerlet2}
 \end{align}
 
 \subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
@@ -846,7 +831,7 @@ $\varphi_1(t)$ and $\varphi_2(t$ respectively , we hav
 error of splitting method in terms of commutator of the
 operators(\ref{introEquation:exponentialOperator}) associated with
 the sub-flow. For operators $hX$ and $hY$ which are associate to
-$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have
+$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have
 \begin{equation}
 \exp (hX + hY) = \exp (hZ)
 \end{equation}
@@ -862,10 +847,9 @@ can obtain
 Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we
 can obtain
 \begin{eqnarray*}
-\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2
-[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
-& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} +
-\ldots )
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\
+                                   &   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
+                                   &   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots )
 \end{eqnarray*}
 Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local
 error of Spring splitting is proportional to $h^3$. The same
@@ -874,7 +858,7 @@ Careful choice of coefficient $a_1 ,\ldot , b_m$ will 
 \varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
 1} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
 \end{equation}
-Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher
+Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher
 order method. Yoshida proposed an elegant way to compose higher
 order methods based on symmetric splitting. Given a symmetric second
 order base method $ \varphi _h^{(2)} $, a fourth-order symmetric
@@ -898,16 +882,299 @@ As a special discipline of molecular modeling, Molecul
 
 \section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
-As a special discipline of molecular modeling, Molecular dynamics
-has proven to be a powerful tool for studying the functions of
-biological systems, providing structural, thermodynamic and
-dynamical information.
-
-\subsection{\label{introSec:mdInit}Initialization}
-
-\subsection{\label{introSec:forceEvaluation}Force Evaluation}
+As one of the principal tools of molecular modeling, Molecular
+dynamics has proven to be a powerful tool for studying the functions
+of biological systems, providing structural, thermodynamic and
+dynamical information. The basic idea of molecular dynamics is that
+macroscopic properties are related to microscopic behavior and
+microscopic behavior can be calculated from the trajectories in
+simulations. For instance, instantaneous temperature of an
+Hamiltonian system of $N$ particle can be measured by
+\[
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
+\]
+where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
+respectively, $f$ is the number of degrees of freedom, and $k_B$ is
+the boltzman constant.
 
-\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
+A typical molecular dynamics run consists of three essential steps:
+\begin{enumerate}
+  \item Initialization
+    \begin{enumerate}
+    \item Preliminary preparation
+    \item Minimization
+    \item Heating
+    \item Equilibration
+    \end{enumerate}
+  \item Production
+  \item Analysis
+\end{enumerate}
+These three individual steps will be covered in the following
+sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
+initialization of a simulation. Sec.~\ref{introSec:production} will
+discusses issues in production run. Sec.~\ref{introSection:Analysis}
+provides the theoretical tools for trajectory analysis.
+
+\subsection{\label{introSec:initialSystemSettings}Initialization}
+
+\subsubsection{Preliminary preparation}
+
+When selecting the starting structure of a molecule for molecular
+simulation, one may retrieve its Cartesian coordinates from public
+databases, such as RCSB Protein Data Bank \textit{etc}. Although
+thousands of crystal structures of molecules are discovered every
+year, many more remain unknown due to the difficulties of
+purification and crystallization. Even for the molecule with known
+structure, some important information is missing. For example, the
+missing hydrogen atom which acts as donor in hydrogen bonding must
+be added. Moreover, in order to include electrostatic interaction,
+one may need to specify the partial charges for individual atoms.
+Under some circumstances, we may even need to prepare the system in
+a special setup. For instance, when studying transport phenomenon in
+membrane system, we may prepare the lipids in bilayer structure
+instead of placing lipids randomly in solvent, since we are not
+interested in self-aggregation and it takes a long time to happen.
+
+\subsubsection{Minimization}
+
+It is quite possible that some of molecules in the system from
+preliminary preparation may be overlapped with each other. This
+close proximity leads to high potential energy which consequently
+jeopardizes any molecular dynamics simulations. To remove these
+steric overlaps, one typically performs energy minimization to find
+a more reasonable conformation. Several energy minimization methods
+have been developed to exploit the energy surface and to locate the
+local minimum. While converging slowly near the minimum, steepest
+descent method is extremely robust when systems are far from
+harmonic. Thus, it is often used to refine structure from
+crystallographic data. Relied on the gradient or hessian, advanced
+methods like conjugate gradient and Newton-Raphson converge rapidly
+to a local minimum, while become unstable if the energy surface is
+far from quadratic. Another factor must be taken into account, when
+choosing energy minimization method, is the size of the system.
+Steepest descent and conjugate gradient can deal with models of any
+size. Because of the limit of computation power to calculate hessian
+matrix and insufficient storage capacity to store them, most
+Newton-Raphson methods can not be used with very large models.
+
+\subsubsection{Heating}
+
+Typically, Heating is performed by assigning random velocities
+according to a Gaussian distribution for a temperature. Beginning at
+a lower temperature and gradually increasing the temperature by
+assigning greater random velocities, we end up with setting the
+temperature of the system to a final temperature at which the
+simulation will be conducted. In heating phase, we should also keep
+the system from drifting or rotating as a whole. Equivalently, the
+net linear momentum and angular momentum of the system should be
+shifted to zero.
+
+\subsubsection{Equilibration}
+
+The purpose of equilibration is to allow the system to evolve
+spontaneously for a period of time and reach equilibrium. The
+procedure is continued until various statistical properties, such as
+temperature, pressure, energy, volume and other structural
+properties \textit{etc}, become independent of time. Strictly
+speaking, minimization and heating are not necessary, provided the
+equilibration process is long enough. However, these steps can serve
+as a means to arrive at an equilibrated structure in an effective
+way.
+
+\subsection{\label{introSection:production}Production}
+
+Production run is the most important steps of the simulation, in
+which the equilibrated structure is used as a starting point and the
+motions of the molecules are collected for later analysis. In order
+to capture the macroscopic properties of the system, the molecular
+dynamics simulation must be performed in correct and efficient way.
+
+The most expensive part of a molecular dynamics simulation is the
+calculation of non-bonded forces, such as van der Waals force and
+Coulombic forces \textit{etc}. For a system of $N$ particles, the
+complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
+which making large simulations prohibitive in the absence of any
+computation saving techniques.
+
+A natural approach to avoid system size issue is to represent the
+bulk behavior by a finite number of the particles. However, this
+approach will suffer from the surface effect. To offset this,
+\textit{Periodic boundary condition} is developed to simulate bulk
+properties with a relatively small number of particles. In this
+method, the simulation box is replicated throughout space to form an
+infinite lattice. During the simulation, when a particle moves in
+the primary cell, its image in other cells move in exactly the same
+direction with exactly the same orientation. Thus, as a particle
+leaves the primary cell, one of its images will enter through the
+opposite face.
+%\begin{figure}
+%\centering
+%\includegraphics[width=\linewidth]{pbcFig.eps}
+%\caption[An illustration of periodic boundary conditions]{A 2-D
+%illustration of periodic boundary conditions. As one particle leaves
+%the right of the simulation box, an image of it enters the left.}
+%\label{introFig:pbc}
+%\end{figure}
+
+%cutoff and minimum image convention
+Another important technique to improve the efficiency of force
+evaluation is to apply cutoff where particles farther than a
+predetermined distance, are not included in the calculation
+\cite{Frenkel1996}. The use of a cutoff radius will cause a
+discontinuity in the potential energy curve. Fortunately, one can
+shift the potential to ensure the potential curve go smoothly to
+zero at the cutoff radius. Cutoff strategy works pretty well for
+Lennard-Jones interaction because of its short range nature.
+However, simply truncating the electrostatic interaction with the
+use of cutoff has been shown to lead to severe artifacts in
+simulations. Ewald summation, in which the slowly conditionally
+convergent Coulomb potential is transformed into direct and
+reciprocal sums with rapid and absolute convergence, has proved to
+minimize the periodicity artifacts in liquid simulations. Taking the
+advantages of the fast Fourier transform (FFT) for calculating
+discrete Fourier transforms, the particle mesh-based methods are
+accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative
+approach is \emph{fast multipole method}, which treats Coulombic
+interaction exactly at short range, and approximate the potential at
+long range through multipolar expansion. In spite of their wide
+acceptances at the molecular simulation community, these two methods
+are hard to be implemented correctly and efficiently. Instead, we
+use a damped and charge-neutralized Coulomb potential method
+developed by Wolf and his coworkers. The shifted Coulomb potential
+for particle $i$ and particle $j$ at distance $r_{rj}$ is given by:
+\begin{equation}
+V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
+r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
+R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
+r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb}
+\end{equation}
+where $\alpha$ is the convergence parameter. Due to the lack of
+inherent periodicity and rapid convergence,this method is extremely
+efficient and easy to implement.
+%\begin{figure}
+%\centering
+%\includegraphics[width=\linewidth]{pbcFig.eps}
+%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.}
+%\label{introFigure:shiftedCoulomb}
+%\end{figure}
+
+%multiple time step
+
+\subsection{\label{introSection:Analysis} Analysis}
+
+Recently, advanced visualization technique are widely applied to
+monitor the motions of molecules. Although the dynamics of the
+system can be described qualitatively from animation, quantitative
+trajectory analysis are more appreciable. According to the
+principles of Statistical Mechanics,
+Sec.~\ref{introSection:statisticalMechanics}, one can compute
+thermodynamics properties, analyze fluctuations of structural
+parameters, and investigate time-dependent processes of the molecule
+from the trajectories.
+
+\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties}
+
+Thermodynamics properties, which can be expressed in terms of some
+function of the coordinates and momenta of all particles in the
+system, can be directly computed from molecular dynamics. The usual
+way to measure the pressure is based on virial theorem of Clausius
+which states that the virial is equal to $-3Nk_BT$. For a system
+with forces between particles, the total virial, $W$, contains the
+contribution from external pressure and interaction between the
+particles:
+\[
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
+f_{ij} } } \right\rangle
+\]
+where $f_{ij}$ is the force between particle $i$ and $j$ at a
+distance $r_{ij}$. Thus, the expression for the pressure is given
+by:
+\begin{equation}
+P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
+< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
+\end{equation}
+
+\subsubsection{\label{introSection:structuralProperties}Structural Properties}
+
+Structural Properties of a simple fluid can be described by a set of
+distribution functions. Among these functions,\emph{pair
+distribution function}, also known as \emph{radial distribution
+function}, is of most fundamental importance to liquid-state theory.
+Pair distribution function can be gathered by Fourier transforming
+raw data from a series of neutron diffraction experiments and
+integrating over the surface factor \cite{Powles73}. The experiment
+result can serve as a criterion to justify the correctness of the
+theory. Moreover, various equilibrium thermodynamic and structural
+properties can also be expressed in terms of radial distribution
+function \cite{allen87:csl}.
+
+A pair distribution functions $g(r)$ gives the probability that a
+particle $i$ will be located at a distance $r$ from a another
+particle $j$ in the system
+\[
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle.
+\]
+Note that the delta function can be replaced by a histogram in
+computer simulation. Figure
+\ref{introFigure:pairDistributionFunction} shows a typical pair
+distribution function for the liquid argon system. The occurrence of
+several peaks in the plot of $g(r)$ suggests that it is more likely
+to find particles at certain radial values than at others. This is a
+result of the attractive interaction at such distances. Because of
+the strong repulsive forces at short distance, the probability of
+locating particles at distances less than about 2.5{\AA} from each
+other is essentially zero.
+
+%\begin{figure}
+%\centering
+%\includegraphics[width=\linewidth]{pdf.eps}
+%\caption[Pair distribution function for the liquid argon
+%]{Pair distribution function for the liquid argon}
+%\label{introFigure:pairDistributionFunction}
+%\end{figure}
+
+\subsubsection{\label{introSection:timeDependentProperties}Time-dependent
+Properties}
+
+Time-dependent properties are usually calculated using \emph{time
+correlation function}, which correlates random variables $A$ and $B$
+at two different time
+\begin{equation}
+C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
+\label{introEquation:timeCorrelationFunction}
+\end{equation}
+If $A$ and $B$ refer to same variable, this kind of correlation
+function is called \emph{auto correlation function}. One example of
+auto correlation function is velocity auto-correlation function
+which is directly related to transport properties of molecular
+liquids:
+\[
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
+\right\rangle } dt
+\]
+where $D$ is diffusion constant. Unlike velocity autocorrelation
+function which is averaging over time origins and over all the
+atoms, dipole autocorrelation are calculated for the entire system.
+The dipole autocorrelation function is given by:
+\[
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
+\right\rangle
+\]
+Here $u_{tot}$ is the net dipole of the entire system and is given
+by
+\[
+u_{tot} (t) = \sum\limits_i {u_i (t)}
+\]
+In principle, many time correlation functions can be related with
+Fourier transforms of the infrared, Raman, and inelastic neutron
+scattering spectra of molecular liquids. In practice, one can
+extract the IR spectrum from the intensity of dipole fluctuation at
+each frequency using the following relationship:
+\[
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
+i2\pi vt} dt}
+\]
 
 \section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
 
@@ -942,19 +1209,19 @@ rotation matrix $A$ and re-formulating Hamiltonian's e
 The break through in geometric literature suggests that, in order to
 develop a long-term integration scheme, one should preserve the
 symplectic structure of the flow. Introducing conjugate momentum to
-rotation matrix $A$ and re-formulating Hamiltonian's equation, a
+rotation matrix $Q$ and re-formulating Hamiltonian's equation, a
 symplectic integrator, RSHAKE, was proposed to evolve the
 Hamiltonian system in a constraint manifold by iteratively
-satisfying the orthogonality constraint $A_t A = 1$. An alternative
+satisfying the orthogonality constraint $Q_T Q = 1$. An alternative
 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: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 ].
@@ -969,13 +1236,13 @@ Q^T Q = 1$, \label{introEquation:orthogonalConstraint}
 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}
+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 . \\
+Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
 \label{introEquation:RBFirstOrderConstraint}
 \end{equation}
 
@@ -987,142 +1254,404 @@ the equations of motion,
  \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}\\
+ \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. These two methods
+are proved to be equivalent. The holonomic constraint and equations
+of motions define a constraint manifold for rigid body
 \[
-M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1}  + J^{ - 1} P^t Q = 0}
-\right\} .
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
+\right\}.
 \]
 
-\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
-
-\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations}
-
-
-\section{\label{introSection:langevinDynamics}Langevin Dynamics}
-
-\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
-
-\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
-
-\begin{equation}
-H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
-\label{introEquation:bathGLE}
-\end{equation}
-where $H_B$ is harmonic bath Hamiltonian,
+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
 \[
-H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
 \]
-and $\Delta U$ is bilinear system-bath coupling,
+the mechanical system subject to a holonomic constraint manifold $M$
+can be re-formulated as a Hamiltonian system on the cotangent space
 \[
-\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
+1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
 \]
-Completing the square,
+
+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
 \[
-H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
-{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
-1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
+V(q,Q) = V(Q X_0 + q).
 \]
-and putting it back into Eq.~\ref{introEquation:bathGLE},
+Hence, the force and torque are given by
 \[
-H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
 \]
-where
+and
 \[
-W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
-}}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
 \]
-Since the first two terms of the new Hamiltonian depend only on the
-system coordinates, we can get the equations of motion for
-Generalized Langevin Dynamics by Hamilton's equations
-\ref{introEquation:motionHamiltonianCoordinate,
-introEquation:motionHamiltonianMomentum},
-\begin{align}
-\dot p &=  - \frac{{\partial H}}{{\partial x}}
-       &= m\ddot x
-       &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)}
-\label{introEquation:Lp5}
-\end{align}
-, and
-\begin{align}
-\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
-                &= m\ddot x_\alpha
-                &= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
-\end{align}
+respectively.
 
-\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
-
+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,
 \[
-L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
+\begin{array}{l}
+ \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
+ Q^T Q = 1 \\
+ \end{array}
 \]
 
+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}
+   0 & { - 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
 \[
-L(x + y) = L(x) + L(y)
+\hat vu = v \times u
 \]
 
+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
 \[
-L(ax) = aL(x)
+\dot Q = Qskew(I^{ - 1} \pi )
 \]
 
+\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}
+   0 & {\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
+\[ R_1  = \left( {\begin{array}{*{20}c}
+   0 & 0 & 0  \\
+   0 & 0 & {\pi _1 }  \\
+   0 & { - \pi _1 } & 0  \\
+\end{array}} \right).
+\]
+The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is
 \[
-L(\dot x) = pL(x) - px(0)
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
+Q(0)e^{\Delta tR_1 }
 \]
-
+with
 \[
-L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
+   0 & 0 & 0  \\
+   0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
+   0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
 \]
-
+To reduce the cost of computing expensive functions in $e^{\Delta
+tR_1 }$, we can use Cayley transformation,
 \[
-L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
+)
 \]
+The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
+manner.
 
-Some relatively important transformation,
+In order to construct a second-order symplectic method, we split the
+angular kinetic Hamiltonian function can into five terms
 \[
-L(\cos at) = \frac{p}{{p^2  + a^2 }}
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
+(\pi _1 )
+\].
+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
 \[
-L(\sin at) = \frac{a}{{p^2  + a^2 }}
+\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
+)
 \]
-
+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
 \[
-L(1) = \frac{1}{p}
+\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
+}}{2})\pi
 \]
+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.
 
-First, the bath coordinates,
+\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,
 \[
-p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
-_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
-}}L(x)
+H = T(p,\pi ) + V(q,Q)
 \]
-\[
-L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
-px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
+The equations of motion corresponding to potential energy and
+kinetic energy are listed in the below table,
+\begin{table}
+\caption{Equations of motion due to Potential and Kinetic Energies}
+\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}
+\end{table}
+A second-order symplectic method is now obtained by the
+composition of the flow maps,
+\[
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
 \]
-Then, the system coordinates,
-\begin{align}
-mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
-\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{\frac{{g_\alpha
-}}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha
-(0)}}{{p^2  + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha
-}}\omega _\alpha ^2 L(x)} \right\}}
-%
- &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
- \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x)
- - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0)
- - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}
-\end{align}
-Then, the inverse transform,
+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 equation 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}Derivation of Generalized Langevin Equation}
+
+Harmonic bath model, in which an effective set of harmonic
+oscillators are used to mimic the effect of a linearly responding
+environment, has been widely used in quantum chemistry and
+statistical mechanics. One of the successful applications of
+Harmonic bath model is the derivation of Deriving Generalized
+Langevin Dynamics. Lets consider a system, in which the degree of
+freedom $x$ is assumed to couple to the bath linearly, giving a
+Hamiltonian of the form
+\begin{equation}
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
+\label{introEquation:bathGLE}.
+\end{equation}
+Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated
+with this degree of freedom, $H_B$ is harmonic bath Hamiltonian,
+\[
+H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  \omega _\alpha ^2 }
+\right\}}
+\]
+where the index $\alpha$ runs over all the bath degrees of freedom,
+$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
+the harmonic bath masses, and $\Delta U$ is bilinear system-bath
+coupling,
+\[
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
+\]
+where $g_\alpha$ are the coupling constants between the bath and the
+coordinate $x$. Introducing
+\[
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\] and combining the last two terms in Equation
+\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath
+Hamiltonian as
+\[
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
+w_\alpha ^2 }}x} \right)^2 } \right\}}
+\]
+Since the first two terms of the new Hamiltonian depend only on the
+system coordinates, we can get the equations of motion for
+Generalized Langevin Dynamics by Hamilton's equations
+\ref{introEquation:motionHamiltonianCoordinate,
+introEquation:motionHamiltonianMomentum},
+\begin{equation}
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
+\sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   -
+\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)},
+\label{introEquation:coorMotionGLE}
+\end{equation}
+and
+\begin{equation}
+m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x_\alpha   -
+\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
+\label{introEquation:bathMotionGLE}
+\end{equation}
+
+In order to derive an equation for $x$, the dynamics of the bath
+variables $x_\alpha$ must be solved exactly first. As an integral
+transform which is particularly useful in solving linear ordinary
+differential equations, Laplace transform is the appropriate tool to
+solve this problem. The basic idea is to transform the difficult
+differential equations into simple algebra problems which can be
+solved easily. Then applying inverse Laplace transform, also known
+as the Bromwich integral, we can retrieve the solutions of the
+original problems.
+
+Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace
+transform of f(t) is a new function defined as
+\[
+L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
+\]
+where  $p$ is real and  $L$ is called the Laplace Transform
+Operator. Below are some important properties of Laplace transform
+\begin{equation}
+\begin{array}{c}
+ L(x + y) = L(x) + L(y) \\
+ L(ax) = aL(x) \\
+ L(\dot x) = pL(x) - px(0) \\
+ L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\
+ L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\
+ \end{array}
+\end{equation}
+
+Applying Laplace transform to the bath coordinates, we obtain
+\[
+\begin{array}{c}
+ p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega _\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) \\
+ L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }} \\
+ \end{array}
+\]
+By the same way, the system coordinates become
+\[
+\begin{array}{c}
+ mL(\ddot x) =  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
+  - \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0) - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}  \\
+ \end{array}
+\]
+
+With the help of some relatively important inverse Laplace
+transformations:
+\[
+\begin{array}{c}
+ L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
+ L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
+ L(1) = \frac{1}{p} \\
+ \end{array}
+\]
+, we obtain
 \begin{align}
 m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
 \sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
@@ -1142,72 +1671,372 @@ t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  =
 (\omega _\alpha  t)} \right\}}
 \end{align}
 
+Introducing a \emph{dynamic friction kernel}
 \begin{equation}
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
+\label{introEquation:dynamicFrictionKernelDefinition}
+\end{equation}
+and \emph{a random force}
+\begin{equation}
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
+\label{introEquation:randomForceDefinition}
+\end{equation}
+the equation of motion can be rewritten as
+\begin{equation}
 m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
 (t)\dot x(t - \tau )d\tau }  + R(t)
 \label{introEuqation:GeneralizedLangevinDynamics}
 \end{equation}
-%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
-%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
+which is known as the \emph{generalized Langevin equation}.
+
+\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel}
+
+One may notice that $R(t)$ depends only on initial conditions, which
+implies it is completely deterministic within the context of a
+harmonic bath. However, it is easy to verify that $R(t)$ is totally
+uncorrelated to $x$ and $\dot x$,
 \[
-\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
-}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
+\begin{array}{l}
+ \left\langle {x(t)R(t)} \right\rangle  = 0, \\
+ \left\langle {\dot x(t)R(t)} \right\rangle  = 0. \\
+ \end{array}
 \]
-For an infinite harmonic bath, we can use the spectral density and
-an integral over frequencies.
+This property is what we expect from a truly random process. As long
+as the model, which is gaussian distribution in general, chosen for
+$R(t)$ is a truly random process, the stochastic nature of the GLE
+still remains.
 
+%dynamic friction kernel
+The convolution integral
 \[
-R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
-- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
-\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
-(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
 \]
-The random forces depend only on initial conditions.
+depends on the entire history of the evolution of $x$, which implies
+that the bath retains memory of previous motions. In other words,
+the bath requires a finite time to respond to change in the motion
+of the system. For a sluggish bath which responds slowly to changes
+in the system coordinate, we may regard $\xi(t)$ as a constant
+$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
+\[
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
+\]
+and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
+\[
+m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
+\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
+\]
+which can be used to describe dynamic caging effect. The other
+extreme is the bath that responds infinitely quickly to motions in
+the system. Thus, $\xi (t)$ can be taken as a $delta$ function in
+time:
+\[
+\xi (t) = 2\xi _0 \delta (t)
+\]
+Hence, the convolution integral becomes
+\[
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
+{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
+\]
+and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
+\begin{equation}
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
+x(t) + R(t) \label{introEquation:LangevinEquation}
+\end{equation}
+which is known as the Langevin equation. The static friction
+coefficient $\xi _0$ can either be calculated from spectral density
+or be determined by Stokes' law for regular shaped particles.A
+briefly review on calculating friction tensor for arbitrary shaped
+particles is given in Sec.~\ref{introSection:frictionTensor}.
 
 \subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
-So we can define a new set of coordinates,
+
+Defining a new set of coordinates,
 \[
 q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
 ^2 }}x(0)
-\]
-This makes
+\],
+we can rewrite $R(T)$ as
 \[
-R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
 \]
 And since the $q$ coordinates are harmonic oscillators,
 \[
-\begin{array}{l}
+\begin{array}{c}
+ \left\langle {q_\alpha ^2 } \right\rangle  = \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
  \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
  \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
+ \left\langle {R(t)R(0)} \right\rangle  = \sum\limits_\alpha  {\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle } }  \\
+  = \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
+  = kT\xi (t) \\
  \end{array}
+\]
+Thus, we recover the \emph{second fluctuation dissipation theorem}
+\begin{equation}
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
+\label{introEquation:secondFluctuationDissipation}.
+\end{equation}
+In effect, it acts as a constraint on the possible ways in which one
+can model the random force and friction kernel.
+
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
+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 resistance (friction) tensor respectively,
+while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $
+{\Xi^{rt} }$ is rotation-translation coupling tensor. When a
+particle moves in a fluid, it may experience friction force or
+torque along the opposite direction of the velocity or angular
+velocity,
+\[
+\left( \begin{array}{l}
+ F_R  \\
+ \tau _R  \\
+ \end{array} \right) =  - \left( {\begin{array}{*{20}c}
+   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
+   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
+\end{array}} \right)\left( \begin{array}{l}
+ v \\
+ w \\
+ \end{array} \right)
+\]
+where $F_r$ is the friction force and $\tau _R$ is the friction
+toque.
 
-\begin{align}
-\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
-{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
-(t)q_\beta  (0)} \right\rangle } }
-%
-&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
-\right\rangle \cos (\omega _\alpha  t)}
-%
-&= kT\xi (t)
-\end{align}
+\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape}
 
-\begin{equation}
-\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-\label{introEquation:secondFluctuationDissipation}
-\end{equation}
+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:hydroynamics}Hydrodynamics}
+Other non-spherical shape, such as cylinder and ellipsoid
+\textit{etc}, are widely used as reference for developing new
+hydrodynamics theory, because their properties can be calculated
+exactly. In 1936, Perrin extended Stokes's law to general ellipsoid,
+also called a triaxial ellipsoid, which is given in Cartesian
+coordinates by
+\[
+\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2
+}} = 1
+\]
+where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately,
+due to the complexity of the elliptic integral, only the ellipsoid
+with the restriction of two axes having to be equal, \textit{i.e.}
+prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved
+exactly. Introducing an elliptic integral parameter $S$ for prolate,
+\[
+S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
+} }}{b},
+\]
+and oblate,
+\[
+S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
+}}{a}
+\],
+one can write down the translational and rotational resistance
+tensors
+\[
+\begin{array}{l}
+ \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
+ \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
+ \end{array},
+\]
+and
+\[
+\begin{array}{l}
+ \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
+ \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
+ \end{array}.
+\]
 
-\subsection{\label{introSection:frictionTensor} Friction Tensor}
-\subsection{\label{introSection:analyticalApproach}Analytical
-Approach}
+\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape}
 
-\subsection{\label{introSection:approximationApproach}Approximation
-Approach}
+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}
 
-\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-Body}
+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
+\begin{equation}
+B = \left( {\begin{array}{*{20}c}
+   {B_{11} } &  \ldots  & {B_{1N} }  \\
+    \vdots  &  \ddots  &  \vdots   \\
+   {B_{N1} } &  \cdots  & {B_{NN} }  \\
+\end{array}} \right),
+\end{equation}
+where $B_{ij}$ is given by
+\[
+B_{ij}  = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij}
+)T_{ij}
+\]
+where $\delta _{ij}$ is Kronecker delta function. Inverting matrix
+$B$, we obtain
 
-\section{\label{introSection:correlationFunctions}Correlation Functions}
+\[
+C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
+   {C_{11} } &  \ldots  & {C_{1N} }  \\
+    \vdots  &  \ddots  &  \vdots   \\
+   {C_{N1} } &  \cdots  & {C_{NN} }  \\
+\end{array}} \right)
+\]
+, which can be partitioned into $N \times N$ $3 \times 3$ block
+$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$
+\[
+U_i  = \left( {\begin{array}{*{20}c}
+   0 & { - z_i } & {y_i }  \\
+   {z_i } & 0 & { - x_i }  \\
+   { - y_i } & {x_i } & 0  \\
+\end{array}} \right)
+\]
+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} } } , \\
+ \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}
+\label{introEquation:ResistanceTensorArbitraryOrigin}
+\end{equation}
+
+The resistance tensor depends on the origin to which they refer. The
+proper location for applying friction force is the center of
+resistance (reaction), at which the trace of rotational resistance
+tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of
+resistance is defined as an unique point of the rigid body at which
+the translation-rotation coupling tensor are symmetric,
+\begin{equation}
+\Xi^{tr}  = \left( {\Xi^{tr} } \right)^T
+\label{introEquation:definitionCR}
+\end{equation}
+Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin},
+we can easily find out that the translational resistance tensor is
+origin independent, while the rotational resistance tensor and
+translation-rotation coupling resistance tensor depend on the
+origin. Given resistance tensor at an arbitrary origin $O$, and a
+vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can
+obtain the resistance tensor at $P$ by
+\begin{equation}
+\begin{array}{l}
+ \Xi _P^{tt}  = \Xi _O^{tt}  \\
+ \Xi _P^{tr}  = \Xi _P^{rt}  = \Xi _O^{tr}  - U_{OP} \Xi _O^{tt}  \\
+ \Xi _P^{rr}  = \Xi _O^{rr}  - U_{OP} \Xi _O^{tt} U_{OP}  + \Xi _O^{tr} U_{OP}  - U_{OP} \Xi _O^{tr} ^{^T }  \\
+ \end{array}
+ \label{introEquation:resistanceTensorTransformation}
+\end{equation}
+where
+\[
+U_{OP}  = \left( {\begin{array}{*{20}c}
+   0 & { - z_{OP} } & {y_{OP} }  \\
+   {z_i } & 0 & { - x_{OP} }  \\
+   { - y_{OP} } & {x_{OP} } & 0  \\
+\end{array}} \right)
+\]
+Using Equations \ref{introEquation:definitionCR} and
+\ref{introEquation:resistanceTensorTransformation}, one can locate
+the position of center of resistance,
+\[
+\left( \begin{array}{l}
+ x_{OR}  \\
+ y_{OR}  \\
+ z_{OR}  \\
+ \end{array} \right) = \left( {\begin{array}{*{20}c}
+   {(\Xi _O^{rr} )_{yy}  + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} }  \\
+   { - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz}  + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} }  \\
+   { - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx}  + (\Xi _O^{rr} )_{yy} }  \\
+\end{array}} \right)^{ - 1} \left( \begin{array}{l}
+ (\Xi _O^{tr} )_{yz}  - (\Xi _O^{tr} )_{zy}  \\
+ (\Xi _O^{tr} )_{zx}  - (\Xi _O^{tr} )_{xz}  \\
+ (\Xi _O^{tr} )_{xy}  - (\Xi _O^{tr} )_{yx}  \\
+ \end{array} \right).
+\]
+where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
+joining center of resistance $R$ and origin $O$.
