--- trunk/tengDissertation/Introduction.tex	2006/04/10 05:35:55	2699
+++ trunk/tengDissertation/Introduction.tex	2006/07/18 16:42:49	2950
@@ -3,38 +3,38 @@ Closely related to Classical Mechanics, Molecular Dyna
 \section{\label{introSection:classicalMechanics}Classical
 Mechanics}
 
-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
-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
-specification of this state; Finally, the specification of the state
-when further combine with the laws of mechanics will also be
-sufficient to predict the future behavior of the system.
+Using equations of motion derived from 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 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 specification of this state; Finally, the
+specification of the state when further combined with the laws of
+mechanics will also be sufficient to predict the future behavior of
+the system.
 
 \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
+$F_{ij}$ 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}
-
 Conservation laws of Newtonian Mechanics play very important roles
 in solving mechanics problems. The linear momentum of a particle is
 conserved if it is free or it experiences no force. The second
@@ -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},
 \[
@@ -59,66 +59,60 @@ thus,
 \]
 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
-that if all forces are conservative, Energy
-\begin{equation}E = T + V \label{introEquation:energyConservation}
+that if all forces are conservative, energy is conserved,
+\begin{equation}E = T + V. \label{introEquation:energyConservation}
 \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}.
+All of these conserved quantities are important factors to determine
+the quality of numerical integration 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 an important limitation: motion can
+only be described in cartesian coordinate systems which make it
+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 systems, 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,
-
-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}.
+possible to base all of mechanics and most of classical physics.
+Hamilton's Principle may be stated as follows: the 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 $U$,
 \begin{equation}
-\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
+\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 ) ,
+L \equiv K - U = L(q_i ,\dot q_i ).
 \label{introEquation:lagrangianDef}
 \end{equation}
-then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
+Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
 \begin{equation}
-\delta \int_{t_1 }^{t_2 } {L dt = 0} ,
+\delta \int_{t_1 }^{t_2 } {L dt = 0} .
 \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 +126,7 @@ 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}
@@ -143,7 +136,6 @@ p_i  = \frac{{\partial L}}{{\partial q_i }}
 p_i  = \frac{{\partial L}}{{\partial q_i }}
 \label{introEquation:generalizedMomentaDot}
 \end{equation}
-
 With the help of the generalized momenta, we may now define a new
 quantity $H$ by the equation
 \begin{equation}
@@ -151,32 +143,30 @@ $L$ is the Lagrangian function for the system.
 \label{introEquation:hamiltonianDefByLagrangian}
 \end{equation}
 where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
-$L$ is the Lagrangian function for the system.
-
-Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
-one can obtain
+$L$ is the Lagrangian function for the system. Differentiating
+Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain
 \begin{equation}
 dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
 \frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
 L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
-L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
+L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1}
 \end{equation}
-Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
-second and fourth terms in the parentheses cancel. Therefore,
+Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second
+and fourth terms in the parentheses cancel. Therefore,
 Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
 \begin{equation}
 dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
-\right)}  - \frac{{\partial L}}{{\partial t}}dt
+\right)}  - \frac{{\partial L}}{{\partial t}}dt .
 \label{introEquation:diffHamiltonian2}
 \end{equation}
 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
@@ -185,34 +175,31 @@ t}}
 t}}
 \label{introEquation:motionHamiltonianTime}
 \end{equation}
-
-Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
+where 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}.
-It follows that Hamilton's equations of motion conserve the total
-Hamiltonian.
+conserves the total energy
+(Eq.~\ref{introEquation:energyConservation}). It follows that
+Hamilton's equations of motion conserve the total Hamiltonian
 \begin{equation}
 \frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
 H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
 }}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
 H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
 \frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
-q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian}
+q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian}
 \end{equation}
 
 \section{\label{introSection:statisticalMechanics}Statistical
@@ -221,101 +208,286 @@ Statistical Mechanics concepts presented in this disse
 The thermodynamic behaviors and properties of Molecular Dynamics
 simulation are governed by the principle of Statistical Mechanics.
 The following section will give a brief introduction to some of the
-Statistical Mechanics concepts presented in this dissertation.
+Statistical Mechanics concepts and theorems presented in this
+dissertation.
 
-\subsection{\label{introSection:ensemble}Ensemble and Phase Space}
+\subsection{\label{introSection:ensemble}Phase Space and Ensemble}
 
-\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
+Mathematically, phase space is the space which represents all
+possible states of a system. 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 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, $x
+=
+(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
+\over q} _1 , \ldots
+,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
+\over q} _f
+,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
+\over p} _1  \ldots
+,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
+\over p} _f )$ , with a unique set of values of $6f$ coordinates and
+momenta is a phase space vector.
+%%%fix me
 
-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
-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}.
+In statistical mechanics, 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 distribution for an ensemble with $f$ degrees of
+freedom is defined as,
 \begin{equation}
-\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty }
-\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma
-{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq
+\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
+\label{introEquation:densityDistribution}
 \end{equation}
-where $\langle A \rangle_t$ is an equilibrium value of a physical
-quantity and $\rho (p(t), q(t))$ is the equilibrium distribution
-function. If an observation is averaged over a sufficiently long
-time (longer than relaxation time), all accessible microstates in
-phase space are assumed to be equally probed, giving 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 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}.
+Governed by the principles of mechanics, the phase points change
+their locations which changes 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,
+\begin{equation}
+\delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
+\label{introEquation:deltaN}
+\end{equation}
+Assuming enough copies of the 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}
+\end{equation}
+gives us an expression for the total number of copies. Hence, the
+probability per unit volume in the phase space can be obtained by,
+\begin{equation}
+\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int
+{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
+\label{introEquation:unitProbability}
+\end{equation}
+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 are
+complex, or stochastic, or even discontinuous, the average
+properties of the ensemble of possibilities as a whole 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,
+\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
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
+\label{introEquation:ensembelAverage}
+\end{equation}
 
+\subsection{\label{introSection:liouville}Liouville's theorem}
+
+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 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
+}}\delta q_1 } \right)\delta q_2  \ldots \delta q_f \delta p_1
+\ldots \delta p_f .
+\end{equation}
+Summing all over the phase space, we obtain
+\begin{equation}
+\frac{{d(\delta N)}}{{dt}} =  - \sum\limits_{i = 1}^f {\left[ {\rho
+\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} +
+\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left(
+{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  + \frac{{\partial
+\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1
+\ldots \delta q_f \delta p_1  \ldots \delta p_f .
+\end{equation}
+Differentiating the equations of motion in Hamiltonian formalism
+(\ref{introEquation:motionHamiltonianCoordinate},
+\ref{introEquation:motionHamiltonianMomentum}), we can show,
+\begin{equation}
+\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }}
++ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
+\end{equation}
+which cancels the first terms of the right hand side. Furthermore,
+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}
+\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f
+{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  +
+\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
+\label{introEquation:liouvilleTheorem}
+\end{equation}
+Liouville's theorem states that the distribution function is
+constant along any trajectory in phase space. In classical
+statistical mechanics, since the number of system copies 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}
+\end{equation}
+In such stationary system, the density of distribution $\rho$ can be
+connected to the Hamiltonian $H$ through Maxwell-Boltzmann
+distribution,
+\begin{equation}
+\rho  \propto e^{ - \beta H}.
+\label{introEquation:densityAndHamiltonian}
+\end{equation}
+
+\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 integral. Thus, the number of phase points
+inside this region is given by,
+\begin{eqnarray}
+\delta N &=& \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f,\\
+\frac{{d(\delta N)}}{{dt}} &=& \frac{{d\rho }}{{dt}}\delta v + \rho
+\frac{d}{{dt}}(\delta v) = 0.
+\end{eqnarray}
+With the help of the stationary assumption
+(Eq.~\ref{introEquation:stationary}), we obtain the principle of
+\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}\textbf{Liouville's Theorem in Other Forms}}
+
+Liouville's theorem can be expressed 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
+bracket $\{F,G\}$ is defined as
+\begin{equation}
+\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial
+F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} -
+\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial
+q_i }}} \right)}.
+\label{introEquation:poissonBracket}
+\end{equation}
+Substituting equations of motion in Hamiltonian formalism
+(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\}.
+\label{introEquation:liouvilleTheromInPoissin}
+\end{equation}
+Moreover, the Liouville operator is defined as
+\begin{equation}
+iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial
+p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial
+H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)}
+\label{introEquation:liouvilleOperator}
+\end{equation}
+In terms of Liouville operator, Liouville's equation can also be
+expressed as
+\begin{equation}
+\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - iL\rho
+\label{introEquation:liouvilleTheoremInOperator}
+\end{equation}
+which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$.
+\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
+
+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
+experiments and computer simulations are usually performed during a
+certain time interval and the measurements are averaged over a
+period of time which is different from the average behavior of
+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
+{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp
+\end{equation}
+where $\langle  A(q , p) \rangle_t$ is an equilibrium value of a
+physical quantity and $\rho (p(t), q(t))$ is the equilibrium
+distribution function. If an observation is averaged over a
+sufficiently long time (longer than the relaxation time), all
+accessible microstates in phase space are assumed to be equally
+probed, giving 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 methods\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 conditions 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, were
+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
-\emph{differentiable manifold} $M$ and a close, non-degenerated,
+\subsection{\label{introSection:symplecticManifold}Manifolds and Bundles}
+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.\cite{Hirsch1997} A \emph{symplectic manifold} is
+defined as a pair $(M, \omega)$ which consists of a
+\emph{differentiable manifold} $M$ and a close, non-degenerate,
 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.
+$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in
+vector field is an example of symplectic form.
+Given vector spaces $V$ and $W$ over same field $F$, $f: V \to W$ is a linear transformation if
+\begin{eqnarray*}
+f(x+y) & = & f(x) + f(y) \\
+f(ax) & = & af(x)
+\end{eqnarray*}
+are always satisfied for any two vectors $x$ and $y$ in $V$ and any scalar $a$ in $F$. One can define the dual vector space $V^*$ of $V$ if any two built-in linear transformations $\phi$ and $\psi$ in $V^*$ satisfy the following definition of addition and scalar multiplication:
+\begin{eqnarray*}
+(\phi+\psi)(x) & = & \phi(x)+\psi(x) \\
+(a\phi)(x) & = & a \phi(x)
+\end{eqnarray*}
+for all $a$ in $F$ and $x$ in $V$. For a manifold $M$, one can define a tangent vector of a tangent space $TM_q$ at every point $q$
+\begin{equation}
+\dot q = \mathop {\lim }\limits_{t \to 0} \frac{{\phi (t) - \phi (0)}}{t}
+\end{equation}
+where $\phi(0)=q$ and $\phi(t) \in M$. One may also define a cotangent space $T^*M_q$ as the dual space of the tangent space $TM_q$. The tangent space and the cotangent space are isomorphic to each other, since they are both real vector spaces with same dimension.
+The union of tangent spaces at every point of $M$ is called the tangent bundle of $M$ and is denoted by $TM$, while cotangent bundle $T^*M$ is defined as the union of the cotangent spaces to $M$.\cite{Jost2002} For a Hamiltonian system with configuration manifold $V$, the $(q,\dot q)$ phase space is the tangent bundle of the configuration manifold $V$, while the cotangent bundle is represented by $(q,p)$.
 
-One of the motivations to study \emph{symplectic manifold} 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
-\begin{equation}
-f(r) = J\nabla _x H(r).
 \end{equation}
-$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric
-matrix
+where $x = x(q,p)$, this system is a canonical Hamiltonian, if
+$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian
+function and $J$ is the skew-symmetric matrix
 \begin{equation}
 J = \left( {\begin{array}{*{20}c}
    0 & I  \\
@@ -326,85 +498,119 @@ system can be rewritten as,
 where $I$ is an identity matrix. Using this notation, Hamiltonian
 system can be rewritten as,
 \begin{equation}
-\frac{d}{{dt}}x = J\nabla _x H(x)
+\frac{d}{{dt}}x = J\nabla _x H(x).
 \label{introEquation:compactHamiltonian}
 \end{equation}In this case, $f$ is
-called a \emph{Hamiltonian vector field}.
-
-Another generalization of Hamiltonian dynamics is Poisson Dynamics,
+called a \emph{Hamiltonian vector field}. 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.
+where the most obvious change being that matrix $J$ now depends on
+$x$.
+
+\subsection{\label{introSection:exactFlow}Exact Propagator}
+
+Let $x(t)$ be the exact solution of the ODE
+system,
 \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)
+\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE}
+\end{equation} we can
+define its exact propagator $\varphi_\tau$:
+\[ x(t+\tau)
+=\varphi_\tau(x(t))
+\]
+where $\tau$ is a fixed time step and $\varphi$ is a map from phase
+space to itself. The propagator 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}
-H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2
-}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right)
+\varphi _\tau   \circ \varphi _{ - \tau }  = I
 \end{equation}
-
-\subsection{\label{introSection:geometricProperties}Geometric Properties}
-Let $x(t)$ be the exact solution of the ODE system,
+Therefore, the exact propagator is self-adjoint,
 \begin{equation}
-\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}
+\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
 \end{equation}
-The exact flow(solution) $\varphi_\tau$ is defined by
-\[
-x(t+\tau) =\varphi_\tau(x(t))
-\]
-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
+In most cases, it is not easy to find the exact propagator
+$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$,
+which is usually called an 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}
 
-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 an
+ODE and its propagator play important roles in numerical studies.
+Many of them can be found in systems which occur naturally in
+applications. Let $\varphi$ be the propagator of Hamiltonian vector
+field, $\varphi$ is a \emph{symplectic} propagator 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
-statistical mechanics. Furthermore, the flow of a Hamiltonian vector
-field on a symplectic manifold can be shown to be a
+under a Hamiltonian propagator, which is the basis for classical
+statistical mechanics. Furthermore, the propagator of a Hamiltonian
+vector 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 that must be preserved by the integrator. It is
+possible to construct a \emph{volume-preserving} propagator for a
+source free ODE ($ \nabla \cdot f = 0 $), if the propagator
+satisfies $ \det d\varphi  = 1$. One can show easily that a
+symplectic propagator will be volume-preserving. Changing the
+variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will
+result in a new system,
 \[
 \dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
 \]
 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.
+In other words, the propagator of this vector field is reversible if
+and only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
+conserved quantity of a general differential function is a function
+$ 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 the chain rule, one may obtain,
+\[
+\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G,
+\]
+which is the condition for conserved quantities. 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) \notag\\
+                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)).
+\label{introEquation:firstIntegral1}
+\end{eqnarray}
+Using poisson bracket notion, Eq.~\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 a conserved
+quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As
+is well known, the Hamiltonian (or energy) H of a Hamiltonian system
+is a conserved quantity, 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
+propagator.
 
 \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
+been successful precisely because of their symplectic nature 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
@@ -412,273 +618,1044 @@ constructed using one of four different methods.
 \item Runge-Kutta methods
 \item Splitting methods
 \end{enumerate}
+Generating functions\cite{Channell1990} tend 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 they are geometrically unstable
+against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
+methods are not suitable for Hamiltonian
+system.\cite{Cartwright1992} 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 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{McLachlan1998,
+Tuckerman1992}
 
-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,
+\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 propagators,
 \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,
+where each of the sub-propagator is chosen such that each represent
+a 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
+propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a
+simple first order expression is then given by the Lie-Trotter
+formula\cite{Trotter1959}
 \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 propagators
+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.
+The Lie-Trotter
+splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces
+local errors proportional to $h^2$, while the Strang splitting gives
+a second-order decomposition,\cite{Strang1968}
+\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 $h^3$. The Strang
+splitting's popularity in molecular simulation community attribute
+to its symmetric property,
+\begin{equation}
+\varphi _h^{ - 1} = \varphi _{ - h}.
+\label{introEquation:timeReversible}
+\end{equation}
 
+\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 the 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(Eq.~\ref{introEquation:SymplecticFlowComposition}),
+time-reversible(Eq.~\ref{introEquation:timeReversible}) and
+volume-preserving (Eq.~\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(t)]. \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.
 
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
 
-\section{\label{introSection:molecularDynamics}Molecular Dynamics}
+\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move.
 
-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.
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
+\end{enumerate}
+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], %
+\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}
 
-\subsection{\label{introSec:mdInit}Initialization}
+\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
 
-\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
+The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used
+to determine the local error of a splitting method in terms of the
+commutator of the
+operators associated
+with the sub-propagator. 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}
+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 commutator of operator $X$ and $Y$ given by
+\[
+[X,Y] = XY - YX .
+\]
+Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
+to the Strang 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 Strang splitting is proportional to $h^3$. The same
+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}
+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,
+\[
+\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
+h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
+\]
+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
+\[
+\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
+\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
+\]
 
-\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
-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.
+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 a
+Hamiltonian system of $N$ particles 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.
 
-Applications of dynamics of rigid bodies.
+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}
+ discusses issues of production runs.
+Sec.~\ref{introSection:Analysis} provides the theoretical tools for
+analysis of trajectories.
 
-\subsection{\label{introSection:lieAlgebra}Lie Algebra}
+\subsection{\label{introSec:initialSystemSettings}Initialization}
 
-\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
+\subsubsection{\textbf{Preliminary preparation}}
 
-\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
+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
+structures, 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 interactions,
+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.
 
-%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
+\subsubsection{\textbf{Minimization}}
 
-\section{\label{introSection:correlationFunctions}Correlation Functions}
+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, the steepest descent method is extremely robust when
+systems are strongly anharmonic. Thus, it is often used to refine
+structures from crystallographic data. Relying on the 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 diagonalize these matrices,
+most Newton-Raphson methods can not be used with very large systems.
 
-\section{\label{introSection:langevinDynamics}Langevin Dynamics}
+\subsubsection{\textbf{Heating}}
 
-\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
+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 setting the temperature of the system to a final temperature
+at which the simulation will be conducted. In the 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.
 
-\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
+\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 mean 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 makes 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 surface effects
+at the edges of the simulation. To offset this, \textit{Periodic
+boundary conditions} (see Fig.~\ref{introFig:pbc}) were 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 cutoffs 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 a 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 advantage of
+fast Fourier transform (FFT) techniques 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}
-H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
-\label{introEquation:bathGLE}
+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 $H_B$ is harmonic bath Hamiltonian,
-\[
-H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
-\]
-and $\Delta U$ is bilinear system-bath coupling,
-\[
-\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
-\]
-Completing the square,
-\[
-H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
-{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
-1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
-\]
-and putting it back into Eq.~\ref{introEquation:bathGLE},
-\[
-H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-w_\alpha ^2 }}x} \right)^2 } \right\}}
-\]
-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 $\alpha$ 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}
 
-\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
+\subsection{\label{introSection:Analysis} Analysis}
 
-\[
-L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
-\]
+According to the principles of
+Statistical Mechanics in
+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:
 \[
-L(x + y) = L(x) + L(y)
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
+f_{ij} } } \right\rangle
 \]
+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 functions 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 the 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
+\begin{equation}
+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}.
+\end{equation}
+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
+functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation
+function which is directly related to transport properties of
+molecular liquids:
+\begin{equation}
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
+\right\rangle } dt
+\end{equation}
+where $D$ is diffusion constant. Unlike the velocity autocorrelation
+function, which is averaged over time origins and over all the
+atoms, the dipole autocorrelation functions is calculated for the
+entire system. The dipole autocorrelation function is given by:
+\begin{equation}
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
+\right\rangle
+\end{equation}
+Here $u_{tot}$ is the net dipole of the entire system and is given
+by
+\begin{equation}
+u_{tot} (t) = \sum\limits_i {u_i (t)}.
+\end{equation}
+In principle, many time correlation functions can be related to
+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 the molecular dipole
+fluctuation at each frequency using the following relationship:
+\begin{equation}
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
+i2\pi vt} dt}.
+\end{equation}
+
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+
+Rigid bodies are frequently involved in the modeling of different
+areas, including engineering, physics and chemistry. For example,
+missiles and vehicles are usually modeled by rigid bodies.  The
+movement of the objects in 3D gaming engines or other physics
+simulators 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 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 these 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 used a nonseparable Hamiltonian
+resulting from the quaternion representation, which prevented the
+symplectic algorithm from being 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,
+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 propagator. 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 bodies developed by
+Dullweber and his coworkers\cite{Dullweber1997} in depth.
+
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
+The Hamiltonian of a rigid body is given by
+\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 vector 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
 \[
-L(ax) = aL(x)
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
 \]
-
+where $I_{ii}$ is the diagonal element of the inertia tensor. This
+constrained Hamiltonian equation is subjected to a holonomic
+constraint,
+\begin{equation}
+Q^T Q = 1, \label{introEquation:orthogonalConstraint}
+\end{equation}
+which is used to ensure the rotation matrix's unitarity. Using
+Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~
+\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}
+Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and
+using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain,
+\begin{equation}
+Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
+\label{introEquation:RBFirstOrderConstraint}
+\end{equation}
+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 the 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
 \[
-L(\dot x) = pL(x) - px(0)
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
+\right\}.
 \]
-
+Unfortunately, this constraint manifold is not $T^* SO(3)$ which is
+a symplectic manifold on Lie rotation group $SO(3)$. However, it
+turns out that under symplectic transformation, the cotangent space
+and the phase space are diffeomorphic. By introducing
 \[
-L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
 \]
-
+the mechanical system subjected to a holonomic constraint manifold $M$
+can be re-formulated as a Hamiltonian system on the cotangent space
 \[
-L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
+1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
 \]
-
-Some relatively important transformation,
+For a body fixed vector $X_i$ with respect to the center of mass of
+the rigid body, its corresponding lab fixed vector $X_i^{lab}$  is
+given as
+\begin{equation}
+X_i^{lab} = Q X_i + q.
+\end{equation}
+Therefore, potential energy $V(q,Q)$ is defined by
 \[
-L(\cos at) = \frac{p}{{p^2  + a^2 }}
+V(q,Q) = V(Q X_0 + q).
 \]
-
+Hence, the force and torque are given by
 \[
-L(\sin at) = \frac{a}{{p^2  + a^2 }}
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
 \]
-
+and
 \[
-L(1) = \frac{1}{p}
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
 \]
-
-First, the bath coordinates,
+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}
+ \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
+ \dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
+ \end{array}
+ \label{introEqaution:RBMotionPI}
+\end{equation}
+as well as holonomic constraints $\Pi J^{ - 1}  + J^{ - 1} \Pi ^t  =
+0$ and $Q^T Q = 1$. 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
 \[
-p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
-_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
-}}L(x)
+\hat vu = v \times u.
 \]
+Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
+matrix,
+\begin{eqnarray}
+(\dot \Pi  - \dot \Pi ^T )&= &(\Pi  - \Pi ^T )(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
+& & + \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{eqnarray}
+Since $\Lambda$ is symmetric, the last term of
+Eq.~\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} Applying the hat-map isomorphism, we obtain the
+equation of motion for angular momentum
+\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
 \[
-L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
-px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
+\dot Q = Qskew(I^{ - 1} \pi ).
 \]
-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,
 
-\begin{align}
-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\}}
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
+Lie-Poisson Integrator for Free Rigid Bodies}
+
+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
+Eq.~\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}
+with
+\[ R_1  = \left( {\begin{array}{*{20}c}
+   0 & 0 & 0  \\
+   0 & 0 & {\pi _1 }  \\
+   0 & { - \pi _1 } & 0  \\
+\end{array}} \right).
+\]
+The solutions of Eq.~\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 the Cayley transformation to obtain a
+single-aixs propagator,
+\begin{eqnarray*}
+e^{\Delta tR_1 }  & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta
+tR_1 ) \\
 %
-&= - \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}
+& \approx & \left( \begin{array}{ccc}
+1 & 0 & 0 \\
+0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4}  & -\frac{\theta}{1+
+\theta^2 / 4} \\
+0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 +
+\theta^2 / 4}
+\end{array}
+\right).
+\end{eqnarray*}
+The propagators 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 into five terms
+\[
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
+(\pi _1 ).
+\]
+By concatenating the propagators 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 functions $F(\pi )$ 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}.\cite{McLachlan1993} 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 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, $H = T(p,\pi ) + V(q,Q)$. The equations
+of motion corresponding to potential energy and kinetic energy are
+listed in Table~\ref{introTable:rbEquations}.
+\begin{table}
+\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
+\label{introTable:rbEquations}
+\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,
+\[
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
+\]
+Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
+sub-propagators which corresponding to force and torque
+respectively,
+\[
+\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
+_{\Delta t/2,\tau }.
+\]
+Since the associated operators of $\varphi _{\Delta t/2,F} $ and
+$\circ \varphi _{\Delta t/2,\tau }$ 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}
-m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
-(t)\dot x(t - \tau )d\tau }  + R(t)
-\label{introEuqation:GeneralizedLangevinDynamics}
+T(p,\pi ) =T^t (p) + T^r (\pi ).
 \end{equation}
-%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
-%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
+where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
+defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore,
+the corresponding propagators are given by
 \[
-\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
-}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
+\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
+_{\Delta t,T^r }.
 \]
-For an infinite harmonic bath, we can use the spectral density and
-an integral over frequencies.
+Finally, we obtain the overall symplectic propagators for freely
+moving rigid bodies
+\begin{eqnarray}
+ \varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \notag\\
+  & & \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 }  \notag\\
+  & & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .
+\label{introEquation:overallRBFlowMaps}
+\end{eqnarray}
 
+\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 the generalized
+Langevin equation will be given first. Following that, we will
+discuss the physical meaning of the terms appearing in the equation.
+
+\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). 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,
 \[
-R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
-- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
-\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
-(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
+H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  x_\alpha ^2 }
+\right\}}
 \]
-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,
+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,
 \[
-q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
-^2 }}x(0)
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
 \]
-This makes
+where $g_\alpha$ are the coupling constants between the bath
+coordinates ($x_ \alpha$) and the system coordinate ($x$).
+Introducing
 \[
-R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
 \]
-And since the $q$ coordinates are harmonic oscillators,
+and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
 \[
-\begin{array}{l}
- \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
- \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
+w_\alpha ^2 }}x} \right)^2 } \right\}}.
+\]
+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, 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 the 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*}
+In the same way, the system coordinates become
+\begin{eqnarray*}
+ mL(\ddot 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\}}  \\
+  & & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}.
+\end{eqnarray*}
+With the help of some relatively important inverse Laplace
+transformations:
+\[
+\begin{array}{c}
+ L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
+ L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
+ L(1) = \frac{1}{p} \\
  \end{array}
 \]
-
-\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 } }
+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 } } \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\}}\\
 %
-&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
-\right\rangle \cos (\omega _\alpha  t)}
-%
-&= kT\xi (t)
-\end{align}
-
+& = & -
+\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{eqnarray*}
+Introducing a \emph{dynamic friction kernel}
 \begin{equation}
-\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-\label{introEquation:secondFluctuationDissipation}
+\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}
+which is known as the \emph{generalized Langevin equation} (GLE).
 
-\section{\label{introSection:hydroynamics}Hydrodynamics}
+\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
 
-\subsection{\label{introSection:frictionTensor} Friction Tensor}
-\subsection{\label{introSection:analyticalApproach}Analytical
-Approach}
+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 $\dot x$, $\left\langle {x(t)R(t)}
+\right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
+0.$ 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
+\[
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
+\]
+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 Eq.~\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 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:
+\[
+\xi (t) = 2\xi _0 \delta (t).
+\]
+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 Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes the
+Langevin equation
+\begin{equation}
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
+x(t) + R(t) \label{introEquation:LangevinEquation}.
+\end{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 brief review on calculating friction tensors for
+arbitrary shaped particles is given in
+Sec.~\ref{introSection:frictionTensor}.
 
-\subsection{\label{introSection:approximationApproach}Approximation
-Approach}
+\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
 
-\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-Body}
+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,
+\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}
+which acts as a constraint on the possible ways in which one can
+model the random force and friction kernel.
