--- trunk/tengDissertation/Introduction.tex	2006/04/11 03:38:09	2700
+++ trunk/tengDissertation/Introduction.tex	2006/06/05 21:24:52	2793
@@ -27,11 +27,11 @@ $F_ij$ be the force that particle $i$ exerts on partic
 \end{equation}
 A point mass interacting with other bodies moves with the
 acceleration along the direction of the force acting on it. Let
-$F_ij$ be the force that particle $i$ exerts on particle $j$, and
-$F_ji$ be the force that particle $j$ exerts on particle $i$.
+$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and
+$F_{ji}$ be the force that particle $j$ exerts on particle $i$.
 Newton��s third law states that
 \begin{equation}
-F_ij = -F_ji
+F_{ij} = -F_{ji}
  \label{introEquation:newtonThirdLaw}
 \end{equation}
 
@@ -93,7 +93,7 @@ the kinetic, $K$, and potential energies, $U$ \cite{to
 The actual trajectory, along which a dynamical system may move from
 one point to another within a specified time, is derived by finding
 the path which minimizes the time integral of the difference between
-the kinetic, $K$, and potential energies, $U$ \cite{tolman79}.
+the kinetic, $K$, and potential energies, $U$ \cite{Tolman1979}.
 \begin{equation}
 \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
 \label{introEquation:halmitonianPrinciple1}
@@ -189,7 +189,7 @@ known as the canonical equations of motions \cite{Gold
 Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
 Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
 equation of motion. Due to their symmetrical formula, they are also
-known as the canonical equations of motions \cite{Goldstein01}.
+known as the canonical equations of motions \cite{Goldstein2001}.
 
 An important difference between Lagrangian approach and the
 Hamiltonian approach is that the Lagrangian is considered to be a
@@ -200,7 +200,7 @@ equations\cite{Marion90}.
 appropriate for application to statistical mechanics and quantum
 mechanics, since it treats the coordinate and its time derivative as
 independent variables and it only works with 1st-order differential
-equations\cite{Marion90}.
+equations\cite{Marion1990}.
 
 In Newtonian Mechanics, a system described by conservative forces
 conserves the total energy \ref{introEquation:energyConservation}.
@@ -315,8 +315,7 @@ partition function like,
 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
@@ -394,8 +393,36 @@ distribution,
 \begin{equation}
 \rho  \propto e^{ - \beta H}
 \label{introEquation:densityAndHamiltonian}
+\end{equation}
+
+\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space}
+Lets consider a region in the phase space,
+\begin{equation}
+\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
+\end{equation}
+If this region is small enough, the density $\rho$ can be regarded
+as uniform over the whole phase space. Thus, the number of phase
+points inside this region is given by,
+\begin{equation}
+\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
+dp_1 } ..dp_f.
+\end{equation}
+
+\begin{equation}
+\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho
+\frac{d}{{dt}}(\delta v) = 0.
 \end{equation}
+With the help of stationary assumption
+(\ref{introEquation:stationary}), we obtain the principle of the
+\emph{conservation of extension in phase space},
+\begin{equation}
+\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
+...dq_f dp_1 } ..dp_f  = 0.
+\label{introEquation:volumePreserving}
+\end{equation}
 
+\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms}
+
 Liouville's theorem can be expresses in a variety of different forms
 which are convenient within different contexts. For any two function
 $F$ and $G$ of the coordinates and momenta of a system, the Poisson
@@ -431,7 +458,6 @@ expressed as
 \label{introEquation:liouvilleTheoremInOperator}
 \end{equation}
 
-
 \subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
 
 Various thermodynamic properties can be calculated from Molecular
@@ -444,7 +470,7 @@ statistical ensemble are identical \cite{Frenkel1996, 
 many-body system in Statistical Mechanics. Fortunately, Ergodic
 Hypothesis is proposed to make a connection between time average and
 ensemble average. It states that time average and average over the
-statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
+statistical ensemble are identical \cite{Frenkel1996, Leach2001}.
 \begin{equation}
 \langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
 \frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
@@ -458,7 +484,7 @@ reasonable, the Monte Carlo techniques\cite{metropolis
 a properly weighted statistical average. This allows the researcher
 freedom of choice when deciding how best to measure a given
 observable. In case an ensemble averaged approach sounds most
-reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be
+reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be
 utilized. Or if the system lends itself to a time averaging
 approach, the Molecular Dynamics techniques in
 Sec.~\ref{introSection:molecularDynamics} will be the best
@@ -472,10 +498,11 @@ issue. The velocity verlet method, which happens to be
 within the equations. Since 1990, geometric integrators, which
 preserve various phase-flow invariants such as symplectic structure,
 volume and time reversal symmetry, are developed to address this
-issue. The velocity verlet method, which happens to be a simple
-example of symplectic integrator, continues to gain its popularity
-in molecular dynamics community. This fact can be partly explained
-by its geometric nature.
+issue\cite{Dullweber1997, McLachlan1998, Leimkuhler1999}. The
+velocity verlet method, which happens to be a simple example of
+symplectic integrator, continues to gain its popularity in molecular
+dynamics community. This fact can be partly explained by its
+geometric nature.
 
 \subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
 A \emph{manifold} is an abstract mathematical space. It locally
@@ -539,28 +566,15 @@ Another generalization of Hamiltonian dynamics is Pois
 \end{equation}In this case, $f$ is
 called a \emph{Hamiltonian vector field}.
 
-Another generalization of Hamiltonian dynamics is Poisson Dynamics,
+Another generalization of Hamiltonian dynamics is Poisson
+Dynamics\cite{Olver1986},
 \begin{equation}
 \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}
 
-\subsection{\label{introSection:geometricProperties}Geometric Properties}
 Let $x(t)$ be the exact solution of the ODE system,
 \begin{equation}
 \frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}
@@ -570,20 +584,44 @@ space to itself. In most cases, it is not easy to find
 x(t+\tau) =\varphi_\tau(x(t))
 \]
 where $\tau$ is a fixed time step and $\varphi$ is a map from phase
-space to itself. In most cases, it is not easy to find the exact
-flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$,
-which is usually called integrator. The order of an integrator
-$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
-order $p$,
+space to itself. The flow has the continuous group property,
 \begin{equation}
+\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
++ \tau _2 } .
+\end{equation}
+In particular,
+\begin{equation}
+\varphi _\tau   \circ \varphi _{ - \tau }  = I
+\end{equation}
+Therefore, the exact flow is self-adjoint,
+\begin{equation}
+\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
+\end{equation}
+The exact flow can also be written in terms of the of an operator,
+\begin{equation}
+\varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
+}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
+\label{introEquation:exponentialOperator}
+\end{equation}
+
+In most cases, it is not easy to find the exact flow $\varphi_\tau$.
+Instead, we use a approximate map, $\psi_\tau$, which is usually
+called integrator. The order of an integrator $\psi_\tau$ is $p$, if
+the Taylor series of $\psi_\tau$ agree to order $p$,
+\begin{equation}
 \psi_tau(x) = x + \tau f(x) + O(\tau^{p+1})
 \end{equation}
 
-The hidden geometric properties of ODE and its flow play important
-roles in numerical studies. Let $\varphi$ be the flow of Hamiltonian
-vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies,
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
+
+The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE
+and its flow play important roles in numerical studies. Many of them
+can be found in systems which occur naturally in applications.
+
+Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is
+a \emph{symplectic} flow if it satisfies,
 \begin{equation}
-'\varphi^T J '\varphi = J.
+{\varphi '}^T J \varphi ' = J.
 \end{equation}
 According to Liouville's theorem, the symplectic volume is invariant
 under a Hamiltonian flow, which is the basis for classical
@@ -591,22 +629,62 @@ symplectomorphism. As to the Poisson system,
 field on a symplectic manifold can be shown to be a
 symplectomorphism. As to the Poisson system,
 \begin{equation}
-'\varphi ^T J '\varphi  = J \circ \varphi
+{\varphi '}^T J \varphi ' = J \circ \varphi
 \end{equation}
-is the property must be preserved by the integrator. It is possible
-to construct a \emph{volume-preserving} flow for a source free($
-\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi  =
-1$. Changing the variables $y = h(x)$ in a
-ODE\ref{introEquation:ODE} will result in a new system,
+is the property must be preserved by the integrator.
+
+It is possible to construct a \emph{volume-preserving} flow for a
+source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $
+\det d\varphi  = 1$. One can show easily that a symplectic flow will
+be volume-preserving.
+
+Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE}
+will result in a new system,
 \[
 \dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
 \]
 The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
 In other words, the flow of this vector field is reversible if and
-only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. When
-designing any numerical methods, one should always try to preserve
-the structural properties of the original ODE and its flow.
+only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
 
+A \emph{first integral}, or conserved quantity of a general
+differential function is a function $ G:R^{2d}  \to R^d $ which is
+constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
+\[
+\frac{{dG(x(t))}}{{dt}} = 0.
+\]
+Using chain rule, one may obtain,
+\[
+\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G,
+\]
+which is the condition for conserving \emph{first integral}. For a
+canonical Hamiltonian system, the time evolution of an arbitrary
+smooth function $G$ is given by,
+
+\begin{eqnarray}
+\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\
+                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
+\label{introEquation:firstIntegral1}
+\end{eqnarray}
+
+
+Using poisson bracket notion, Equation
+\ref{introEquation:firstIntegral1} can be rewritten as
+\[
+\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
+\]
+Therefore, the sufficient condition for $G$ to be the \emph{first
+integral} of a Hamiltonian system is
+\[
+\left\{ {G,H} \right\} = 0.
+\]
+As well known, the Hamiltonian (or energy) H of a Hamiltonian system
+is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
+0$.
+
+When designing any numerical methods, one should always try to
+preserve the structural properties of the original ODE and its flow.
+
 \subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
 A lot of well established and very effective numerical methods have
 been successful precisely because of their symplecticities even
@@ -621,197 +699,974 @@ Generating function tends to lead to methods which are
 \item Splitting methods
 \end{enumerate}
 
-Generating function tends to lead to methods which are cumbersome
-and difficult to use\cite{}. In dissipative systems, variational
-methods can capture the decay of energy accurately\cite{}. Since
-their geometrically unstable nature against non-Hamiltonian
-perturbations, ordinary implicit Runge-Kutta methods are not
-suitable for Hamiltonian system. Recently, various high-order
-explicit Runge--Kutta methods have been developed to overcome this
-instability \cite{}. However, due to computational penalty involved
-in implementing the Runge-Kutta methods, they do not attract too
-much attention from Molecular Dynamics community. Instead, splitting
-have been widely accepted since they exploit natural decompositions
-of the system\cite{Tuckerman92}. The main idea behind splitting
-methods is to decompose the discrete $\varphi_h$ as a composition of
-simpler flows,
+Generating function\cite{Channell1990} tends to lead to methods
+which are cumbersome and difficult to use. In dissipative systems,
+variational methods can capture the decay of energy
+accurately\cite{Kane2000}. Since their geometrically unstable nature
+against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
+methods are not suitable for Hamiltonian system. Recently, various
+high-order explicit Runge-Kutta methods
+\cite{Owren1992,Chen2003}have been developed to overcome this
+instability. However, due to computational penalty involved in
+implementing the Runge-Kutta methods, they do not attract too much
+attention from Molecular Dynamics community. Instead, splitting have
+been widely accepted since they exploit natural decompositions of
+the system\cite{Tuckerman1992, McLachlan1998}.
+
+\subsubsection{\label{introSection:splittingMethod}Splitting Method}
+
+The main idea behind splitting methods is to decompose the discrete
+$\varphi_h$ as a composition of simpler flows,
 \begin{equation}
 \varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
 \varphi _{h_n }
 \label{introEquation:FlowDecomposition}
 \end{equation}
 where each of the sub-flow is chosen such that each represent a
-simpler integration of the system. Let $\phi$ and $\psi$ both be
-symplectic maps, it is easy to show that any composition of
-symplectic flows yields a symplectic map,
+simpler integration of the system.
+
+Suppose that a Hamiltonian system takes the form,
+\[
+H = H_1 + H_2.
+\]
+Here, $H_1$ and $H_2$ may represent different physical processes of
+the system. For instance, they may relate to kinetic and potential
+energy respectively, which is a natural decomposition of the
+problem. If $H_1$ and $H_2$ can be integrated using exact flows
+$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first
+order is then given by the Lie-Trotter formula
 \begin{equation}
+\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
+\label{introEquation:firstOrderSplitting}
+\end{equation}
+where $\varphi _h$ is the result of applying the corresponding
+continuous $\varphi _i$ over a time $h$. By definition, as
+$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
+must follow that each operator $\varphi_i(t)$ is a symplectic map.
+It is easy to show that any composition of symplectic flows yields a
+symplectic map,
+\begin{equation}
 (\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
-'\phi ' = \phi '^T J\phi ' = J.
+'\phi ' = \phi '^T J\phi ' = J,
  \label{introEquation:SymplecticFlowComposition}
 \end{equation}
-Suppose that a Hamiltonian system has a form with $H = T + V$
+where $\phi$ and $\psi$ both are symplectic maps. Thus operator
+splitting in this context automatically generates a symplectic map.
 
-\section{\label{introSection:molecularDynamics}Molecular Dynamics}
+The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting})
+introduces local errors proportional to $h^2$, while Strang
+splitting gives a second-order decomposition,
+\begin{equation}
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
+_{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
+symmetric property,
+\begin{equation}
+\varphi _h^{ - 1} = \varphi _{ - h}.
+\label{introEquation:timeReversible}
+\end{equation}
 
-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.
+\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method}
+The classical equation for a system consisting of interacting
+particles can be written in Hamiltonian form,
+\[
+H = T + V
+\]
+where $T$ is the kinetic energy and $V$ is the potential energy.
+Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one
+obtains the following:
+\begin{align}
+q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
+    \frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
+\label{introEquation:Lp10a} \\%
+%
+\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
+    \biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
+\label{introEquation:Lp10b}
+\end{align}
+where $F(t)$ is the force at time $t$. This integration scheme is
+known as \emph{velocity verlet} which is
+symplectic(\ref{introEquation:SymplecticFlowComposition}),
+time-reversible(\ref{introEquation:timeReversible}) and
+volume-preserving (\ref{introEquation:volumePreserving}). These
+geometric properties attribute to its long-time stability and its
+popularity in the community. However, the most commonly used
+velocity verlet integration scheme is written as below,
+\begin{align}
+\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
+    \dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
+%
+q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
+    \label{introEquation:Lp9b}\\%
+%
+\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
+    \frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c}
+\end{align}
+From the preceding splitting, one can see that the integration of
+the equations of motion would follow:
+\begin{enumerate}
+\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
 
-\subsection{\label{introSec:mdInit}Initialization}
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
 
-\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
+\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move.
 
-\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
+\end{enumerate}
 
-A rigid body is a body in which the distance between any two given
-points of a rigid body remains constant regardless of external
-forces exerted on it. A rigid body therefore conserves its shape
-during its motion.
+Simply switching the order of splitting and composing, a new
+integrator, the \emph{position verlet} integrator, can be generated,
+\begin{align}
+\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) +
+\frac{{\Delta t}}{{2m}}\dot q(0)} \right], %
+\label{introEquation:positionVerlet1} \\%
+%
+q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
+q(\Delta t)} \right]. %
+ \label{introEquation:positionVerlet2}
+\end{align}
 
-Applications of dynamics of rigid bodies.
+\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
 
-\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}
-
-%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
-
-\section{\label{introSection:correlationFunctions}Correlation Functions}
-
-\section{\label{introSection:langevinDynamics}Langevin Dynamics}
-
-\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
-
-\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
-
+Baker-Campbell-Hausdorff formula can be used to determine the local
+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
 \begin{equation}
-H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
-\label{introEquation:bathGLE}
+\exp (hX + hY) = \exp (hZ)
 \end{equation}
-where $H_B$ is harmonic bath Hamiltonian,
+where
+\begin{equation}
+hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left(
+{[X,[X,Y]] + [Y,[Y,X]]} \right) +  \ldots .
+\end{equation}
+Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by
 \[
-H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
+[X,Y] = XY - YX .
 \]
-and $\Delta U$ is bilinear system-bath coupling,
+Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} 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 + \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
+procedure can be applied to general splitting,  of the form
+\begin{equation}
+\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
+order method. Yoshida proposed an elegant way to compose higher
+order methods based on symmetric splitting\cite{Yoshida1990}. Given
+a symmetric second order base method $ \varphi _h^{(2)} $, a
+fourth-order symmetric method can be constructed by composing,
 \[
-\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
+\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
+h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
 \]
-Completing the square,
+where $ \alpha  =  - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta
+= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric
+integrator $ \varphi _h^{(2n + 2)}$ can be composed by
+\begin{equation}
+\varphi _h^{(2n + 2)}  = \varphi _{\alpha h}^{(2n)}  \circ \varphi
+_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)}
+\end{equation}
+, if the weights are chosen as
 \[
-H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
-{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
-1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
+\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
 \]
-and putting it back into Eq.~\ref{introEquation:bathGLE},
+
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
+
+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
 \[
-H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
 \]
-where
-\[
-W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
-}}{{2m_\alpha  w_\alpha ^2 }}} x^2
-\]
-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{introEq: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}
+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.
+
+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{introSection:laplaceTransform}The Laplace Transform}
+\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 step 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} (see Fig.~\ref{introFig:pbc})
+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]{pbc.eps}
+\caption[An illustration of periodic boundary conditions]{A 2-D
+illustration of periodic boundary conditions. As one particle leaves
+the left of the simulation box, an image of it enters the right.}
+\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\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
+$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast
+multipole method}\cite{Greengard1987, Greengard1994}, 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\cite{Wolf1999}. 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]{shifted_coulomb.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:
 \[
-L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
+f_{ij} } } \right\rangle
 \]
+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{Powles1973}. 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{Allen1987}.
+
+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
 \[
-L(x + y) = L(x) + L(y)
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle.
 \]
+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:
 \[
-L(ax) = aL(x)
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
+\right\rangle } dt
 \]
-
+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:
 \[
-L(\dot x) = pL(x) - px(0)
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
+\right\rangle
 \]
-
+Here $u_{tot}$ is the net dipole of the entire system and is given
+by
 \[
-L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
+u_{tot} (t) = \sum\limits_i {u_i (t)}
 \]
+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}
+
+Rigid bodies are frequently involved in the modeling of different
+areas, from engineering, physics, to chemistry. For example,
+missiles and vehicle are usually modeled by rigid bodies.  The
+movement of the objects in 3D gaming engine or other physics
+simulator is governed by the rigid body dynamics. In molecular
+simulation, rigid body is used to simplify the model in
+protein-protein docking study\cite{Gray2003}.
+
+It is very important to develop stable and efficient methods to
+integrate the equations of motion of orientational degrees of
+freedom. Euler angles are the nature choice to describe the
+rotational degrees of freedom. However, due to its singularity, the
+numerical integration of corresponding equations of motion is very
+inefficient and inaccurate. Although an alternative integrator using
+different sets of Euler angles can overcome this
+difficulty\cite{Barojas1973}, the computational penalty and the lost
+of angular momentum conservation still remain. A singularity free
+representation utilizing quaternions was developed by Evans in
+1977\cite{Evans1977}. Unfortunately, this approach suffer from the
+nonseparable Hamiltonian resulted from quaternion representation,
+which prevents the symplectic algorithm to be utilized. Another
+different approach is to apply holonomic constraints to the atoms
+belonging to the rigid body. Each atom moves independently under the
+normal forces deriving from potential energy and constraint forces
+which are used to guarantee the rigidness. However, due to their
+iterative nature, SHAKE and Rattle algorithm converge very slowly
+when the number of constraint increases\cite{Ryckaert1977,
+Andersen1983}.
+
+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 $Q$ and re-formulating Hamiltonian's equation, a
+symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve
+the Hamiltonian system in a constraint manifold by iteratively
+satisfying the orthogonality constraint $Q_T Q = 1$. An alternative
+method using quaternion representation was developed by
+Omelyan\cite{Omelyan1998}. 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{Dullweber1997} in depth.
+
+\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
 \[
-L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
 \]
+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}
 
-Some relatively important transformation,
+Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
+\ref{introEquation:motionHamiltonianMomentum}), one can write down
+the equations of motion,
 \[
-L(\cos at) = \frac{p}{{p^2  + a^2 }}
+\begin{array}{c}
+ \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\
+ \frac{{dp}}{{dt}} =  - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
+ \frac{{dQ}}{{dt}} = PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
+ \frac{{dP}}{{dt}} =  - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\
+ \end{array}
 \]
 
+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. These two methods
+are proved to be equivalent. The holonomic constraint and equations
+of motions define a constraint manifold for rigid body
 \[
-L(\sin at) = \frac{a}{{p^2  + a^2 }}
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
+\right\}.
 \]
 
+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
 \[
-L(1) = \frac{1}{p}
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
 \]
+the mechanical system subject to a holonomic constraint manifold $M$
+can be re-formulated as a Hamiltonian system on the cotangent space
+\[
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
+1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
+\]
 
-First, the bath coordinates,
+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
 \[
-p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
-_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
-}}L(x)
+V(q,Q) = V(Q X_0 + q).
 \]
+Hence, the force and torque are given by
 \[
-L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
-px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
 \]
-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,
+and
+\[
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
+\]
+respectively.
 
-\begin{align}
-m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
+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}
+   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
+\[
+\hat vu = v \times u
+\]
+
+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{Kol1997, Omelyan1998}.
+
+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}
+   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
+\[
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
+Q(0)e^{\Delta tR_1 }
+\]
+with
+\[
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
+   0 & 0 & 0  \\
+   0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
+   0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
+\]
+To reduce the cost of computing expensive functions in $e^{\Delta
+tR_1 }$, we can use Cayley transformation,
+\[
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
+)
+\]
+The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
+manner.
+
+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{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,
+\[
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
+\]
+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{eqnarray*}
+ 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{eqnarray*}
+
+
+Applying Laplace transform to the bath coordinates, we obtain
+\begin{eqnarray*}
+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{eqnarray*}
+
+By the same way, the system coordinates become
+\begin{eqnarray*}
+ mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
+  & & \mbox{} - \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{eqnarray*}
+
+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
+\[
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
 \sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
 }}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
 _\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
 - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
 (\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
 _\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
-%
-&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
+\]
+\[
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
 {\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
 }}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
 t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
@@ -819,72 +1674,378 @@ t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  =
 \omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
 \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
 (\omega _\alpha  t)} \right\}}
-\end{align}
+\]
 
+Introducing a \emph{dynamic friction kernel}
 \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. $W(x) =  - kT\ln p(x)$
-\[
 \xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
 }}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
-\]
-For an infinite harmonic bath, we can use the spectral density and
-an integral over frequencies.
-
-\[
+\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)
+(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}
+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 $\dot x$,
+\[
+\begin{array}{l}
+ \left\langle {x(t)R(t)} \right\rangle  = 0, \\
+ \left\langle {\dot x(t)R(t)} \right\rangle  = 0. \\
+ \end{array}
 \]
-The random forces depend only on initial conditions.
+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
+\[
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
+\]
+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{eqnarray*}
+ \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{eqnarray*}
+
+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.
+
+\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance 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.
+
+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\cite{Perrin1934, Perrin1936}
+\[
+\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2
+}} = 1
+\]
+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
+\[
 \begin{array}{l}
- \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
- \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
- \end{array}
+ \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
+ \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
+ \end{array},
 \]
+and
+\[
+\begin{array}{l}
+ \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
+ \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
+ \end{array}.
+\]
 
-\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:resistanceTensorRegularArbitrary}The Resistance 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{Wegener1979} as well as the intrinsic coupling between
+translational and rotational motion of rigid body, 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 where the molecule of interest was modeled by
+combinations of spheres(beads)\cite{Carrasco1999} 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}
-\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-\label{introEquation:secondFluctuationDissipation}
+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{Rotne1969} and improved by
+Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977},
+\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}
 
-\section{\label{introSection:hydroynamics}Hydrodynamics}
+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
 
-\subsection{\label{introSection:frictionTensor} Friction Tensor}
-\subsection{\label{introSection:analyticalApproach}Analytical
-Approach}
+\[
+C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
+   {C_{11} } &  \ldots  & {C_{1N} }  \\
+    \vdots  &  \ddots  &  \vdots   \\
+   {C_{N1} } &  \cdots  & {C_{NN} }  \\
+\end{array}} \right)
+\]
+, 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}
 
-\subsection{\label{introSection:approximationApproach}Approximation
-Approach}
+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,
+\begin{eqnarray*}
+ \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) \\
+\end{eqnarray*}
 
-\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-Body}
+
+
+where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
+joining center of resistance $R$ and origin $O$.
