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+Closely related to Classical Mechanics, Molecular Dynamics
+simulations are carried out by integrating the equations of motion
+for a given system of particles. There are three fundamental ideas
-<
+behind classical mechanics. Firstly, One can determine the state of
->
+behind classical mechanics. Firstly, one can determine the state of
+a mechanical system at any time of interest; Secondly, all the
+mechanical properties of the system at that time can be determined
+by combining the knowledge of the properties of the system with the
+\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}
+\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},
+\[
+\]
+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
+\end{equation}
+ is conserved. All of these conserved quantities are
+important factors to determine the quality of numerical integration
-<
+scheme for rigid body \cite{Dullweber1997}.
->
+schemes for rigid bodies \cite{Dullweber1997}.
+\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
-<
+Newtonian Mechanics suffers from two important limitations: it
-<
+describes their motion in special cartesian coordinate systems.
-<
+Another limitation of Newtonian mechanics becomes obvious when we
-<
+try to describe systems with large numbers of particles. It becomes
-<
+very difficult to predict the properties of the system by carrying
-<
+out calculations involving the each individual interaction between
-<
+all the particles, even if we know all of the details of the
-<
+interaction. In order to overcome some of the practical difficulties
-<
+which arise in attempts to apply Newton's equation to complex
-<
+system, alternative procedures may be developed.
->
+Newtonian Mechanics suffers from two important limitations: motions
->
+can only be described in cartesian coordinate systems. Moreover, It
->
+become impossible to predict analytically the properties of the
->
+system even if we know all of the details of the interaction. In
->
+order to overcome some of the practical difficulties which arise in
->
+attempts to apply Newton's equation to complex system, approximate
->
+numerical procedures may be developed.
-<
+\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
-<
+Principle}
->
+\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's
->
+Principle}}
+Hamilton introduced the dynamical principle upon which it is
-<
+possible to base all of mechanics and, indeed, most of classical
-<
+physics. Hamilton's Principle may be stated as follow,
->
+possible to base all of mechanics and most of classical physics.
->
+Hamilton's Principle may be stated as follows,
+The actual trajectory, along which a dynamical system may move from
+one point to another within a specified time, is derived by finding
+the path which minimizes the time integral of the difference between
-<
+the kinetic, $K$, and potential energies, $U$ \cite{tolman79}.
->
+the kinetic, $K$, and potential energies, $U$.
+\begin{equation}
+\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
+\label{introEquation:halmitonianPrinciple1}
+\end{equation}
+For simple mechanical systems, where the forces acting on the
-<
+different part are derivable from a potential and the velocities are
-<
+small compared with that of light, the Lagrangian function $L$ can
-<
+be define as the difference between the kinetic energy of the system
-<
+and its potential energy,
->
+different parts are derivable from a potential, the Lagrangian
->
+function $L$ can be defined as the difference between the kinetic
->
+energy of the system and its potential energy,
+\begin{equation}
+L \equiv K - U = L(q_i ,\dot q_i ) ,
+\label{introEquation:lagrangianDef}
+\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
+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}
+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
+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}.
+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 theorem presented in this
->
+dissertation.
-<
+\subsection{\label{introSection:ensemble}Ensemble and Phase Space}
->
+\subsection{\label{introSection:ensemble}Phase Space and Ensemble}
->
->
+Mathematically, phase space is the space which represents all
->
+possible states. Each possible state of the system corresponds to
->
+one unique point in the phase space. For mechanical systems, the
->
+phase space usually consists of all possible values of position and
->
+momentum variables. Consider a dynamic system of $f$ particles in a
->
+cartesian space, where each of the $6f$ coordinates and momenta is
->
+assigned to one of $6f$ mutually orthogonal axes, the phase space of
->
+this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots
->
+,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$
->
+coordinates and momenta is a phase space vector.
->
->
+%%%fix me
->
+A microscopic state or microstate of a classical system is
->
+specification of the complete phase space vector of a system at any
->
+instant in time. An ensemble is defined as a collection of systems
->
+sharing one or more macroscopic characteristics but each being in a
->
+unique microstate. The complete ensemble is specified by giving all
->
+systems or microstates consistent with the common macroscopic
->
+characteristics of the ensemble. Although the state of each
->
+individual system in the ensemble could be precisely described at
->
+any instance in time by a suitable phase space vector, when using
->
+ensembles for statistical purposes, there is no need to maintain
->
+distinctions between individual systems, since the numbers of
->
+systems at any time in the different states which correspond to
->
+different regions of the phase space are more interesting. Moreover,
->
+in the point of view of statistical mechanics, one would prefer to
->
+use ensembles containing a large enough population of separate
->
+members so that the numbers of systems in such different states can
->
+be regarded as changing continuously as we traverse different
->
+regions of the phase space. The condition of an ensemble at any time
->
+can be regarded as appropriately specified by the density $\rho$
->
+with which representative points are distributed over the phase
->
+space. The density distribution for an ensemble with $f$ degrees of
->
+freedom is defined as,
->
+\begin{equation}
->
+\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
->
+\label{introEquation:densityDistribution}
->
+\end{equation}
->
+Governed by the principles of mechanics, the phase points change
->
+their locations which would change the density at any time at phase
->
+space. Hence, the density distribution is also to be taken as a
->
+function of the time.
-+
+The number of systems $\delta N$ at time $t$ can be determined by,
-+
+\begin{equation}
-+
+\delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
-+
+\label{introEquation:deltaN}
-+
+\end{equation}
-+
+Assuming a large enough population of systems, 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 the systems. Hence,
-+
+the probability per unit 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 is
-+
+complex, or stochastic, or even discontinuous, the average
-+
+properties of the ensemble of possibilities as a whole remaining
-+
+well defined. For a classical system in thermal equilibrium with its
-+
+environment, the ensemble average of a mechanical quantity, $\langle
-+
+A(q , p) \rangle_t$, takes the form of an integral over the phase
-+
+space of the system,
-+
+\begin{equation}
-+
+\langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
-+
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
-+
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}
-+
+\label{introEquation:ensembelAverage}
-+
+\end{equation}
-+
-+
+There are several different types of ensembles with different
-+
+statistical characteristics. As a function of macroscopic
-+
+parameters, such as temperature \textit{etc}, the partition function
-+
+can be used to describe the statistical properties of a system in
-+
+thermodynamic equilibrium.
-+
-+
+As an ensemble of systems, each of which is known to be thermally
-+
+isolated and conserve energy, the Microcanonical ensemble (NVE) has
-+
+a partition function like,
-+
+\begin{equation}
-+
+\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
-+
+\end{equation}
-+
+A canonical ensemble (NVT)is an ensemble of systems, each of which
-+
+can share its energy with a large heat reservoir. The distribution
-+
+of the total energy amongst the possible dynamical states is given
-+
+by the partition function,
-+
+\begin{equation}
-+
+\Omega (N,V,T) = e^{ - \beta A}
-+
+\label{introEquation:NVTPartition}
-+
+\end{equation}
-+
+Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
-+
+TS$. Since most experiments are carried out under constant pressure
-+
+condition, the isothermal-isobaric ensemble (NPT) plays a very
-+
+important role in molecular simulations. The isothermal-isobaric
-+
+ensemble allow the system to exchange energy with a heat bath of
-+
+temperature $T$ and to change the volume as well. Its partition
-+
+function is given as
-+
+\begin{equation}
-+
+\Delta (N,P,T) =  - e^{\beta G}.
-+
+ \label{introEquation:NPTPartition}
-+
+\end{equation}
-+
+Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy.
-+
-+
+\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 members in an ensemble is
-+
+huge and constant, we can assume the local density has no reason
-+
+(other than classical mechanics) to change,
-+
+\begin{equation}
-+
+\frac{{\partial \rho }}{{\partial t}} = 0.
-+
+\label{introEquation:stationary}
-+
+\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{equation}
-+
+\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
-+
+dp_1 } ..dp_f.
-+
+\end{equation}
-+
-+
+\begin{equation}
-+
+\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho
-+
+\frac{d}{{dt}}(\delta v) = 0.
-+
+\end{equation}
-+
+With the help of stationary assumption
-+
+(\ref{introEquation:stationary}), we obtain the principle of the
-+
+\emph{conservation of 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 expresses in a variety of different forms
-+
+which are convenient within different contexts. For any two function
-+
+$F$ and $G$ of the coordinates and momenta of a system, the Poisson
-+
+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}
-+
+\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 of
-<
+experiment and computer simulation are usually performed during a
->
+simulation and the quality of the underlying model. However, both
->
+experiments and computer simulations are usually performed during a
+certain time interval and the measurements are averaged over a
+period of them which is different from the average behavior of
-<
+many-body system in Statistical Mechanics. Fortunately, Ergodic
-<
+Hypothesis is proposed to make a connection between time average and
-<
+ensemble average. It states that time average and average over the
-<
+statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
->
+many-body system in Statistical Mechanics. Fortunately, the Ergodic
->
+Hypothesis makes a connection between time average and the ensemble
->
+average. It states that the time average and average over the
->
+statistical ensemble are identical \cite{Frenkel1996, Leach2001}.
+\begin{equation}
-<
+\langle A \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
->
+\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 $\langle A \rangle_t$ 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}.
->
+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 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{Metropolis1949} can be
->
+utilized. Or if the system lends itself to a time averaging
->
+approach, the Molecular Dynamics techniques in
->
+Sec.~\ref{introSection:molecularDynamics} will be the best
->
+choice\cite{Frenkel1996}.
+\section{\label{introSection:geometricIntegratos}Geometric Integrators}
-<
+A variety of numerical integrators were proposed to simulate the
-<
+motions. They usually begin with an initial conditionals and move
-<
+the objects in the direction governed by the differential equations.
-<
+However, most of them ignore the hidden physical law contained
-<
+within the equations. Since 1990, geometric integrators, which
-<
+preserve various phase-flow invariants such as symplectic structure,
-<
+volume and time reversal symmetry, are developed to address this
-<
+issue. The velocity verlet method, which happens to be a simple
-<
+example of symplectic integrator, continues to gain its popularity
-<
+in molecular dynamics community. This fact can be partly explained
-<
+by its geometric nature.
->
+A variety of numerical integrators have been proposed to simulate
->
+the motions of atoms in MD simulation. They usually begin with
->
+initial conditionals and move the objects in the direction governed
->
+by the differential equations. However, most of them ignore the
->
+hidden physical laws contained within the equations. Since 1990,
->
+geometric integrators, which preserve various phase-flow invariants
->
+such as symplectic structure, volume and time reversal symmetry, are
->
+developed to address this issue\cite{Dullweber1997, McLachlan1998,
->
+Leimkuhler1999}. The velocity Verlet method, which happens to be a
->
+simple example of symplectic integrator, continues to gain
->
+popularity in the molecular dynamics community. This fact can be
->
+partly explained by its geometric nature.
-<
+\subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
-<
+A \emph{manifold} is an abstract mathematical space. It locally
-<
+looks like Euclidean space, but when viewed globally, it may have
-<
+more complicate structure. A good example of manifold is the surface
-<
+of Earth. It seems to be flat locally, but it is round if viewed as
-<
+a whole. A \emph{differentiable manifold} (also known as
-<
+\emph{smooth manifold}) is a manifold with an open cover in which
-<
+the covering neighborhoods are all smoothly isomorphic to one
-<
+another. In other words,it is possible to apply calculus on
-<
+\emph{differentiable manifold}. A \emph{symplectic manifold} is
-<
+defined as a pair $(M, \omega)$ which consisting of a
->
+\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds}
->
+A \emph{manifold} is an abstract mathematical space. It looks
->
+locally like Euclidean space, but when viewed globally, it may have
->
+more complicated structure. A good example of manifold is the
->
+surface of Earth. It seems to be flat locally, but it is round if
->
+viewed as a whole. A \emph{differentiable manifold} (also known as
->
+\emph{smooth manifold}) is a manifold on which it is possible to
->
+apply calculus on \emph{differentiable manifold}. A \emph{symplectic
->
+manifold} is defined as a pair $(M, \omega)$ which consists of a
+\emph{differentiable manifold} $M$ and a close, non-degenerated,
+bilinear symplectic form, $\omega$. A symplectic form on a vector
+space $V$ is a function $\omega(x, y)$ which satisfies
+$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
+\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
-<
+$\omega(x, x) = 0$. Cross product operation in vector field is an
-<
+example of symplectic form.
->
+$\omega(x, x) = 0$. The cross product operation in vector field is
->
+an example of symplectic form.
-<
+One of the motivations to study \emph{symplectic manifold} in
->
+One of the motivations to study \emph{symplectic manifolds} in
+Hamiltonian Mechanics is that a symplectic manifold can represent
+all possible configurations of the system and the phase space of the
+system can be described by it's cotangent bundle. Every symplectic
+manifold is even dimensional. For instance, in Hamilton equations,
+coordinate and momentum always appear in pairs.
-–
+Let  $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map
-–
+\[
-–
+f : M \rightarrow N
-–
+\]
-–
+is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and
-–
+the \emph{pullback} of $\eta$ under f is equal to $\omega$.
-–
+Canonical transformation is an example of symplectomorphism in
-–
+classical mechanics.
-–
+\subsection{\label{introSection:ODE}Ordinary Differential Equations}
-<
+For a ordinary differential system defined as
->
+For an ordinary differential system defined as
+\begin{equation}
+\dot x = f(x)
+\end{equation}
-<
+where $x = x(q,p)^T$, this system is canonical Hamiltonian, if
->
+where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if
+\begin{equation}
+f(r) = J\nabla _x H(r).
+\end{equation}
+\end{equation}In this case, $f$ is
+called a \emph{Hamiltonian vector field}.
-<
+Another generalization of Hamiltonian dynamics is Poisson Dynamics,
->
+Another generalization of Hamiltonian dynamics is Poisson
->
+Dynamics\cite{Olver1986},
+\begin{equation}
+\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
+\end{equation}
+The most obvious change being that matrix $J$ now depends on $x$.
-–
+The free rigid body is an example of Poisson system (actually a
-–
+Lie-Poisson system) with Hamiltonian function of angular kinetic
-–
+energy.
-–
+\begin{equation}
-–
+J(\pi ) = \left( {\begin{array}{*{20}c}
-–
+& {\pi _3 } & { - \pi _2 }  \\
-–
+   { - \pi _3 } & 0 & {\pi _1 }  \\
-–
+   {\pi _2 } & { - \pi _1 } & 0  \\
-–
+\end{array}} \right)
-–
+\end{equation}
-<
+\begin{equation}
-<
+H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2
-<
+}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right)
-<
+\end{equation}
->
+\subsection{\label{introSection:exactFlow}Exact Flow}
-–
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
+Let $x(t)$ be the exact solution of the ODE system,
+\begin{equation}
+\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}
+x(t+\tau) =\varphi_\tau(x(t))
+\]
+where $\tau$ is a fixed time step and $\varphi$ is a map from phase
-<
+space to itself. In most cases, it is not easy to find the exact
-<
+flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$,
-<
+which is usually called integrator. The order of an integrator
-<
+$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
-<
+order $p$,
->
+space to itself. The flow has the continuous group property,
+\begin{equation}
-<
+\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1})
->
+\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
->
++ \tau _2 } .
+\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,
->
+In particular,
+\begin{equation}
-<
+'\varphi^T J '\varphi = J.
->
+\varphi _\tau   \circ \varphi _{ - \tau }  = I
+\end{equation}
-+
+Therefore, the exact flow is self-adjoint,
-+
+\begin{equation}
-+
+\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
-+
+\end{equation}
-+
+The exact flow can also be written in terms of the of an operator,
-+
+\begin{equation}
-+
+\varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
-+
+}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
-+
+\label{introEquation:exponentialOperator}
-+
+\end{equation}
-+
-+
+In most cases, it is not easy to find the exact flow $\varphi_\tau$.
-+
+Instead, we use an approximate map, $\psi_\tau$, which is usually
-+
+called integrator. The order of an integrator $\psi_\tau$ is $p$, if
-+
+the Taylor series of $\psi_\tau$ agree to order $p$,
-+
+\begin{equation}
-+
+\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1})
-+
+\end{equation}
-+
-+
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
-+
-+
+The hidden geometric properties\cite{Budd1999, Marsden1998} of an
-+
+ODE and its flow play important roles in numerical studies. Many of
-+
+them can be found in systems which occur naturally in applications.
-+
-+
+Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is
-+
+a \emph{symplectic} flow if it satisfies,
-+
+\begin{equation}
-+
+{\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
+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  =
-<
+$. 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} flow for a
->
+source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $
->
+\det d\varphi  = 1$. One can show easily that a symplectic flow will
->
+be volume-preserving.
->
->
+Changing the variables $y = h(x)$ in 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.
->
+only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
-+
+A \emph{first integral}, or conserved quantity of a general
-+
+differential function is a function $ G:R^{2d}  \to R^d $ which is
-+
+constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
-+
+\[
-+
+\frac{{dG(x(t))}}{{dt}} = 0.
-+
+\]
-+
+Using chain rule, one may obtain,
-+
+\[
-+
+\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G,
-+
+\]
-+
+which is the condition for conserving \emph{first integral}. For a
-+
+canonical Hamiltonian system, the time evolution of an arbitrary
-+
+smooth function $G$ is given by,
-+
-+
+\begin{eqnarray}
-+
+\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\
-+
+                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
-+
+\label{introEquation:firstIntegral1}
-+
+\end{eqnarray}
-+
-+
-+
+Using poisson bracket notion, Equation
-+
+\ref{introEquation:firstIntegral1} can be rewritten as
-+
+\[
-+
+\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
-+
+\]
-+
+Therefore, the sufficient condition for $G$ to be the \emph{first
-+
+integral} of a Hamiltonian system is
-+
+\[
-+
+\left\{ {G,H} \right\} = 0.
-+
+\]
-+
+As well known, the Hamiltonian (or energy) H of a Hamiltonian system
-+
+is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
-+
+$.
-+
-+
+When designing any numerical methods, one should always try to
-+
+preserve the structural properties of the original ODE and its flow.
-+
+\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
+A lot of well established and very effective numerical methods have
+been successful precisely because of their symplecticities even
+though this fact was not recognized when they were first
-<
+constructed. The most famous example is leapfrog methods in
-<
+molecular dynamics. In general, symplectic integrators can be
->
+constructed. The most famous example is the Verlet-leapfrog method
->
+in molecular dynamics. In general, symplectic integrators can be
+constructed using one of four different methods.
+\begin{enumerate}
+\item Generating functions
+\item Splitting methods
+\end{enumerate}
-<
+Generating function tends to lead to methods which are cumbersome
-<
+and difficult to use\cite{}. In dissipative systems, variational
-<
+methods can capture the decay of energy accurately\cite{}. Since
-<
+their geometrically unstable nature against non-Hamiltonian
-<
+perturbations, ordinary implicit Runge-Kutta methods are not
-<
+suitable for Hamiltonian system. Recently, various high-order
-<
+explicit Runge--Kutta methods have been developed to overcome this
-<
+instability \cite{}. However, due to computational penalty involved
-<
+in implementing the Runge-Kutta methods, they do not attract too
-<
+much attention from Molecular Dynamics community. Instead, splitting
-<
+have been widely accepted since they exploit natural decompositions
-<
+of the system\cite{Tuckerman92}. The main idea behind splitting
-<
+methods is to decompose the discrete $\varphi_h$ as a composition of
-<
+simpler flows,
->
+Generating function\cite{Channell1990} tends to lead to methods
->
+which are cumbersome and difficult to use. In dissipative systems,
->
+variational methods can capture the decay of energy
->
+accurately\cite{Kane2000}. Since their geometrically unstable nature
->
+against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
->
+methods are not suitable for Hamiltonian system. Recently, various
->
+high-order explicit Runge-Kutta methods
->
+\cite{Owren1992,Chen2003}have been developed to overcome this
->
+instability. However, due to computational penalty involved in
->
+implementing the Runge-Kutta methods, they have not attracted much
->
+attention from the Molecular Dynamics community. Instead, splitting
->
+methods have been widely accepted since they exploit natural
->
+decompositions of the system\cite{Tuckerman1992, McLachlan1998}.
->
->
+\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
->
->
+The main idea behind splitting methods is to decompose the discrete
->
+$\varphi_h$ as a composition of simpler flows,
+\begin{equation}
+\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
+\varphi _{h_n }
+\label{introEquation:FlowDecomposition}
+\end{equation}
+where each of the sub-flow is chosen such that each represent a
-<
+simpler integration of the system. Let $\phi$ and $\psi$ both be
-<
+symplectic maps, it is easy to show that any composition of
-<
+symplectic flows yields a symplectic map,
->
+simpler integration of the system.
->
->
+Suppose that a Hamiltonian system takes the form,
->
+\[
->
+H = H_1 + H_2.
->
+\]
->
+Here, $H_1$ and $H_2$ may represent different physical processes of
->
+the system. For instance, they may relate to kinetic and potential
->
+energy respectively, which is a natural decomposition of the
->
+problem. If $H_1$ and $H_2$ can be integrated using exact flows
->
+$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first
->
+order expression is then given by the Lie-Trotter formula
+\begin{equation}
-+
+\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
-+
+\label{introEquation:firstOrderSplitting}
-+
+\end{equation}
-+
+where $\varphi _h$ is the result of applying the corresponding
-+
+continuous $\varphi _i$ over a time $h$. By definition, as
-+
+$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
-+
+must follow that each operator $\varphi_i(t)$ is a symplectic map.
-+
+It is easy to show that any composition of symplectic flows yields a
-+
+symplectic map,
-+
+\begin{equation}
+(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
-<
+'\phi ' = \phi '^T J\phi ' = J.
->
+'\phi ' = \phi '^T J\phi ' = J,
+ \label{introEquation:SymplecticFlowComposition}
+\end{equation}
-<
+Suppose that a Hamiltonian system has a form with $H = T + V$
->
+where $\phi$ and $\psi$ both are symplectic maps. Thus operator
->
+splitting in this context automatically generates a symplectic map.
-+
+The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting})
-+
+introduces local errors proportional to $h^2$, while Strang
-+
+splitting gives a second-order decomposition,
-+
+\begin{equation}
-+
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
-+
+_{1,h/2} , \label{introEquation:secondOrderSplitting}
-+
+\end{equation}
-+
+which has a local error proportional to $h^3$. The Sprang
-+
+splitting's popularity in molecular simulation community attribute
-+
+to its symmetric property,
-+
+\begin{equation}
-+
+\varphi _h^{ - 1} = \varphi _{ - h}.
-+
+\label{introEquation:timeReversible}
-+
+\end{equation}
-+
+\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(\ref{introEquation:SymplecticFlowComposition}),
-+
+time-reversible(\ref{introEquation:timeReversible}) and
-+
+volume-preserving (\ref{introEquation:volumePreserving}). These
-+
+geometric properties attribute to its long-time stability and its
-+
+popularity in the community. However, the most commonly used
-+
+velocity verlet integration scheme is written as below,
-+
+\begin{align}
-+
+\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
-+
+    \dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
-+
+%
-+
+q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
-+
+    \label{introEquation:Lp9b}\\%
-+
+%
-+
+\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
-+
+    \frac{\Delta t}{2m}\, F[q(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.
-<
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
->
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
-<
+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 Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move.
-<
+\subsection{\label{introSec:mdInit}Initialization}
->
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
->
+\end{enumerate}
-<
+\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
-<
-<
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
->
+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}
-<
+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.
->
+\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
-<
+Applications of dynamics of rigid bodies.
->
+The Baker-Campbell-Hausdorff formula can be used to determine the
->
+local error of splitting method in terms of the commutator of the
->
+operators(\ref{introEquation:exponentialOperator}) associated with
->
+the sub-flow. For operators $hX$ and $hY$ which are 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 commutators of operator $X$ and $Y$ given by
->
+\[
->
+[X,Y] = XY - YX .
->
+\]
->
+Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
->
+to the Sprang splitting, we can obtain
->
+\begin{eqnarray*}
->
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\
->
+                                   &   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
->
+                                   &   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots )
->
+\end{eqnarray*}
->
+Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0,\] the dominant local
->
+error of Spring splitting is proportional to $h^3$. The same
->
+procedure can be applied to a general splitting,  of the form
->
+\begin{equation}
->
+\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
->
+} 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
->
+\[
->
+\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
->
+\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
->
+\]
-<
+\subsection{\label{introSection:lieAlgebra}Lie Algebra}
->
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
-<
+\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body 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 an
->
+Hamiltonian system of $N$ particle can be measured by
->
+\[
->
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
->
+\]
->
+where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
->
+respectively, $f$ is the number of degrees of freedom, and $k_B$ is
->
+the boltzman constant.
-<
+\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
->
+A typical molecular dynamics run consists of three essential steps:
->
+\begin{enumerate}
->
+  \item Initialization
->
+    \begin{enumerate}
->
+    \item Preliminary preparation
->
+    \item Minimization
->
+    \item Heating
->
+    \item Equilibration
->
+    \end{enumerate}
->
+  \item Production
->
+  \item Analysis
->
+\end{enumerate}
->
+These three individual steps will be covered in the following
->
+sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
->
+initialization of a simulation. Sec.~\ref{introSection:production}
->
+will discusse issues in production run.
->
+Sec.~\ref{introSection:Analysis} provides the theoretical tools for
->
+trajectory analysis.
-<
+%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
->
+\subsection{\label{introSec:initialSystemSettings}Initialization}
-<
+\section{\label{introSection:correlationFunctions}Correlation Functions}
->
+\subsubsection{\textbf{Preliminary preparation}}
-<
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
->
+When selecting the starting structure of a molecule for molecular
->
+simulation, one may retrieve its Cartesian coordinates from public
->
+databases, such as RCSB Protein Data Bank \textit{etc}. Although
->
+thousands of crystal structures of molecules are discovered every
->
+year, many more remain unknown due to the difficulties of
->
+purification and crystallization. Even for molecules with known
->
+structure, some important information is missing. For example, a
->
+missing hydrogen atom which acts as donor in hydrogen bonding must
->
+be added. Moreover, in order to include electrostatic interaction,
->
+one may need to specify the partial charges for individual atoms.
->
+Under some circumstances, we may even need to prepare the system in
->
+a special configuration. For instance, when studying transport
->
+phenomenon in membrane systems, we may prepare the lipids in a
->
+bilayer structure instead of placing lipids randomly in solvent,
->
+since we are not interested in the slow self-aggregation process.
-<
+\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
->
+\subsubsection{\textbf{Minimization}}
-<
+\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
->
+It is quite possible that some of molecules in the system from
->
+preliminary preparation may be overlapping with each other. This
->
+close proximity leads to high initial potential energy which
->
+consequently jeopardizes any molecular dynamics simulations. To
->
+remove these steric overlaps, one typically performs energy
->
+minimization to find a more reasonable conformation. Several energy
->
+minimization methods have been developed to exploit the energy
->
+surface and to locate the local minimum. While converging slowly
->
+near the minimum, steepest descent method is extremely robust when
->
+systems are strongly anharmonic. Thus, it is often used to refine
->
+structure from crystallographic data. Relied on the gradient or
->
+hessian, advanced methods like Newton-Raphson converge rapidly to a
->
+local minimum, but become unstable if the energy surface is far from
->
+quadratic. Another factor that must be taken into account, when
->
+choosing energy minimization method, is the size of the system.
->
+Steepest descent and conjugate gradient can deal with models of any
->
+size. Because of the limits on computer memory to store the hessian
->
+matrix and the computing power needed to diagonalized these
->
+matrices, most Newton-Raphson methods can not be used with very
->
+large systems.
-+
+\subsubsection{\textbf{Heating}}
-+
-+
+Typically, Heating is performed by assigning random velocities
-+
+according to a Maxwell-Boltzman distribution for a desired
-+
+temperature. Beginning at a lower temperature and gradually
-+
+increasing the temperature by assigning larger random velocities, we
-+
+end up with setting the temperature of the system to a final
-+
+temperature at which the simulation will be conducted. In heating
-+
+phase, we should also keep the system from drifting or rotating as a
-+
+whole. To do this, the net linear momentum and angular momentum of
-+
+the system is shifted to zero after each resampling from the Maxwell
-+
+-Boltzman distribution.
-+
-+
+\subsubsection{\textbf{Equilibration}}
-+
-+
+The purpose of equilibration is to allow the system to evolve
-+
+spontaneously for a period of time and reach equilibrium. The
-+
+procedure is continued until various statistical properties, such as
-+
+temperature, pressure, energy, volume and other structural
-+
+properties \textit{etc}, become independent of time. Strictly
-+
+speaking, minimization and heating are not necessary, provided the
-+
+equilibration process is long enough. However, these steps can serve
-+
+as a means to arrive at an equilibrated structure in an effective
-+
+way.
-+
-+
+\subsection{\label{introSection:production}Production}
-+
-+
+The production run is the most important step of the simulation, in
-+
+which the equilibrated structure is used as a starting point and the
-+
+motions of the molecules are collected for later analysis. In order
-+
+to capture the macroscopic properties of the system, the molecular
-+
+dynamics simulation must be performed by sampling correctly and
-+
+efficiently from the relevant thermodynamic ensemble.
-+
-+
+The most expensive part of a molecular dynamics simulation is the
-+
+calculation of non-bonded forces, such as van der Waals force and
-+
+Coulombic forces \textit{etc}. For a system of $N$ particles, the
-+
+complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
-+
+which making large simulations prohibitive in the absence of any
-+
+algorithmic tricks.
-+
-+
+A natural approach to avoid system size issues is to represent the
-+
+bulk behavior by a finite number of the particles. However, this
-+
+approach will suffer from the surface effect at the edges of the
-+
+simulation. To offset this, \textit{Periodic boundary conditions}
-+
+(see Fig.~\ref{introFig:pbc}) is developed to simulate bulk
-+
+properties with a relatively small number of particles. In this
-+
+method, the simulation box is replicated throughout space to form an
-+
+infinite lattice. During the simulation, when a particle moves in
-+
+the primary cell, its image in other cells move in exactly the same
-+
+direction with exactly the same orientation. Thus, as a particle
-+
+leaves the primary cell, one of its images will enter through the
-+
+opposite face.
-+
+\begin{figure}
-+
+\centering
-+
+\includegraphics[width=\linewidth]{pbc.eps}
-+
+\caption[An illustration of periodic boundary conditions]{A 2-D
-+
+illustration of periodic boundary conditions. As one particle leaves
-+
+the left of the simulation box, an image of it enters the right.}
-+
+\label{introFig:pbc}
-+
+\end{figure}
-+
-+
+%cutoff and minimum image convention
-+
+Another important technique to improve the efficiency of force
-+
+evaluation is to apply spherical cutoff where particles farther than
-+
+a predetermined distance are not included in the calculation
-+
+\cite{Frenkel1996}. The use of a cutoff radius will cause a
-+
+discontinuity in the potential energy curve. Fortunately, one can
-+
+shift simple radial potential to ensure the potential curve go
-+
+smoothly to zero at the cutoff radius. The cutoff strategy works
-+
+well for Lennard-Jones interaction because of its short range
-+
+nature. However, simply truncating the electrostatic interaction
-+
+with the use of cutoffs has been shown to lead to severe artifacts
-+
+in simulations. The Ewald summation, in which the slowly decaying
-+
+Coulomb potential is transformed into direct and reciprocal sums
-+
+with rapid and absolute convergence, has proved to minimize the
-+
+periodicity artifacts in liquid simulations. Taking the advantages
-+
+of the fast Fourier transform (FFT) for calculating discrete Fourier
-+
+transforms, the particle mesh-based
-+
+methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
-+
+$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the
-+
+\emph{fast multipole method}\cite{Greengard1987, Greengard1994},
-+
+which treats Coulombic interactions exactly at short range, and
-+
+approximate the potential at long range through multipolar
-+
+expansion. In spite of their wide acceptance at the molecular
-+
+simulation community, these two methods are difficult to implement
-+
+correctly and efficiently. Instead, we use a damped and
-+
+charge-neutralized Coulomb potential method developed by Wolf and
-+
+his coworkers\cite{Wolf1999}. The shifted Coulomb potential for
-+
+particle $i$ and particle $j$ at distance $r_{rj}$ is given by:
+\begin{equation}
-<
+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,
->
+where $\alpha$ is the convergence parameter. Due to the lack of
->
+inherent periodicity and rapid convergence,this method is extremely
->
+efficient and easy to implement.
->
+\begin{figure}
->
+\centering
->
+\includegraphics[width=\linewidth]{shifted_coulomb.eps}
->
+\caption[An illustration of shifted Coulomb potential]{An
->
+illustration of shifted Coulomb potential.}
->
+\label{introFigure:shiftedCoulomb}
->
+\end{figure}
->
->
+%multiple time step
->
->
+\subsection{\label{introSection:Analysis} Analysis}
->
->
+Recently, advanced visualization technique have become applied to
->
+monitor the motions of molecules. Although the dynamics of the
->
+system can be described qualitatively from animation, quantitative
->
+trajectory analysis are more useful. According to the principles of
->
+Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics},
->
+one can compute thermodynamic properties, analyze fluctuations of
->
+structural parameters, and investigate time-dependent processes of
->
+the molecule from the trajectories.
->
->
+\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}
->
->
+Thermodynamic properties, which can be expressed in terms of some
->
+function of the coordinates and momenta of all particles in the
->
+system, can be directly computed from molecular dynamics. The usual
->
+way to measure the pressure is based on virial theorem of Clausius
->
+which states that the virial is equal to $-3Nk_BT$. For a system
->
+with forces between particles, the total virial, $W$, contains the
->
+contribution from external pressure and interaction between the
->
+particles:
+\[
-<
+H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-<
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
->
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
->
+f_{ij} } } \right\rangle
+\]
-<
+and $\Delta U$ is bilinear system-bath coupling,
->
+where $f_{ij}$ is the force between particle $i$ and $j$ at a
->
+distance $r_{ij}$. Thus, the expression for the pressure is given
->
+by:
->
+\begin{equation}
->
+P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
->
+< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
->
+\end{equation}
->
->
+\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}
->
->
+Structural Properties of a simple fluid can be described by a set of
->
+distribution functions. Among these functions,the \emph{pair
->
+distribution function}, also known as \emph{radial distribution
->
+function}, is of most fundamental importance to liquid theory.
->
+Experimentally, pair distribution function can be gathered by
->
+Fourier transforming raw data from a series of neutron diffraction
->
+experiments and integrating over the surface factor
->
+\cite{Powles1973}. The experimental results can serve as a criterion
->
+to justify the correctness of a liquid model. Moreover, various
->
+equilibrium thermodynamic and structural properties can also be
->
+expressed in terms of radial distribution function \cite{Allen1987}.
->
->
+The pair distribution functions $g(r)$ gives the probability that a
->
+particle $i$ will be located at a distance $r$ from a another
->
+particle $j$ in the system
+\[
-<
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
->
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
->
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
->
+(r)}{\rho}.
+\]
-<
+Completing the square,
->
+Note that the delta function can be replaced by a histogram in
->
+computer simulation. Peaks in $g(r)$ represent solvent shells, and
->
+the height of these peaks gradually decreases to 1 as the liquid of
->
+large distance approaches the bulk density.
->
->
->
+\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
->
+Properties}}
->
->
+Time-dependent properties are usually calculated using \emph{time
->
+correlation functions}, which correlate random variables $A$ and $B$
->
+at two different times,
->
+\begin{equation}
->
+C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
->
+\label{introEquation:timeCorrelationFunction}
->
+\end{equation}
->
+If $A$ and $B$ refer to same variable, this kind of correlation
->
+function is called an \emph{autocorrelation function}. One example
->
+of an auto correlation function is the velocity auto-correlation
->
+function which is directly related to transport properties of
->
+molecular liquids:
+\[
-<
+H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
-<
+{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-<
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-<
+w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
-<
+}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
->
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
->
+\right\rangle } dt
+\]
-<
+and putting it back into Eq.~\ref{introEquation:bathGLE},
->
+where $D$ is diffusion constant. Unlike the velocity autocorrelation
->
+function, which is averaging over time origins and over all the
->
+atoms, the dipole autocorrelation functions are calculated for the
->
+entire system. The dipole autocorrelation function is given by:
+\[
-<
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-<
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-<
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-<
+w_\alpha ^2 }}x} \right)^2 } \right\}}
->
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
->
+\right\rangle
+\]
-<
+where
->
+Here $u_{tot}$ is the net dipole of the entire system and is given
->
+by
+\[
-<
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
-<
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
->
+u_{tot} (t) = \sum\limits_i {u_i (t)}
+\]
-<
+Since the first two terms of the new Hamiltonian depend only on the
-<
+system coordinates, we can get the equations of motion for
-<
+Generalized Langevin Dynamics by Hamilton's equations
-<
+\ref{introEquation:motionHamiltonianCoordinate,
-<
+introEquation:motionHamiltonianMomentum},
-<
+\begin{align}
-<
+\dot p &=  - \frac{{\partial H}}{{\partial x}}
-<
+       &= m\ddot x
-<
+       &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)}
-<
+\label{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}
->
+In principle, many time correlation functions can be related with
->
+Fourier transforms of the infrared, Raman, and inelastic neutron
->
+scattering spectra of molecular liquids. In practice, one can
->
+extract the IR spectrum from the intensity of dipole fluctuation at
->
+each frequency using the following relationship:
->
+\[
->
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
->
+i2\pi vt} dt}
->
+\]
-<
+\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
->
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
-+
+Rigid bodies are frequently involved in the modeling of different
-+
+areas, from engineering, physics, to chemistry. For example,
-+
+missiles and vehicle are usually modeled by rigid bodies.  The
-+
+movement of the objects in 3D gaming engine or other physics
-+
+simulator is governed by 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 motion is very inefficient and inaccurate. Although an
-+
+alternative integrator using multiple sets of Euler angles can
-+
+overcome this difficulty\cite{Barojas1973}, the computational
-+
+penalty and the loss of angular momentum conservation still remain.
-+
+A singularity-free representation utilizing quaternions was
-+
+developed by Evans in 1977\cite{Evans1977}. Unfortunately, this
-+
+approach uses a nonseparable Hamiltonian resulting from the
-+
+quaternion representation, which prevents the symplectic algorithm
-+
+to be utilized. Another different approach is to apply holonomic
-+
+constraints to the atoms belonging to the rigid body. Each atom
-+
+moves independently under the normal forces deriving from potential
-+
+energy and constraint forces which are used to guarantee the
-+
+rigidness. However, due to their iterative nature, the SHAKE and
-+
+Rattle algorithms also converge very slowly when the number of
-+
+constraints increases\cite{Ryckaert1977, Andersen1983}.
-+
-+
+A break-through in geometric literature suggests that, in order to
-+
+develop a long-term integration scheme, one should preserve the
-+
+symplectic structure of the flow. By introducing a conjugate
-+
+momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's
-+
+equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was
-+
+proposed to evolve the Hamiltonian system in a constraint manifold
-+
+by iteratively satisfying the orthogonality constraint $Q^T Q = 1$.
-+
+An alternative method using the quaternion representation was
-+
+developed by Omelyan\cite{Omelyan1998}. However, both of these
-+
+methods are iterative and inefficient. In this section, we descibe a
-+
+symplectic Lie-Poisson integrator for rigid body developed by
-+
+Dullweber and his coworkers\cite{Dullweber1997} in depth.
-+
-+
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
-+
+The motion of a rigid body is Hamiltonian with the Hamiltonian
-+
+function
-+
+\begin{equation}
-+
+H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
-+
+V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
-+
+\label{introEquation:RBHamiltonian}
-+
+\end{equation}
-+
+Here, $q$ and $Q$  are the position and rotation matrix for the
-+
+rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ , and
-+
+$J$, a diagonal matrix, is defined by
+\[
-<
+L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
->
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
+\]
-+
+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 rotation matrix's unitarity. Differentiating
-+
+\ref{introEquation:orthogonalConstraint} and using Equation
-+
+\ref{introEquation:RBMotionMomentum}, one may obtain,
-+
+\begin{equation}
-+
+Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
-+
+\label{introEquation:RBFirstOrderConstraint}
-+
+\end{equation}
-+
+Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
-+
+\ref{introEquation:motionHamiltonianMomentum}), one can write down
-+
+the equations of motion,
-+
-+
+\begin{eqnarray}
-+
+ \frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\
-+
+ \frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
-+
+ \frac{{dQ}}{{dt}} & = & PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
-+
+ \frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
-+
+\end{eqnarray}
-+
-+
+In general, there are two ways to satisfy the holonomic constraints.
-+
+We can use a constraint force provided by a Lagrange multiplier on
-+
+the normal manifold to keep the motion on constraint space. Or we
-+
+can simply evolve the system on the constraint manifold. These two
-+
+methods have been proved to be equivalent. The holonomic constraint
-+
+and equations of motions define a constraint manifold for rigid
-+
+bodies
+\[
-<
+L(x + y) = L(x) + L(y)
->
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
->
+\right\}.
+\]
-+
+Unfortunately, this constraint manifold is not the cotangent bundle
-+
+$T_{\star}SO(3)$. However, it turns out that under symplectic
-+
+transformation, the cotangent space and the phase space are
-+
+diffeomorphic. By introducing
+\[
-<
+L(ax) = aL(x)
->
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
+\]
-<
->
+the mechanical system subject to a holonomic constraint manifold $M$
->
+can be re-formulated as a Hamiltonian system on the cotangent space
+\[
-<
+L(\dot x) = pL(x) - px(0)
->
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
->
+,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
+\]
-+
+For a body fixed vector $X_i$ with respect to the center of mass of
-+
+the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
-+
+given as
-+
+\begin{equation}
-+
+X_i^{lab} = Q X_i + q.
-+
+\end{equation}
-+
+Therefore, potential energy $V(q,Q)$ is defined by
+\[
-<
+L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
->
+V(q,Q) = V(Q X_0 + q).
+\]
-<
->
+Hence, the force and torque are given by
+\[
-<
+L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
->
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
+\]
-<
-<
+Some relatively important transformation,
->
+and
+\[
-<
+L(\cos at) = \frac{p}{{p^2  + a^2 }}
->
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
+\]
-+
+respectively.
-+
+As a common choice to describe the rotation dynamics of the rigid
-+
+body, the angular momentum on the body fixed frame $\Pi  = Q^t P$ is
-+
+introduced to rewrite the equations of motion,
-+
+\begin{equation}
-+
+ \begin{array}{l}
-+
+ \mathop \Pi \limits^ \bullet   = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda  \\
-+
+ \mathop Q\limits^{{\rm{   }} \bullet }  = Q\Pi {\rm{ }}J^{ - 1}  \\
-+
+ \end{array}
-+
+ \label{introEqaution:RBMotionPI}
-+
+\end{equation}
-+
+, as well as holonomic constraints,
+\[
-<
+L(\sin at) = \frac{a}{{p^2  + a^2 }}
->
+\begin{array}{l}
->
+ \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
->
+ Q^T Q = 1 \\
->
+ \end{array}
+\]
-+
+For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
-+
+so(3)^ \star$, the hat-map isomorphism,
-+
+\begin{equation}
-+
+v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
-+
+{\begin{array}{*{20}c}
-+
+& { - v_3 } & {v_2 }  \\
-+
+   {v_3 } & 0 & { - v_1 }  \\
-+
+   { - v_2 } & {v_1 } & 0  \\
-+
+\end{array}} \right),
-+
+\label{introEquation:hatmapIsomorphism}
-+
+\end{equation}
-+
+will let us associate the matrix products with traditional vector
-+
+operations
+\[
-<
+L(1) = \frac{1}{p}
->
+\hat vu = v \times u
+\]
-+
+Using \ref{introEqaution:RBMotionPI}, one can construct a skew
-+
+matrix,
-+
+\begin{equation}
-+
+(\mathop \Pi \limits^ \bullet   - \mathop \Pi \limits^ {\bullet  ^T}
-+
+){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi  + \Pi J^{
-+
+- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]} -
-+
+(\Lambda  - \Lambda ^T ) . \label{introEquation:skewMatrixPI}
-+
+\end{equation}
-+
+Since $\Lambda$ is symmetric, the last term of Equation
-+
+\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
-+
+multiplier $\Lambda$ is absent from the equations of motion. This
-+
+unique property eliminates the requirement of iterations which can
-+
+not be avoided in other methods\cite{Kol1997, Omelyan1998}.
-<
+First, the bath coordinates,
->
+Applying the hat-map isomorphism, we obtain the equation of motion
->
+for angular momentum on body frame
->
+\begin{equation}
->
+\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
->
+F_i (r,Q)} \right) \times X_i }.
->
+\label{introEquation:bodyAngularMotion}
->
+\end{equation}
->
+In the same manner, the equation of motion for rotation matrix is
->
+given by
+\[
-<
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
-<
+_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
-<
+}}L(x)
->
+\dot Q = Qskew(I^{ - 1} \pi )
+\]
-+
-+
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
-+
+Lie-Poisson Integrator for Free Rigid Body}
-+
-+
+If there are no external forces exerted on the rigid body, the only
-+
+contribution to the rotational motion is from the kinetic energy
-+
+(the first term of \ref{introEquation:bodyAngularMotion}). The free
-+
+rigid body is an example of a Lie-Poisson system with Hamiltonian
-+
+function
-+
+\begin{equation}
-+
+T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
-+
+\label{introEquation:rotationalKineticRB}
-+
+\end{equation}
-+
+where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
-+
+Lie-Poisson structure matrix,
-+
+\begin{equation}
-+
+J(\pi ) = \left( {\begin{array}{*{20}c}
-+
+& {\pi _3 } & { - \pi _2 }  \\
-+
+   { - \pi _3 } & 0 & {\pi _1 }  \\
-+
+   {\pi _2 } & { - \pi _1 } & 0  \\
-+
+\end{array}} \right)
-+
+\end{equation}
-+
+Thus, the dynamics of free rigid body is governed by
-+
+\begin{equation}
-+
+\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
-+
+\end{equation}
-+
-+
+One may notice that each $T_i^r$ in Equation
-+
+\ref{introEquation:rotationalKineticRB} can be solved exactly. For
-+
+instance, the equations of motion due to $T_1^r$ are given by
-+
+\begin{equation}
-+
+\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
-+
+\label{introEqaution:RBMotionSingleTerm}
-+
+\end{equation}
-+
+where
-+
+\[ R_1  = \left( {\begin{array}{*{20}c}
-+
+& 0 & 0  \\
-+
+& 0 & {\pi _1 }  \\
-+
+& { - \pi _1 } & 0  \\
-+
+\end{array}} \right).
-+
+\]
-+
+The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is
+\[
-<
+L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
-<
+px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
->
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
->
+Q(0)e^{\Delta tR_1 }
+\]
-<
+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,
->
+with
->
+\[
->
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
->
+& 0 & 0  \\
->
+& {\cos \theta _1 } & {\sin \theta _1 }  \\
->
+& { - \sin \theta _1 } & {\cos \theta _1 }  \\
->
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
->
+\]
->
+To reduce the cost of computing expensive functions in $e^{\Delta
->
+tR_1 }$, we can use Cayley transformation to obtain a single-aixs
->
+propagator,
->
+\[
->
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
->
+)
->
+\]
->
+The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
->
+manner. In order to construct a second-order symplectic method, we
->
+split the angular kinetic Hamiltonian function can into five terms
->
+\[
->
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
->
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
->
+(\pi _1 ).
->
+\]
->
+By concatenating the propagators corresponding to these five terms,
->
+we can obtain an symplectic integrator,
->
+\[
->
+\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
->
+\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
->
+\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
->
+_1 }.
->
+\]
-<
+\begin{align}
-<
+m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
->
+The non-canonical Lie-Poisson bracket ${F, G}$ of two function
->
+$F(\pi )$ and $G(\pi )$ is defined by
->
+\[
->
+\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
->
+)
->
+\]
->
+If the Poisson bracket of a function $F$ with an arbitrary smooth
->
+function $G$ is zero, $F$ is a \emph{Casimir}, which is the
->
+conserved quantity in Poisson system. We can easily verify that the
->
+norm of the angular momentum, $\parallel \pi
->
+\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel
->
+\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
->
+then by the chain rule
->
+\[
->
+\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
->
+}}{2})\pi
->
+\]
->
+Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
->
+\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
->
+Lie-Poisson integrator is found to be 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 the below table,
->
+\begin{table}
->
+\caption{Equations of motion due to Potential and Kinetic Energies}
->
+\begin{center}
->
+\begin{tabular}{|l|l|}
->
+  \hline
->
+  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
->
+  Potential & Kinetic \\
->
+  $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
->
+  $\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
->
+  $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
->
+  $ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
->
+  \hline
->
+\end{tabular}
->
+\end{center}
->
+\end{table}
->
+A second-order symplectic method is now obtained by the composition
->
+of the position and velocity propagators,
->
+\[
->
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
->
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
->
+\]
->
+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}
->
+T(p,\pi ) =T^t (p) + T^r (\pi ).
->
+\end{equation}
->
+where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
->
+defined by \ref{introEquation:rotationalKineticRB}. Therefore, the
->
+corresponding propagators are given by
->
+\[
->
+\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
->
+_{\Delta t,T^r }.
->
+\]
->
+Finally, we obtain the overall symplectic propagators for freely
->
+moving rigid bodies
->
+\begin{equation}
->
+\begin{array}{c}
->
+ \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
->
+  \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \\
->
+  \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
->
+ \end{array}
->
+\label{introEquation:overallRBFlowMaps}
->
+\end{equation}
->
->
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
->
+As an alternative to newtonian dynamics, Langevin dynamics, which
->
+mimics a simple heat bath with stochastic and dissipative forces,
->
+has been applied in a variety of studies. This section will review
->
+the theory of Langevin dynamics. A brief derivation of generalized
->
+Langevin equation will be given first. Following that, we will
->
+discuss the physical meaning of the terms appearing in the equation
->
+as well as the calculation of friction tensor from hydrodynamics
->
+theory.
->
->
+\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
->
->
+A harmonic bath model, in which an effective set of harmonic
->
+oscillators are used to mimic the effect of a linearly responding
->
+environment, has been widely used in quantum chemistry and
->
+statistical mechanics. One of the successful applications of
->
+Harmonic bath model is the derivation of the Generalized Langevin
->
+Dynamics (GLE). Lets consider a system, in which the degree of
->
+freedom $x$ is assumed to couple to the bath linearly, giving a
->
+Hamiltonian of the form
->
+\begin{equation}
->
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
->
+\label{introEquation:bathGLE}.
->
+\end{equation}
->
+Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated
->
+with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian,
->
+\[
->
+H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
->
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  \omega _\alpha ^2 }
->
+\right\}}
->
+\]
->
+where the index $\alpha$ runs over all the bath degrees of freedom,
->
+$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
->
+the harmonic bath masses, and $\Delta U$ is a bilinear system-bath
->
+coupling,
->
+\[
->
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
->
+\]
->
+where $g_\alpha$ are the coupling constants between the bath
->
+coordinates ($x_ \alpha$) and the system coordinate ($x$).
->
+Introducing
->
+\[
->
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
->
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
->
+\] and combining the last two terms in Equation
->
+\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath
->
+Hamiltonian as
->
+\[
->
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
->
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
->
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
->
+w_\alpha ^2 }}x} \right)^2 } \right\}}
->
+\]
->
+Since the first two terms of the new Hamiltonian depend only on the
->
+system coordinates, we can get the equations of motion for
->
+Generalized Langevin Dynamics by Hamilton's equations,
->
+\begin{equation}
->
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
->
+\sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   -
->
+\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)},
->
+\label{introEquation:coorMotionGLE}
->
+\end{equation}
->
+and
->
+\begin{equation}
->
+m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x_\alpha   -
->
+\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
->
+\label{introEquation:bathMotionGLE}
->
+\end{equation}
->
->
+In order to derive an equation for $x$, the dynamics of the bath
->
+variables $x_\alpha$ must be solved exactly first. As an integral
->
+transform which is particularly useful in solving linear ordinary
->
+differential equations,the Laplace transform is the appropriate tool
->
+to solve this problem. The basic idea is to transform the difficult
->
+differential equations into simple algebra problems which can be
->
+solved easily. Then, by applying the inverse Laplace transform, also
->
+known as the Bromwich integral, we can retrieve the solutions of the
->
+original problems.
->
->
+Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace
->
+transform of f(t) is a new function defined as
->
+\[
->
+L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
->
+\]
->
+where  $p$ is real and  $L$ is called the Laplace Transform
->
+Operator. Below are some important properties of Laplace transform
->
->
+\begin{eqnarray*}
->
+ L(x + y)  & = & L(x) + L(y) \\
->
+ L(ax)     & = & aL(x) \\
->
+ L(\dot x) & = & pL(x) - px(0) \\
->
+ L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
->
+ L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
->
+ \end{eqnarray*}
->
->
->
+Applying the Laplace transform to the bath coordinates, we obtain
->
+\begin{eqnarray*}
->
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) & = & - \omega _\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) \\
->
+L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }} \\
->
+\end{eqnarray*}
->
->
+By the same way, the system coordinates become
->
+\begin{eqnarray*}
->
+ mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
->
+  & & \mbox{} - \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0) - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}  \\
->
+\end{eqnarray*}
->
->
+With the help of some relatively important inverse Laplace
->
+transformations:
->
+\[
->
+\begin{array}{c}
->
+ L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
->
+ L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
->
+ L(1) = \frac{1}{p} \\
->
+ \end{array}
->
+\]
->
+, we obtain
->
+\begin{eqnarray*}
->
+m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
+\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
-<
+_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
-<
+- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
-<
+(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
-<
+_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
-<
+%
-<
+&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
->
+_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
->
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
->
+x_\alpha (0) - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}}
->
+\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
->
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
->
+\end{eqnarray*}
->
+\begin{eqnarray*}
->
+m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
+{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
-<
+t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
-<
+{\left[ {g_\alpha  x_\alpha  (0) - \frac{{g_\alpha  }}{{m_\alpha
-<
+\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
-<
+\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
-<
+(\omega _\alpha  t)} \right\}}
-<
+\end{align}
-<
->
+t)\dot x(t - \tau )d} \tau }  \\
->
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
->
+x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha  }}}
->
+\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
->
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
->
+\end{eqnarray*}
->
+Introducing a \emph{dynamic friction kernel}
+\begin{equation}
-+
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
-+
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
-+
+\label{introEquation:dynamicFrictionKernelDefinition}
-+
+\end{equation}
-+
+and \emph{a random force}
-+
+\begin{equation}
-+
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
-+
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
-+
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
-+
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
-+
+\label{introEquation:randomForceDefinition}
-+
+\end{equation}
-+
+the equation of motion can be rewritten as
-+
+\begin{equation}
+m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
+(t)\dot x(t - \tau )d\tau }  + R(t)
+\label{introEuqation:GeneralizedLangevinDynamics}
+\end{equation}
-<
+%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
-<
+%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
->
+which is known as the \emph{generalized Langevin equation}.
->
->
+\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
->
->
+One may notice that $R(t)$ depends only on initial conditions, which
->
+implies it is completely deterministic within the context of a
->
+harmonic bath. However, it is easy to verify that $R(t)$ is totally
->
+uncorrelated to $x$ and $\dot x$,
+\[
-<
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
-<
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
->
+\begin{array}{l}
->
+ \left\langle {x(t)R(t)} \right\rangle  = 0, \\
->
+ \left\langle {\dot x(t)R(t)} \right\rangle  = 0. \\
->
+ \end{array}
+\]
-<
+For an infinite harmonic bath, we can use the spectral density and
-<
+an integral over frequencies.
->
+This property is what we expect from a truly random process. As long
->
+as the model chosen for $R(t)$ was a gaussian distribution in
->
+general, the stochastic nature of the GLE still remains.
-+
+%dynamic friction kernel
-+
+The convolution integral
+\[
-<
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
-<
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
-<
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
-<
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
+\]
-<
+The random forces depend only on initial conditions.
-<
-<
+\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
-<
+So we can define a new set of coordinates,
->
+depends on the entire history of the evolution of $x$, which implies
->
+that the bath retains memory of previous motions. In other words,
->
+the bath requires a finite time to respond to change in the motion
->
+of the system. For a sluggish bath which responds slowly to changes
->
+in the system coordinate, we may regard $\xi(t)$ as a constant
->
+$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
+\[
-<
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
-<
+^2 }}x(0)
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
+\]
-<
+This makes
->
+and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
+\[
-<
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
->
+m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
->
+\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
+\]
-<
+And since the $q$ coordinates are harmonic oscillators,
->
+which can be used to describe the effect of dynamic caging in
->
+viscous solvents. The other extreme is the bath that responds
->
+infinitely quickly to motions in the system. Thus, $\xi (t)$ can be
->
+taken as a $delta$ function in time:
+\[
-<
+\begin{array}{l}
-<
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
-<
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
-<
+ \end{array}
->
+\xi (t) = 2\xi _0 \delta (t)
+\]
-<
-<
+\begin{align}
-<
+\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
-<
+{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
-<
+(t)q_\beta  (0)} \right\rangle } }
-<
+%
-<
+&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
-<
+\right\rangle \cos (\omega _\alpha  t)}
-<
+%
-<
+&= kT\xi (t)
-<
+\end{align}
-<
->
+Hence, the convolution integral becomes
->
+\[
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
->
+{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
->
+\]
->
+and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
+\begin{equation}
-<
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-<
+\label{introEquation:secondFluctuationDissipation}
->
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
->
+x(t) + R(t) \label{introEquation:LangevinEquation}
+\end{equation}
-+
+which is known as the Langevin equation. The static friction
-+
+coefficient $\xi _0$ can either be calculated from spectral density
-+
+or be determined by Stokes' law for regular shaped particles. A
-+
+briefly review on calculating friction tensor for arbitrary shaped
-+
+particles is given in Sec.~\ref{introSection:frictionTensor}.
-<
+\section{\label{introSection:hydroynamics}Hydrodynamics}
->
+\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
-<
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
-<
+\subsection{\label{introSection:analyticalApproach}Analytical
-<
+Approach}
->
+Defining a new set of coordinates,
->
+\[
->
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
->
+^2 }}x(0)
->
+\],
->
+we can rewrite $R(T)$ as
->
+\[
->
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
->
+\]
->
+And since the $q$ coordinates are harmonic oscillators,
-<
+\subsection{\label{introSection:approximationApproach}Approximation
-<
+Approach}
->
+\begin{eqnarray*}
->
+ \left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
->
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
->
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
->
+ \left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha  {\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle } }  \\
->
+  & = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
->
+  & = &kT\xi (t) \\
->
+\end{eqnarray*}
-<
+\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-<
+Body}
->
+Thus, we recover the \emph{second fluctuation dissipation theorem}
->
+\begin{equation}
->
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
->
+\label{introEquation:secondFluctuationDissipation}.
->
+\end{equation}
->
+In effect, it acts as a constraint on the possible ways in which one
->
+can model the random force and friction kernel.

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+Removed lines
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+Added lines
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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2699 by tim, Mon Apr 10 05:35:55 2006 UTC vs. Revision 2882 by tim, Fri Jun 23 21:33:52 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2699 by tim, Mon Apr 10 05:35:55 2006 UTC vs.
Revision 2882 by tim, Fri Jun 23 21:33:52 2006 UTC