--- trunk/tengDissertation/Introduction.tex	2006/04/13 04:47:47	2706
+++ trunk/tengDissertation/Introduction.tex	2006/06/23 20:21:54	2881
@@ -6,7 +6,7 @@ behind classical mechanics. Firstly, One can determine
 Closely related to Classical Mechanics, Molecular Dynamics
 simulations are carried out by integrating the equations of motion
 for a given system of particles. There are three fundamental ideas
-behind classical mechanics. Firstly, One can determine the state of
+behind classical mechanics. Firstly, one can determine the state of
 a mechanical system at any time of interest; Secondly, all the
 mechanical properties of the system at that time can be determined
 by combining the knowledge of the properties of the system with the
@@ -17,19 +17,19 @@ Newton¡¯s first law defines a class of inertial frames
 \subsection{\label{introSection:newtonian}Newtonian Mechanics}
 The discovery of Newton's three laws of mechanics which govern the
 motion of particles is the foundation of the classical mechanics.
-Newton¡¯s first law defines a class of inertial frames. Inertial
+Newton's first law defines a class of inertial frames. Inertial
 frames are reference frames where a particle not interacting with
 other bodies will move with constant speed in the same direction.
-With respect to inertial frames Newton¡¯s second law has the form
+With respect to inertial frames, Newton's second law has the form
 \begin{equation}
-F = \frac {dp}{dt} = \frac {mv}{dt}
+F = \frac {dp}{dt} = \frac {mdv}{dt}
 \label{introEquation:newtonSecondLaw}
 \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 $i$.
-Newton¡¯s third law states that
+Newton's third law states that
 \begin{equation}
 F_{ij} = -F_{ji}
  \label{introEquation:newtonThirdLaw}
@@ -46,7 +46,7 @@ N \equiv r \times F \label{introEquation:torqueDefinit
 \end{equation}
 The torque $\tau$ with respect to the same origin is defined to be
 \begin{equation}
-N \equiv r \times F \label{introEquation:torqueDefinition}
+\tau \equiv r \times F \label{introEquation:torqueDefinition}
 \end{equation}
 Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
 \[
@@ -59,7 +59,7 @@ thus,
 \]
 thus,
 \begin{equation}
-\dot L = r \times \dot p = N
+\dot L = r \times \dot p = \tau
 \end{equation}
 If there are no external torques acting on a body, the angular
 momentum of it is conserved. The last conservation theorem state
@@ -68,42 +68,38 @@ scheme for rigid body \cite{Dullweber1997}.
 \end{equation}
  is conserved. All of these conserved quantities are
 important factors to determine the quality of numerical integration
-scheme for rigid body \cite{Dullweber1997}.
+schemes for rigid bodies \cite{Dullweber1997}.
 
 \subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
 
-Newtonian Mechanics suffers from two important limitations: it
-describes their motion in special cartesian coordinate systems.
-Another limitation of Newtonian mechanics becomes obvious when we
-try to describe systems with large numbers of particles. It becomes
-very difficult to predict the properties of the system by carrying
-out calculations involving the each individual interaction between
-all the particles, even if we know all of the details of the
-interaction. In order to overcome some of the practical difficulties
-which arise in attempts to apply Newton's equation to complex
-system, alternative procedures may be developed.
+Newtonian Mechanics suffers from two important limitations: motions
+can only be described in cartesian coordinate systems. Moreover, It
+become impossible to predict analytically the properties of the
+system even if we know all of the details of the interaction. In
+order to overcome some of the practical difficulties which arise in
+attempts to apply Newton's equation to complex system, approximate
+numerical procedures may be developed.
 
-\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
-Principle}
+\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's
+Principle}}
 
 Hamilton introduced the dynamical principle upon which it is
-possible to base all of mechanics and, indeed, most of classical
-physics. Hamilton's Principle may be stated as follow,
+possible to base all of mechanics and most of classical physics.
+Hamilton's Principle may be stated as follows,
 
 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, $U$.
 \begin{equation}
 \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
 \label{introEquation:halmitonianPrinciple1}
 \end{equation}
 
 For simple mechanical systems, where the forces acting on the
-different part are derivable from a potential and the velocities are
-small compared with that of light, the Lagrangian function $L$ can
-be define as the difference between the kinetic energy of the system
-and its potential energy,
+different parts are derivable from a potential, the Lagrangian
+function $L$ can be defined as the difference between the kinetic
+energy of the system and its potential energy,
 \begin{equation}
 L \equiv K - U = L(q_i ,\dot q_i ) ,
 \label{introEquation:lagrangianDef}
@@ -114,11 +110,11 @@ then Eq.~\ref{introEquation:halmitonianPrinciple1} bec
 \label{introEquation:halmitonianPrinciple2}
 \end{equation}
 
-\subsubsection{\label{introSection:equationOfMotionLagrangian}The
-Equations of Motion in Lagrangian Mechanics}
+\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The
+Equations of Motion in Lagrangian Mechanics}}
 
-For a holonomic system of $f$ degrees of freedom, the equations of
-motion in the Lagrangian form is
+For a system of $f$ degrees of freedom, the equations of motion in
+the Lagrangian form is
 \begin{equation}
 \frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
 \frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
@@ -132,8 +128,7 @@ independent of generalized velocities, the generalized
 Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
 introduced by William Rowan Hamilton in 1833 as a re-formulation of
 classical mechanics. If the potential energy of a system is
-independent of generalized velocities, the generalized momenta can
-be defined as
+independent of velocities, the momenta can be defined as
 \begin{equation}
 p_i = \frac{\partial L}{\partial \dot q_i}
 \label{introEquation:generalizedMomenta}
@@ -172,11 +167,11 @@ find
 By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
 find
 \begin{equation}
-\frac{{\partial H}}{{\partial p_k }} = q_k
+\frac{{\partial H}}{{\partial p_k }} = \dot {q_k}
 \label{introEquation:motionHamiltonianCoordinate}
 \end{equation}
 \begin{equation}
-\frac{{\partial H}}{{\partial q_k }} =  - p_k
+\frac{{\partial H}}{{\partial q_k }} =  - \dot {p_k}
 \label{introEquation:motionHamiltonianMomentum}
 \end{equation}
 and
@@ -189,18 +184,17 @@ 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
-function of the generalized velocities $\dot q_i$ and the
-generalized coordinates $q_i$, while the Hamiltonian is considered
-to be a function of the generalized momenta $p_i$ and the conjugate
-generalized coordinate $q_i$. Hamiltonian Mechanics is more
-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}.
+function of the generalized velocities $\dot q_i$ and coordinates
+$q_i$, while the Hamiltonian is considered to be a function of the
+generalized momenta $p_i$ and the conjugate coordinates $q_i$.
+Hamiltonian Mechanics is more 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{Marion1990}.
 
 In Newtonian Mechanics, a system described by conservative forces
 conserves the total energy \ref{introEquation:energyConservation}.
@@ -230,13 +224,14 @@ momentum variables. Consider a dynamic system in a car
 possible states. Each possible state of the system corresponds to
 one unique point in the phase space. For mechanical systems, the
 phase space usually consists of all possible values of position and
-momentum variables. Consider a dynamic system in a cartesian space,
-where each of the $6f$ coordinates and momenta is assigned to one of
-$6f$ mutually orthogonal axes, the phase space of this system is a
-$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 ,
-\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and
-momenta is a phase space vector.
+momentum variables. Consider a dynamic system of $f$ particles in a
+cartesian space, where each of the $6f$ coordinates and momenta is
+assigned to one of $6f$ mutually orthogonal axes, the phase space of
+this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots
+,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$
+coordinates and momenta is a phase space vector.
 
+%%%fix me
 A microscopic state or microstate of a classical system is
 specification of the complete phase space vector of a system at any
 instant in time. An ensemble is defined as a collection of systems
@@ -257,15 +252,15 @@ space. The density of distribution for an ensemble wit
 regions of the phase space. The condition of an ensemble at any time
 can be regarded as appropriately specified by the density $\rho$
 with which representative points are distributed over the phase
-space. The density of distribution for an ensemble with $f$ degrees
-of freedom is defined as,
+space. The density distribution for an ensemble with $f$ degrees of
+freedom is defined as,
 \begin{equation}
 \rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
 \label{introEquation:densityDistribution}
 \end{equation}
 Governed by the principles of mechanics, the phase points change
-their value which would change the density at any time at phase
-space. Hence, the density of distribution is also to be taken as a
+their locations which would change the density at any time at phase
+space. Hence, the density distribution is also to be taken as a
 function of the time.
 
 The number of systems $\delta N$ at time $t$ can be determined by,
@@ -273,10 +268,10 @@ Assuming a large enough population of systems are expl
 \delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
 \label{introEquation:deltaN}
 \end{equation}
-Assuming a large enough population of systems are exploited, we can
-sufficiently approximate $\delta N$ without introducing
-discontinuity when we go from one region in the phase space to
-another. By integrating over the whole phase space,
+Assuming a large enough population of systems, we can sufficiently
+approximate $\delta N$ without introducing discontinuity when we go
+from one region in the phase space to another. By integrating over
+the whole phase space,
 \begin{equation}
 N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
 \label{introEquation:totalNumberSystem}
@@ -288,16 +283,16 @@ With the help of Equation(\ref{introEquation:unitProba
 {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
 \label{introEquation:unitProbability}
 \end{equation}
-With the help of Equation(\ref{introEquation:unitProbability}) and
-the knowledge of the system, it is possible to calculate the average
+With the help of Eq.~\ref{introEquation:unitProbability} and the
+knowledge of the system, it is possible to calculate the average
 value of any desired quantity which depends on the coordinates and
 momenta of the system. Even when the dynamics of the real system is
 complex, or stochastic, or even discontinuous, the average
-properties of the ensemble of possibilities as a whole may still
-remain well defined. For a classical system in thermal equilibrium
-with its environment, the ensemble average of a mechanical quantity,
-$\langle A(q , p) \rangle_t$, takes the form of an integral over the
-phase space of the system,
+properties of the ensemble of possibilities as a whole remaining
+well defined. For a classical system in thermal equilibrium with its
+environment, the ensemble average of a mechanical quantity, $\langle
+A(q , p) \rangle_t$, takes the form of an integral over the phase
+space of the system,
 \begin{equation}
 \langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
 (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
@@ -307,17 +302,17 @@ parameters, such as temperature \textit{etc}, partitio
 
 There are several different types of ensembles with different
 statistical characteristics. As a function of macroscopic
-parameters, such as temperature \textit{etc}, partition function can
-be used to describe the statistical properties of a system in
+parameters, such as temperature \textit{etc}, the partition function
+can be used to describe the statistical properties of a system in
 thermodynamic equilibrium.
 
 As an ensemble of systems, each of which is known to be thermally
-isolated and conserve energy, Microcanonical ensemble(NVE) has a
-partition function like,
+isolated and conserve energy, the Microcanonical ensemble (NVE) has
+a partition function like,
 \begin{equation}
 \Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
 \end{equation}
-A canonical ensemble(NVT)is an ensemble of systems, each of which
+A canonical ensemble (NVT)is an ensemble of systems, each of which
 can share its energy with a large heat reservoir. The distribution
 of the total energy amongst the possible dynamical states is given
 by the partition function,
@@ -326,11 +321,12 @@ TS$. Since most experiment are carried out under const
 \label{introEquation:NVTPartition}
 \end{equation}
 Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
-TS$. Since most experiment are carried out under constant pressure
-condition, isothermal-isobaric ensemble(NPT) play a very important
-role in molecular simulation. The isothermal-isobaric ensemble allow
-the system to exchange energy with a heat bath of temperature $T$
-and to change the volume as well. Its partition function is given as
+TS$. Since most experiments are carried out under constant pressure
+condition, the isothermal-isobaric ensemble (NPT) plays a very
+important role in molecular simulations. The isothermal-isobaric
+ensemble allow the system to exchange energy with a heat bath of
+temperature $T$ and to change the volume as well. Its partition
+function is given as
 \begin{equation}
 \Delta (N,P,T) =  - e^{\beta G}.
  \label{introEquation:NPTPartition}
@@ -339,13 +335,13 @@ The Liouville's theorem is the foundation on which sta
 
 \subsection{\label{introSection:liouville}Liouville's theorem}
 
-The Liouville's theorem is the foundation on which statistical
-mechanics rests. It describes the time evolution of phase space
+Liouville's theorem is the foundation on which statistical mechanics
+rests. It describes the time evolution of the phase space
 distribution function. In order to calculate the rate of change of
-$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we
-consider the two faces perpendicular to the $q_1$ axis, which are
-located at $q_1$ and $q_1 + \delta q_1$, the number of phase points
-leaving the opposite face is given by the expression,
+$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider
+the two faces perpendicular to the $q_1$ axis, which are located at
+$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the
+opposite face is given by the expression,
 \begin{equation}
 \left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
 \right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
@@ -369,7 +365,7 @@ divining $ \delta q_1  \ldots \delta q_f \delta p_1  \
 + \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
 \end{equation}
 which cancels the first terms of the right hand side. Furthermore,
-divining $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
+dividing $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
 p_f $ in both sides, we can write out Liouville's theorem in a
 simple form,
 \begin{equation}
@@ -381,8 +377,9 @@ statistical mechanics, since the number of particles i
 
 Liouville's theorem states that the distribution function is
 constant along any trajectory in phase space. In classical
-statistical mechanics, since the number of particles in the system
-is huge, we may be able to believe the system is stationary,
+statistical mechanics, since the number of members in an ensemble is
+huge and constant, we can assume the local density has no reason
+(other than classical mechanics) to change,
 \begin{equation}
 \frac{{\partial \rho }}{{\partial t}} = 0.
 \label{introEquation:stationary}
@@ -395,14 +392,14 @@ distribution,
 \label{introEquation:densityAndHamiltonian}
 \end{equation}
 
-\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space}
+\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{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,
+as uniform over the whole integral. 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.
@@ -414,14 +411,14 @@ With the help of stationary assumption
 \end{equation}
 With the help of stationary assumption
 (\ref{introEquation:stationary}), we obtain the principle of the
-\emph{conservation of extension in phase space},
+\emph{conservation of volume 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}
+\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{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
@@ -435,10 +432,10 @@ Substituting equations of motion in Hamiltonian formal
 \label{introEquation:poissonBracket}
 \end{equation}
 Substituting equations of motion in Hamiltonian formalism(
-\ref{introEquation:motionHamiltonianCoordinate} ,
-\ref{introEquation:motionHamiltonianMomentum} ) into
-(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's
-theorem using Poisson bracket notion,
+Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
+Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into
+(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite
+Liouville's theorem using Poisson bracket notion,
 \begin{equation}
 \left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
 {\rho ,H} \right\}.
@@ -463,14 +460,14 @@ simulation and the quality of the underlying model. Ho
 Various thermodynamic properties can be calculated from Molecular
 Dynamics simulation. By comparing experimental values with the
 calculated properties, one can determine the accuracy of the
-simulation and the quality of the underlying model. However, both of
-experiment and computer simulation are usually performed during a
+simulation and the quality of the underlying model. However, both
+experiments and computer simulations are usually performed during a
 certain time interval and the measurements are averaged over a
 period of them which is different from the average behavior of
-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}.
+many-body system in Statistical Mechanics. Fortunately, the Ergodic
+Hypothesis makes a connection between time average and the ensemble
+average. It states that the time average and average over the
+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
@@ -484,67 +481,57 @@ 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
 choice\cite{Frenkel1996}.
 
 \section{\label{introSection:geometricIntegratos}Geometric Integrators}
-A variety of numerical integrators were proposed to simulate the
-motions. They usually begin with an initial conditionals and move
-the objects in the direction governed by the differential equations.
-However, most of them ignore the hidden physical law contained
-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.
+A variety of numerical integrators have been proposed to simulate
+the motions of atoms in MD simulation. They usually begin with
+initial conditionals and move the objects in the direction governed
+by the differential equations. However, most of them ignore the
+hidden physical laws contained 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\cite{Dullweber1997, McLachlan1998,
+Leimkuhler1999}. The velocity Verlet method, which happens to be a
+simple example of symplectic integrator, continues to gain
+popularity in the 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
-looks like Euclidean space, but when viewed globally, it may have
-more complicate structure. A good example of manifold is the surface
-of Earth. It seems to be flat locally, but it is round if viewed as
-a whole. A \emph{differentiable manifold} (also known as
-\emph{smooth manifold}) is a manifold with an open cover in which
-the covering neighborhoods are all smoothly isomorphic to one
-another. In other words,it is possible to apply calculus on
-\emph{differentiable manifold}. A \emph{symplectic manifold} is
-defined as a pair $(M, \omega)$ which consisting of a
+\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds}
+A \emph{manifold} is an abstract mathematical space. It looks
+locally like Euclidean space, but when viewed globally, it may have
+more complicated structure. A good example of manifold is the
+surface of Earth. It seems to be flat locally, but it is round if
+viewed as a whole. A \emph{differentiable manifold} (also known as
+\emph{smooth manifold}) is a manifold on which it is possible to
+apply calculus on \emph{differentiable manifold}. A \emph{symplectic
+manifold} is defined as a pair $(M, \omega)$ which consists of a
 \emph{differentiable manifold} $M$ and a close, non-degenerated,
 bilinear symplectic form, $\omega$. A symplectic form on a vector
 space $V$ is a function $\omega(x, y)$ which satisfies
 $\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
 \lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
-$\omega(x, x) = 0$. Cross product operation in vector field is an
-example of symplectic form.
+$\omega(x, x) = 0$. The cross product operation in vector field is
+an example of symplectic form.
 
-One of the motivations to study \emph{symplectic manifold} in
+One of the motivations to study \emph{symplectic manifolds} in
 Hamiltonian Mechanics is that a symplectic manifold can represent
 all possible configurations of the system and the phase space of the
 system can be described by it's cotangent bundle. Every symplectic
 manifold is even dimensional. For instance, in Hamilton equations,
 coordinate and momentum always appear in pairs.
 
-Let  $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map
-\[
-f : M \rightarrow N
-\]
-is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and
-the \emph{pullback} of $\eta$ under f is equal to $\omega$.
-Canonical transformation is an example of symplectomorphism in
-classical mechanics.
-
 \subsection{\label{introSection:ODE}Ordinary Differential Equations}
 
-For a ordinary differential system defined as
+For an ordinary differential system defined as
 \begin{equation}
 \dot x = f(x)
 \end{equation}
-where $x = x(q,p)^T$, this system is canonical Hamiltonian, if
+where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if
 \begin{equation}
 f(r) = J\nabla _x H(r).
 \end{equation}
@@ -565,27 +552,13 @@ 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}
 
 Let $x(t)$ be the exact solution of the ODE system,
@@ -618,18 +591,18 @@ Instead, we use a approximate map, $\psi_\tau$, which 
 \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
+Instead, we use an 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})
+\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1})
 \end{equation}
 
 \subsection{\label{introSection:geometricProperties}Geometric Properties}
 
-The hidden geometric properties of ODE and its flow play important
-roles in numerical studies. Many of them can be found in systems
-which occur naturally in applications.
+The hidden geometric properties\cite{Budd1999, Marsden1998} of an
+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,
@@ -644,15 +617,15 @@ is the property must be preserved by the integrator.
 \begin{equation}
 {\varphi '}^T J \varphi ' = J \circ \varphi
 \end{equation}
-is the property must be preserved by the integrator.
+is the property that 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 $
+source free ODE ($ \nabla \cdot f = 0 $), 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,
+Changing the variables $y = h(x)$ in an ODE
+(Eq.~\ref{introEquation:ODE}) will result in a new system,
 \[
 \dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
 \]
@@ -673,13 +646,14 @@ smooth function $G$ is given by,
 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{equation}
-\begin{array}{c}
- \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)). \\
- \end{array}
+
+\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{equation}
+\end{eqnarray}
+
+
 Using poisson bracket notion, Equation
 \ref{introEquation:firstIntegral1} can be rewritten as
 \[
@@ -694,16 +668,15 @@ is a \emph{first integral}, which is due to the fact $
 is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
 0$.
 
-
- When designing any numerical methods, one should always try to
+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
 though this fact was not recognized when they were first
-constructed. The most famous example is leapfrog methods in
-molecular dynamics. In general, symplectic integrators can be
+constructed. The most famous example is the Verlet-leapfrog method
+in molecular dynamics. In general, symplectic integrators can be
 constructed using one of four different methods.
 \begin{enumerate}
 \item Generating functions
@@ -712,20 +685,21 @@ 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. In dissipative systems, variational methods
-can capture the decay of energy accurately. 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
+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{Tuckerman92}.
+implementing the Runge-Kutta methods, they have not attracted much
+attention from the Molecular Dynamics community. Instead, splitting
+methods have been widely accepted since they exploit natural
+decompositions of the system\cite{Tuckerman1992, McLachlan1998}.
 
-\subsubsection{\label{introSection:splittingMethod}Splitting Method}
+\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
 
 The main idea behind splitting methods is to decompose the discrete
 $\varphi_h$ as a composition of simpler flows,
@@ -746,7 +720,7 @@ order is then given by the Lie-Trotter formula
 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
+order expression is then given by the Lie-Trotter formula
 \begin{equation}
 \varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
 \label{introEquation:firstOrderSplitting}
@@ -772,22 +746,22 @@ which has a local error proportional to $h^3$. Sprang 
 \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,
+which has a local error proportional to $h^3$. The 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}
+\end{equation},appendixFig:architecture
 
-\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method}
+\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the 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
+Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one
 obtains the following:
 \begin{align}
 q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
@@ -814,7 +788,7 @@ q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{
     \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}
+    \frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c}
 \end{align}
 From the preceding splitting, one can see that the integration of
 the equations of motion would follow:
@@ -823,13 +797,14 @@ the equations of motion would follow:
 
 \item Use the half step velocities to move positions one whole step, $\Delta t$.
 
-\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move.
+\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move.
 
 \item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
 \end{enumerate}
 
-Simply switching the order of splitting and composing, a new
-integrator, the \emph{position verlet} integrator, can be generated,
+By simply switching the order of the propagators in the 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], %
@@ -837,16 +812,16 @@ 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}
+\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
 
-Baker-Campbell-Hausdorff formula can be used to determine the local
-error of splitting method in terms of commutator of the
+The Baker-Campbell-Hausdorff formula can be used to determine the
+local error of splitting method in terms of the 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
+the sub-flow. For operators $hX$ and $hY$ which are associated with
+$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have
 \begin{equation}
 \exp (hX + hY) = \exp (hZ)
 \end{equation}
@@ -859,26 +834,25 @@ Applying Baker-Campbell-Hausdorff formula to Sprang sp
 \[
 [X,Y] = XY - YX .
 \]
-Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we
-can obtain
+Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
+to the 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
+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
+procedure can be applied to a 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. Given a symmetric second
-order base method $ \varphi _h^{(2)} $, a fourth-order symmetric
-method can be constructed by composing,
+A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher
+order methods. 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,
 \[
 \varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
 h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
@@ -888,9 +862,9 @@ _{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)}
 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)}
+_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)},
 \end{equation}
-, if the weights are chosen as
+if the weights are chosen as
 \[
 \alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
 \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
@@ -898,63 +872,342 @@ 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.
+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{introSec:mdInit}Initialization}
+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{introSection:production}
+will discusse issues in production run.
+Sec.~\ref{introSection:Analysis} provides the theoretical tools for
+trajectory analysis.
 
-\subsection{\label{introSec:forceEvaluation}Force Evaluation}
+\subsection{\label{introSec:initialSystemSettings}Initialization}
 
-\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
+\subsubsection{\textbf{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 molecules with known
+structure, some important information is missing. For example, a
+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 configuration. For instance, when studying transport
+phenomenon in membrane systems, we may prepare the lipids in a
+bilayer structure instead of placing lipids randomly in solvent,
+since we are not interested in the slow self-aggregation process.
+
+\subsubsection{\textbf{Minimization}}
+
+It is quite possible that some of molecules in the system from
+preliminary preparation may be overlapping with each other. This
+close proximity leads to high initial 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 strongly anharmonic. Thus, it is often used to refine
+structure from crystallographic data. Relied on the gradient or
+hessian, advanced methods like Newton-Raphson converge rapidly to a
+local minimum, but become unstable if the energy surface is far from
+quadratic. Another factor that 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 limits on computer memory to store the hessian
+matrix and the computing power needed to diagonalized these
+matrices, most Newton-Raphson methods can not be used with very
+large systems.
+
+\subsubsection{\textbf{Heating}}
+
+Typically, Heating is performed by assigning random velocities
+according to a Maxwell-Boltzman distribution for a desired
+temperature. Beginning at a lower temperature and gradually
+increasing the temperature by assigning larger 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. To do this, the net linear momentum and angular momentum of
+the system is shifted to zero after each resampling from the Maxwell
+-Boltzman distribution.
+
+\subsubsection{\textbf{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}
+
+The 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 by sampling correctly and
+efficiently from the relevant thermodynamic ensemble.
+
+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
+algorithmic tricks.
+
+A natural approach to avoid system size issues is to represent the
+bulk behavior by a finite number of the particles. However, this
+approach will suffer from the surface effect at the edges of the
+simulation. To offset this, \textit{Periodic boundary conditions}
+(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 spherical 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 simple radial potential to ensure the potential curve go
+smoothly to zero at the cutoff radius. The cutoff strategy works
+well for Lennard-Jones interaction because of its short range
+nature. However, simply truncating the electrostatic interaction
+with the use of cutoffs has been shown to lead to severe artifacts
+in simulations. The Ewald summation, in which the slowly decaying
+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 the
+\emph{fast multipole method}\cite{Greengard1987, Greengard1994},
+which treats Coulombic interactions exactly at short range, and
+approximate the potential at long range through multipolar
+expansion. In spite of their wide acceptance at the molecular
+simulation community, these two methods are difficult to implement
+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 have become applied to
+monitor the motions of molecules. Although the dynamics of the
+system can be described qualitatively from animation, quantitative
+trajectory analysis are more useful. According to the principles of
+Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics},
+one can compute thermodynamic properties, analyze fluctuations of
+structural parameters, and investigate time-dependent processes of
+the molecule from the trajectories.
+
+\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}
+
+Thermodynamic 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}\textbf{Structural Properties}}
+
+Structural Properties of a simple fluid can be described by a set of
+distribution functions. Among these functions,the \emph{pair
+distribution function}, also known as \emph{radial distribution
+function}, is of most fundamental importance to liquid theory.
+Experimentally, 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 experimental results can serve as a criterion
+to justify the correctness of a liquid model. Moreover, various
+equilibrium thermodynamic and structural properties can also be
+expressed in terms of radial distribution function \cite{Allen1987}.
+
+The 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
+\[
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
+(r)}{\rho}.
+\]
+Note that the delta function can be replaced by a histogram in
+computer simulation. Peaks in $g(r)$ represent solvent shells, and
+the height of these peaks gradually decreases to 1 as the liquid of
+large distance approaches the bulk density.
+
+
+\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
+Properties}}
+
+Time-dependent properties are usually calculated using \emph{time
+correlation functions}, which correlate random variables $A$ and $B$
+at two different times,
+\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 an \emph{autocorrelation function}. One example
+of an auto correlation function is the 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 the velocity autocorrelation
+function, which is averaging over time origins and over all the
+atoms, the dipole autocorrelation functions 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}
 
 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{Gray03}}.
+simulator is governed by rigid body dynamics. In molecular
+simulations, rigid bodies are used to simplify protein-protein
+docking studies\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{},
-the computational penalty and the lost of angular momentum
-conservation still remain. A singularity free representation
-utilizing quaternions was developed by Evans in 1977. Unfortunately,
-this approach suffer from the nonseparable Hamiltonian resulted from
+integrate the equations of motion for orientational degrees of
+freedom. Euler angles are the natural choice to describe the
+rotational degrees of freedom. However, due to $\frac {1}{sin
+\theta}$ singularities, the numerical integration of corresponding
+equations of motion is very inefficient and inaccurate. Although an
+alternative integrator using multiple sets of Euler angles can
+overcome this difficulty\cite{Barojas1973}, the computational
+penalty and the loss of angular momentum conservation still remain.
+A singularity-free representation utilizing quaternions was
+developed by Evans in 1977\cite{Evans1977}. Unfortunately, this
+approach uses a nonseparable Hamiltonian resulting from the
 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.
-
-The break through in geometric literature suggests that, in order to
+rigidness. However, due to their iterative nature, the SHAKE and
+Rattle algorithms also converge very slowly when the number of
+constraints increases\cite{Ryckaert1977, Andersen1983}.
+
+A 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
-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
-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{}.
+symplectic structure of the flow. By introducing a conjugate
+momentum to the 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 the quaternion representation was
+developed by Omelyan\cite{Omelyan1998}. However, both of these
+methods are iterative and inefficient. In this section, we descibe a
+symplectic Lie-Poisson integrator 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}
-
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
+The motion of a 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 ].
@@ -967,247 +1220,546 @@ constrained Hamiltonian equation subjects to a holonom
 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,
+constrained Hamiltonian equation is subjected 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,
+which is used to ensure rotation matrix's unitarity. 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}
 
 Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
 \ref{introEquation:motionHamiltonianMomentum}), one can write down
 the equations of motion,
+
+\begin{eqnarray}
+ \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{eqnarray}
+
+In general, there are two ways to satisfy the holonomic constraints.
+We can use a constraint force provided by a Lagrange multiplier on
+the normal manifold to keep the motion on constraint space. Or we
+can simply evolve the system on the constraint manifold. These two
+methods have been proved to be equivalent. The holonomic constraint
+and equations of motions define a constraint manifold for rigid
+bodies
 \[
-\begin{array}{c}
- \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\
- \frac{{dp}}{{dt}} =  - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
- \frac{{dQ}}{{dt}} = PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
- \frac{{dP}}{{dt}} =  - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\
- \end{array}
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
+\right\}.
 \]
 
-
+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. By introducing
 \[
-M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1}  + J^{ - 1} P^t Q = 0}
-\right\} .
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
 \]
+the mechanical system subject to a holonomic constraint manifold $M$
+can be re-formulated as a Hamiltonian system on the cotangent space
+\[
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
+1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
+\]
 
-\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}
-
+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}
-H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
-\label{introEquation:bathGLE}
+X_i^{lab} = Q X_i + q.
 \end{equation}
-where $H_B$ is harmonic bath Hamiltonian,
+Therefore, potential energy $V(q,Q)$ is defined by
 \[
-H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
+V(q,Q) = V(Q X_0 + q).
 \]
-and $\Delta U$ is bilinear system-bath coupling,
+Hence, the force and torque are given by
 \[
-\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
 \]
-Completing the square,
+and
 \[
-H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
-{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
-1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
 \]
-and putting it back into Eq.~\ref{introEquation:bathGLE},
+respectively.
+
+As a common choice to describe the rotation dynamics of the rigid
+body, the angular momentum on the body fixed 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,
 \[
-H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}
+\begin{array}{l}
+ \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
+ Q^T Q = 1 \\
+ \end{array}
 \]
-where
+
+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
 \[
-W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
-}}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\hat vu = v \times u
 \]
-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}
+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 eliminates the requirement of iterations which can
+not be avoided in other methods\cite{Kol1997, Omelyan1998}.
 
-\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
-
+Applying the 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
 \[
-L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
+\dot Q = Qskew(I^{ - 1} \pi )
 \]
 
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
+Lie-Poisson Integrator for Free Rigid Body}
+
+If there are no external forces exerted on the rigid body, the only
+contribution to the rotational motion is from the kinetic energy
+(the first term of \ref{introEquation:bodyAngularMotion}). The free
+rigid body is an example of a 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
 \[
-L(x + y) = L(x) + L(y)
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
+Q(0)e^{\Delta tR_1 }
 \]
-
+with
 \[
-L(ax) = aL(x)
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
+   0 & 0 & 0  \\
+   0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
+   0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
 \]
-
+To reduce the cost of computing expensive functions in $e^{\Delta
+tR_1 }$, we can use Cayley transformation to obtain a single-aixs
+propagator,
 \[
-L(\dot x) = pL(x) - px(0)
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
+)
 \]
-
+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
 \[
-L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
+(\pi _1 ).
 \]
-
+By concatenating the propagators corresponding to these five terms,
+we can obtain an symplectic integrator,
 \[
-L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
+\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
+\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
+\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
+_1 }.
 \]
 
-Some relatively important transformation,
+The non-canonical Lie-Poisson bracket ${F, G}$ of two function
+$F(\pi )$ and $G(\pi )$ is defined by
 \[
-L(\cos at) = \frac{p}{{p^2  + a^2 }}
+\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
+)
 \]
-
+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
 \[
-L(\sin at) = \frac{a}{{p^2  + a^2 }}
+\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
+}}{2})\pi
 \]
+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 both extremely efficient and
+stable. These properties 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,
 \[
-L(1) = \frac{1}{p}
+H = T(p,\pi ) + V(q,Q)
 \]
-
-First, the bath coordinates,
+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 position and velocity propagators,
 \[
-p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
-_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
-}}L(x)
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
 \]
+Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
+sub-propagators which corresponding to force and torque
+respectively,
 \[
-L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
-px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
+\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
+_{\Delta t/2,\tau }.
 \]
-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,
+Since the associated operators of $\varphi _{\Delta t/2,F} $ and
+$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order
+inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the
+kinetic energy 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 propagators are given by
+\[
+\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
+_{\Delta t,T^r }.
+\]
+Finally, we obtain the overall symplectic propagators for freely
+moving rigid bodies
+\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}
 
-\begin{align}
-m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
+\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. A brief derivation of generalized
+Langevin equation will be given first. Following 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}
+
+A 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 the Generalized Langevin
+Dynamics (GLE). 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 $x$, $m$ is the mass associated
+with this degree of freedom, $H_B$ is a 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 a 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
+coordinates ($x_ \alpha$) and the system 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,
+\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,the 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, by applying the 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 the 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
+\begin{eqnarray*}
+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
+_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\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\}}
+\end{eqnarray*}
+\begin{eqnarray*}
+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\{
-{\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\}}
-\end{align}
-
+t)\dot x(t - \tau )d} \tau }  \\
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\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\}}
+\end{eqnarray*}
+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}\textbf{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 chosen for $R(t)$ was a gaussian distribution in
+general, 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.
-
-\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
-So we can define a new set of coordinates,
+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
 \[
-q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
-^2 }}x(0)
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
 \]
-This makes
+and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
 \[
-R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
+m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
+\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
 \]
-And since the $q$ coordinates are harmonic oscillators,
+which can be used to describe the effect of dynamic caging in
+viscous solvents. 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:
 \[
-\begin{array}{l}
- \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
- \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
- \end{array}
+\xi (t) = 2\xi _0 \delta (t)
 \]
-
-\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}
-
+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}
-\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-\label{introEquation:secondFluctuationDissipation}
+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}.
 
-\section{\label{introSection:hydroynamics}Hydrodynamics}
+\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
 
-\subsection{\label{introSection:frictionTensor} Friction Tensor}
-\subsection{\label{introSection:analyticalApproach}Analytical
-Approach}
+Defining a new set of coordinates,
+\[
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
+^2 }}x(0)
+\],
+we can rewrite $R(T)$ as
+\[
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
+\]
+And since the $q$ coordinates are harmonic oscillators,
 
-\subsection{\label{introSection:approximationApproach}Approximation
-Approach}
+\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*}
 
-\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-Body}
-
-\section{\label{introSection:correlationFunctions}Correlation Functions}
+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.