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+\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
-<
+\section{\label{introSection:classicalMechanics}Classical Mechanics}
->
+\section{\label{introSection:classicalMechanics}Classical
->
+Mechanics}
-<
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
->
+Using equations of motion derived from Classical Mechanics,
->
+Molecular Dynamics simulations are carried out by integrating the
->
+equations of motion for a given system of particles. There are three
->
+fundamental ideas behind classical mechanics. Firstly, one can
->
+determine the state of a mechanical system at any time of interest;
->
+Secondly, all the mechanical properties of the system at that time
->
+can be determined by combining the knowledge of the properties of
->
+the system with the specification of this state; Finally, the
->
+specification of the state when further combined with the laws of
->
+mechanics will also be sufficient to predict the future behavior of
->
+the system.
-<
+\section{\label{introSection:statisticalMechanics}Statistical Mechanics}
->
+\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
->
+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
->
+\begin{equation}
->
+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
->
+\begin{equation}
->
+F_{ij} = -F_{ji}.
->
+ \label{introEquation:newtonThirdLaw}
->
+\end{equation}
->
+Conservation laws of Newtonian Mechanics play very important roles
->
+in solving mechanics problems. The linear momentum of a particle is
->
+conserved if it is free or it experiences no force. The second
->
+conservation theorem concerns the angular momentum of a particle.
->
+The angular momentum $L$ of a particle with respect to an origin
->
+from which $r$ is measured is defined to be
->
+\begin{equation}
->
+L \equiv r \times p \label{introEquation:angularMomentumDefinition}
->
+\end{equation}
->
+The torque $\tau$ with respect to the same origin is defined to be
->
+\begin{equation}
->
+\tau \equiv r \times F \label{introEquation:torqueDefinition}
->
+\end{equation}
->
+Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
->
+\[
->
+\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times
->
+\dot p)
->
+\]
->
+since
->
+\[
->
+\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0
->
+\]
->
+thus,
->
+\begin{equation}
->
+\dot L = r \times \dot p = \tau
->
+\end{equation}
->
+If there are no external torques acting on a body, the angular
->
+momentum of it is conserved. The last conservation theorem state
->
+that if all forces are conservative, energy is conserved,
->
+\begin{equation}E = T + V. \label{introEquation:energyConservation}
->
+\end{equation}
->
+All of these conserved quantities are important factors to determine
->
+the quality of numerical integration schemes for rigid
->
+bodies.\cite{Dullweber1997}
-<
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
->
+\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
-<
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
->
+Newtonian Mechanics suffers from an important limitation: motion can
->
+only be described in cartesian coordinate systems which make it
->
+impossible to predict analytically the properties of the system even
->
+if we know all of the details of the interaction. In order to
->
+overcome some of the practical difficulties which arise in attempts
->
+to apply Newton's equation to complex systems, approximate numerical
->
+procedures may be developed.
-<
+\section{\label{introSection:hydroynamics}Hydrodynamics}
->
+\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's
->
+Principle}}
->
->
+Hamilton introduced the dynamical principle upon which it is
->
+possible to base all of mechanics and most of classical physics.
->
+Hamilton's Principle may be stated as follows: the trajectory, along
->
+which a dynamical system may move from one point to another within a
->
+specified time, is derived by finding the path which minimizes the
->
+time integral of the difference between the kinetic $K$, and
->
+potential energies $U$,
->
+\begin{equation}
->
+\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}.
->
+\label{introEquation:halmitonianPrinciple1}
->
+\end{equation}
->
+For simple mechanical systems, where the forces acting on the
->
+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}
->
+\end{equation}
->
+Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
->
+\begin{equation}
->
+\delta \int_{t_1 }^{t_2 } {L dt = 0} .
->
+\label{introEquation:halmitonianPrinciple2}
->
+\end{equation}
->
->
+\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The
->
+Equations of Motion in Lagrangian Mechanics}}
->
->
+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
->
+\label{introEquation:eqMotionLagrangian}
->
+\end{equation}
->
+where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
->
+generalized velocity.
->
->
+\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
->
->
+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 velocities, the momenta can be defined as
->
+\begin{equation}
->
+p_i = \frac{\partial L}{\partial \dot q_i}
->
+\label{introEquation:generalizedMomenta}
->
+\end{equation}
->
+The Lagrange equations of motion are then expressed by
->
+\begin{equation}
->
+p_i  = \frac{{\partial L}}{{\partial q_i }}
->
+\label{introEquation:generalizedMomentaDot}
->
+\end{equation}
->
+With the help of the generalized momenta, we may now define a new
->
+quantity $H$ by the equation
->
+\begin{equation}
->
+H = \sum\limits_k {p_k \dot q_k }  - L ,
->
+\label{introEquation:hamiltonianDefByLagrangian}
->
+\end{equation}
->
+where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
->
+$L$ is the Lagrangian function for the system. Differentiating
->
+Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain
->
+\begin{equation}
->
+dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
->
+\frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
->
+L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
->
+L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1}
->
+\end{equation}
->
+Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second
->
+and fourth terms in the parentheses cancel. Therefore,
->
+Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
->
+\begin{equation}
->
+dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
->
+\right)}  - \frac{{\partial L}}{{\partial t}}dt .
->
+\label{introEquation:diffHamiltonian2}
->
+\end{equation}
->
+By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
->
+find
->
+\begin{equation}
->
+\frac{{\partial H}}{{\partial p_k }} = \dot {q_k}
->
+\label{introEquation:motionHamiltonianCoordinate}
->
+\end{equation}
->
+\begin{equation}
->
+\frac{{\partial H}}{{\partial q_k }} =  - \dot {p_k}
->
+\label{introEquation:motionHamiltonianMomentum}
->
+\end{equation}
->
+and
->
+\begin{equation}
->
+\frac{{\partial H}}{{\partial t}} =  - \frac{{\partial L}}{{\partial
->
+t}}
->
+\label{introEquation:motionHamiltonianTime}
->
+\end{equation}
->
+where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
->
+Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
->
+equation of motion. Due to their symmetrical formula, they are also
->
+known as the canonical equations of motions.\cite{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 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
->
+(Eq.~\ref{introEquation:energyConservation}). It follows that
->
+Hamilton's equations of motion conserve the total Hamiltonian
->
+\begin{equation}
->
+\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
->
+H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
->
+}}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
->
+H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
->
+\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
->
+q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian}
->
+\end{equation}
->
->
+\section{\label{introSection:statisticalMechanics}Statistical
->
+Mechanics}
->
->
+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 and theorems presented in this
->
+dissertation.
->
->
+\subsection{\label{introSection:ensemble}Phase Space and Ensemble}
->
->
+Mathematically, phase space is the space which represents all
->
+possible states of a system. Each possible state of the system
->
+corresponds to one unique point in the phase space. For mechanical
->
+systems, the phase space usually consists of all possible values of
->
+position and momentum variables. Consider a dynamic system of $f$
->
+particles in a cartesian space, where each of the $6f$ coordinates
->
+and momenta is assigned to one of $6f$ mutually orthogonal axes, the
->
+phase space of this system is a $6f$ dimensional space. A point, $x
->
+=
->
+(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
->
+\over q} _1 , \ldots
->
+,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
->
+\over q} _f
->
+,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
->
+\over p} _1  \ldots
->
+,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
->
+\over p} _f )$ , with a unique set of values of $6f$ coordinates and
->
+momenta is a phase space vector.
->
+%%%fix me
->
->
+In statistical mechanics, the condition of an ensemble at any time
->
+can be regarded as appropriately specified by the density $\rho$
->
+with which representative points are distributed over the phase
->
+space. The density distribution for an ensemble with $f$ degrees of
->
+freedom is defined as,
->
+\begin{equation}
->
+\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 changes the density at any time at phase
->
+space. Hence, the density distribution is also to be taken as a
->
+function of the time. The number of systems $\delta N$ at time $t$
->
+can be determined by,
->
+\begin{equation}
->
+\delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
->
+\label{introEquation:deltaN}
->
+\end{equation}
->
+Assuming enough copies of the systems, we can sufficiently
->
+approximate $\delta N$ without introducing discontinuity when we go
->
+from one region in the phase space to another. By integrating over
->
+the whole phase space,
->
+\begin{equation}
->
+N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
->
+\label{introEquation:totalNumberSystem}
->
+\end{equation}
->
+gives us an expression for the total number of copies. Hence, the
->
+probability per unit volume in the phase space can be obtained by,
->
+\begin{equation}
->
+\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int
->
+{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
->
+\label{introEquation:unitProbability}
->
+\end{equation}
->
+With the help of Eq.~\ref{introEquation:unitProbability} and the
->
+knowledge of the system, it is possible to calculate the average
->
+value of any desired quantity which depends on the coordinates and
->
+momenta of the system. Even when the dynamics of the real system are
->
+complex, or stochastic, or even discontinuous, the average
->
+properties of the ensemble of possibilities as a whole remain well
->
+defined. For a classical system in thermal equilibrium with its
->
+environment, the ensemble average of a mechanical quantity, $\langle
->
+A(q , p) \rangle_t$, takes the form of an integral over the phase
->
+space of the system,
->
+\begin{equation}
->
+\langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
->
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
->
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
->
+\label{introEquation:ensembelAverage}
->
+\end{equation}
->
->
+\subsection{\label{introSection:liouville}Liouville's theorem}
->
->
+Liouville's theorem is the foundation on which statistical mechanics
->
+rests. It describes the time evolution of the phase space
->
+distribution function. In order to calculate the rate of change of
->
+$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider
->
+the two faces perpendicular to the $q_1$ axis, which are located at
->
+$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the
->
+opposite face is given by the expression,
->
+\begin{equation}
->
+\left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
->
+\right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
->
+}}\delta q_1 } \right)\delta q_2  \ldots \delta q_f \delta p_1
->
+\ldots \delta p_f .
->
+\end{equation}
->
+Summing all over the phase space, we obtain
->
+\begin{equation}
->
+\frac{{d(\delta N)}}{{dt}} =  - \sum\limits_{i = 1}^f {\left[ {\rho
->
+\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} +
->
+\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left(
->
+{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  + \frac{{\partial
->
+\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1
->
+\ldots \delta q_f \delta p_1  \ldots \delta p_f .
->
+\end{equation}
->
+Differentiating the equations of motion in Hamiltonian formalism
->
+(\ref{introEquation:motionHamiltonianCoordinate},
->
+\ref{introEquation:motionHamiltonianMomentum}), we can show,
->
+\begin{equation}
->
+\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }}
->
++ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
->
+\end{equation}
->
+which cancels the first terms of the right hand side. Furthermore,
->
+dividing $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
->
+p_f $ in both sides, we can write out Liouville's theorem in a
->
+simple form,
->
+\begin{equation}
->
+\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f
->
+{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  +
->
+\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
->
+\label{introEquation:liouvilleTheorem}
->
+\end{equation}
->
+Liouville's theorem states that the distribution function is
->
+constant along any trajectory in phase space. In classical
->
+statistical mechanics, since the number of system copies in an
->
+ensemble is huge and constant, we can assume the local density has
->
+no reason (other than classical mechanics) to change,
->
+\begin{equation}
->
+\frac{{\partial \rho }}{{\partial t}} = 0.
->
+\label{introEquation:stationary}
->
+\end{equation}
->
+In such stationary system, the density of distribution $\rho$ can be
->
+connected to the Hamiltonian $H$ through Maxwell-Boltzmann
->
+distribution,
->
+\begin{equation}
->
+\rho  \propto e^{ - \beta H}
->
+\label{introEquation:densityAndHamiltonian}
->
+\end{equation}
->
->
+\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}}
->
+Lets consider a region in the phase space,
->
+\begin{equation}
->
+\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
->
+\end{equation}
->
+If this region is small enough, the density $\rho$ can be regarded
->
+as uniform over the whole integral. Thus, the number of phase points
->
+inside this region is given by,
->
+\begin{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 the stationary assumption
->
+(Eq.~\ref{introEquation:stationary}), we obtain the principle of
->
+\emph{conservation of volume in phase space},
->
+\begin{equation}
->
+\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
->
+...dq_f dp_1 } ..dp_f  = 0.
->
+\label{introEquation:volumePreserving}
->
+\end{equation}
->
->
+\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}}
->
->
+Liouville's theorem can be expressed in a variety of different forms
->
+which are convenient within different contexts. For any two function
->
+$F$ and $G$ of the coordinates and momenta of a system, the Poisson
->
+bracket $\{F,G\}$ is defined as
->
+\begin{equation}
->
+\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial
->
+F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} -
->
+\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial
->
+q_i }}} \right)}.
->
+\label{introEquation:poissonBracket}
->
+\end{equation}
->
+Substituting equations of motion in Hamiltonian formalism
->
+(Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
->
+Eq.~\ref{introEquation:motionHamiltonianMomentum}) into
->
+(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite
->
+Liouville's theorem using Poisson bracket notion,
->
+\begin{equation}
->
+\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
->
+{\rho ,H} \right\}.
->
+\label{introEquation:liouvilleTheromInPoissin}
->
+\end{equation}
->
+Moreover, the Liouville operator is defined as
->
+\begin{equation}
->
+iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial
->
+p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial
->
+H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)}
->
+\label{introEquation:liouvilleOperator}
->
+\end{equation}
->
+In terms of Liouville operator, Liouville's equation can also be
->
+expressed as
->
+\begin{equation}
->
+\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - iL\rho
->
+\label{introEquation:liouvilleTheoremInOperator}
->
+\end{equation}
->
+which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$.
->
+\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
->
->
+Various thermodynamic properties can be calculated from Molecular
->
+Dynamics simulation. By comparing experimental values with the
->
+calculated properties, one can determine the accuracy of the
->
+simulation and the quality of the underlying model. However, both
->
+experiments and computer simulations are usually performed during a
->
+certain time interval and the measurements are averaged over a
->
+period of time which is different from the average behavior of
->
+many-body system in Statistical Mechanics. Fortunately, the Ergodic
->
+Hypothesis makes a connection between time average and the ensemble
->
+average. It states that the time average and average over the
->
+statistical ensemble are identical:\cite{Frenkel1996, Leach2001}
->
+\begin{equation}
->
+\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
->
+\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
->
+{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp
->
+\end{equation}
->
+where $\langle  A(q , p) \rangle_t$ is an equilibrium value of a
->
+physical quantity and $\rho (p(t), q(t))$ is the equilibrium
->
+distribution function. If an observation is averaged over a
->
+sufficiently long time (longer than the relaxation time), all
->
+accessible microstates in phase space are assumed to be equally
->
+probed, giving a properly weighted statistical average. This allows
->
+the researcher freedom of choice when deciding how best to measure a
->
+given observable. In case an ensemble averaged approach sounds most
->
+reasonable, the Monte Carlo methods\cite{Metropolis1949} can be
->
+utilized. Or if the system lends itself to a time averaging
->
+approach, the Molecular Dynamics techniques in
->
+Sec.~\ref{introSection:molecularDynamics} will be the best
->
+choice.\cite{Frenkel1996}
->
->
+\section{\label{introSection:geometricIntegratos}Geometric Integrators}
->
+A variety of numerical integrators have been proposed to simulate
->
+the motions of atoms in MD simulation. They usually begin with
->
+initial conditions and move the objects in the direction governed by
->
+the differential equations. However, most of them ignore the hidden
->
+physical laws contained within the equations. Since 1990, geometric
->
+integrators, which preserve various phase-flow invariants such as
->
+symplectic structure, volume and time reversal symmetry, were
->
+developed to address this issue.\cite{Dullweber1997, McLachlan1998,
->
+Leimkuhler1999} The velocity Verlet method, which happens to be a
->
+simple example of symplectic integrator, continues to gain
->
+popularity in the molecular dynamics community. This fact can be
->
+partly explained by its geometric nature.
->
->
+\subsection{\label{introSection:symplecticManifold}Symplectic 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.\cite{Hirsch1997} A \emph{symplectic manifold} is
->
+defined as a pair $(M, \omega)$ which consists of a
->
+\emph{differentiable manifold} $M$ and a close, non-degenerate,
->
+bilinear symplectic form, $\omega$. A symplectic form on a vector
->
+space $V$ is a function $\omega(x, y)$ which satisfies
->
+$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
->
+\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
->
+$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in
->
+vector field is an example of symplectic form. 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.\cite{Jost2002} Every
->
+symplectic manifold is even dimensional. For instance, in Hamilton
->
+equations, coordinate and momentum always appear in pairs.
->
->
+\subsection{\label{introSection:ODE}Ordinary Differential Equations}
->
->
+For an ordinary differential system defined as
->
+\begin{equation}
->
+\dot x = f(x)
->
+\end{equation}
->
+where $x = x(q,p)$, this system is a canonical Hamiltonian, if
->
+$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian
->
+function and $J$ is the skew-symmetric matrix
->
+\begin{equation}
->
+J = \left( {\begin{array}{*{20}c}
->
+& I  \\
->
+   { - I} & 0  \\
->
+\end{array}} \right)
->
+\label{introEquation:canonicalMatrix}
->
+\end{equation}
->
+where $I$ is an identity matrix. Using this notation, Hamiltonian
->
+system can be rewritten as,
->
+\begin{equation}
->
+\frac{d}{{dt}}x = J\nabla _x H(x).
->
+\label{introEquation:compactHamiltonian}
->
+\end{equation}In this case, $f$ is
->
+called a \emph{Hamiltonian vector field}. Another generalization of
->
+Hamiltonian dynamics is Poisson Dynamics,\cite{Olver1986}
->
+\begin{equation}
->
+\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
->
+\end{equation}
->
+where the most obvious change being that matrix $J$ now depends on
->
+$x$.
->
->
+\subsection{\label{introSection:exactFlow}Exact Propagator}
->
->
+Let $x(t)$ be the exact solution of the ODE
->
+system,
->
+\begin{equation}
->
+\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE}
->
+\end{equation} we can
->
+define its exact propagator $\varphi_\tau$:
->
+\[ x(t+\tau)
->
+=\varphi_\tau(x(t))
->
+\]
->
+where $\tau$ is a fixed time step and $\varphi$ is a map from phase
->
+space to itself. The propagator has the continuous group property,
->
+\begin{equation}
->
+\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
->
++ \tau _2 } .
->
+\end{equation}
->
+In particular,
->
+\begin{equation}
->
+\varphi _\tau   \circ \varphi _{ - \tau }  = I
->
+\end{equation}
->
+Therefore, the exact propagator is self-adjoint,
->
+\begin{equation}
->
+\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
->
+\end{equation}
->
+The exact propagator can also be written as 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 propagator
->
+$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$,
->
+which is usually called an integrator. The order of an integrator
->
+$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
->
+order $p$,
->
+\begin{equation}
->
+\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1})
->
+\end{equation}
->
->
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
->
->
+The hidden geometric properties\cite{Budd1999, Marsden1998} of an
->
+ODE and its propagator play important roles in numerical studies.
->
+Many of them can be found in systems which occur naturally in
->
+applications. Let $\varphi$ be the propagator of Hamiltonian vector
->
+field, $\varphi$ is a \emph{symplectic} propagator if it satisfies,
->
+\begin{equation}
->
+{\varphi '}^T J \varphi ' = J.
->
+\end{equation}
->
+According to Liouville's theorem, the symplectic volume is invariant
->
+under a Hamiltonian propagator, which is the basis for classical
->
+statistical mechanics. Furthermore, the propagator of a Hamiltonian
->
+vector field on a symplectic manifold can be shown to be a
->
+symplectomorphism. As to the Poisson system,
->
+\begin{equation}
->
+{\varphi '}^T J \varphi ' = J \circ \varphi
->
+\end{equation}
->
+is the property that must be preserved by the integrator. It is
->
+possible to construct a \emph{volume-preserving} propagator for a
->
+source free ODE ($ \nabla \cdot f = 0 $), if the propagator
->
+satisfies $ \det d\varphi  = 1$. One can show easily that a
->
+symplectic propagator will be volume-preserving. Changing the
->
+variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will
->
+result in a new system,
->
+\[
->
+\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
->
+\]
->
+The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
->
+In other words, the propagator of this vector field is reversible if
->
+and only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
->
+conserved quantity of a general differential function is a function
->
+$ G:R^{2d}  \to R^d $ which is constant for all solutions of the ODE
->
+$\frac{{dx}}{{dt}} = f(x)$ ,
->
+\[
->
+\frac{{dG(x(t))}}{{dt}} = 0.
->
+\]
->
+Using the chain rule, one may obtain,
->
+\[
->
+\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G,
->
+\]
->
+which is the condition for conserved quantities. For a canonical
->
+Hamiltonian system, the time evolution of an arbitrary smooth
->
+function $G$ is given by,
->
+\begin{eqnarray}
->
+\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\
->
+                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)).
->
+\label{introEquation:firstIntegral1}
->
+\end{eqnarray}
->
+Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1}
->
+can be rewritten as
->
+\[
->
+\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
->
+\]
->
+Therefore, the sufficient condition for $G$ to be a conserved
->
+quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As
->
+is well known, the Hamiltonian (or energy) H of a Hamiltonian system
->
+is a conserved quantity, which is due to the fact $\{ H,H\}  = 0$.
->
+When designing any numerical methods, one should always try to
->
+preserve the structural properties of the original ODE and its
->
+propagator.
->
->
+\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
->
+A lot of well established and very effective numerical methods have
->
+been successful precisely because of their symplectic nature even
->
+though this fact was not recognized when they were first
->
+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 Variational methods
->
+\item Runge-Kutta methods
->
+\item Splitting methods
->
+\end{enumerate}
->
+Generating functions\cite{Channell1990} tend to lead to methods
->
+which are cumbersome and difficult to use. In dissipative systems,
->
+variational methods can capture the decay of energy
->
+accurately.\cite{Kane2000} Since they are geometrically unstable
->
+against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
->
+methods are not suitable for Hamiltonian system. Recently, various
->
+high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003}
->
+have been developed to overcome this instability. However, due to
->
+computational penalty involved in implementing the Runge-Kutta
->
+methods, they have not attracted much attention from the Molecular
->
+Dynamics community. Instead, splitting methods have been widely
->
+accepted since they exploit natural decompositions of the
->
+system.\cite{McLachlan1998, Tuckerman1992}
->
->
+\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
->
->
+The main idea behind splitting methods is to decompose the discrete
->
+$\varphi_h$ as a composition of simpler propagators,
->
+\begin{equation}
->
+\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
->
+\varphi _{h_n }
->
+\label{introEquation:FlowDecomposition}
->
+\end{equation}
->
+where each of the sub-propagator is chosen such that each represent
->
+a simpler integration of the system. Suppose that a Hamiltonian
->
+system takes the form,
->
+\[
->
+H = H_1 + H_2.
->
+\]
->
+Here, $H_1$ and $H_2$ may represent different physical processes of
->
+the system. For instance, they may relate to kinetic and potential
->
+energy respectively, which is a natural decomposition of the
->
+problem. If $H_1$ and $H_2$ can be integrated using exact
->
+propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a
->
+simple first order expression is then given by the Lie-Trotter
->
+formula
->
+\begin{equation}
->
+\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
->
+\label{introEquation:firstOrderSplitting}
->
+\end{equation}
->
+where $\varphi _h$ is the result of applying the corresponding
->
+continuous $\varphi _i$ over a time $h$. By definition, as
->
+$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
->
+must follow that each operator $\varphi_i(t)$ is a symplectic map.
->
+It is easy to show that any composition of symplectic propagators
->
+yields a symplectic map,
->
+\begin{equation}
->
+(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
->
+'\phi ' = \phi '^T J\phi ' = J,
->
+ \label{introEquation:SymplecticFlowComposition}
->
+\end{equation}
->
+where $\phi$ and $\psi$ both are symplectic maps. Thus operator
->
+splitting in this context automatically generates a symplectic map.
->
+The Lie-Trotter
->
+splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces
->
+local errors proportional to $h^2$, while the Strang splitting gives
->
+a second-order decomposition,\cite{Strang1968}
->
+\begin{equation}
->
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
->
+_{1,h/2} , \label{introEquation:secondOrderSplitting}
->
+\end{equation}
->
+which has a local error proportional to $h^3$. The Strang
->
+splitting's popularity in molecular simulation community attribute
->
+to its symmetric property,
->
+\begin{equation}
->
+\varphi _h^{ - 1} = \varphi _{ - h}.
->
+\label{introEquation:timeReversible}
->
+\end{equation}
->
->
+\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}}
->
+The classical equation for a system consisting of interacting
->
+particles can be written in Hamiltonian form,
->
+\[
->
+H = T + V
->
+\]
->
+where $T$ is the kinetic energy and $V$ is the potential energy.
->
+Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one
->
+obtains the following:
->
+\begin{align}
->
+q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
->
+    \frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
->
+\label{introEquation:Lp10a} \\%
->
+%
->
+\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
->
+    \biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
->
+\label{introEquation:Lp10b}
->
+\end{align}
->
+where $F(t)$ is the force at time $t$. This integration scheme is
->
+known as \emph{velocity verlet} which is
->
+symplectic(Eq.~\ref{introEquation:SymplecticFlowComposition}),
->
+time-reversible(Eq.~\ref{introEquation:timeReversible}) and
->
+volume-preserving (Eq.~\ref{introEquation:volumePreserving}). These
->
+geometric properties attribute to its long-time stability and its
->
+popularity in the community. However, the most commonly used
->
+velocity verlet integration scheme is written as below,
->
+\begin{align}
->
+\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
->
+    \dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
->
+%
->
+q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
->
+    \label{introEquation:Lp9b}\\%
->
+%
->
+\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
->
+    \frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c}
->
+\end{align}
->
+From the preceding splitting, one can see that the integration of
->
+the equations of motion would follow:
->
+\begin{enumerate}
->
+\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
->
->
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
->
->
+\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move.
->
->
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
->
+\end{enumerate}
->
+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}
->
->
+\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
->
->
+The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used
->
+to determine the local error of a splitting method in terms of the
->
+commutator of the
->
+operators(Eq.~\ref{introEquation:exponentialOperator}) associated
->
+with the sub-propagator. For operators $hX$ and $hY$ which are
->
+associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we
->
+have
->
+\begin{equation}
->
+\exp (hX + hY) = \exp (hZ)
->
+\end{equation}
->
+where
->
+\begin{equation}
->
+hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left(
->
+{[X,[X,Y]] + [Y,[Y,X]]} \right) +  \ldots .
->
+\end{equation}
->
+Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by
->
+\[
->
+[X,Y] = XY - YX .
->
+\]
->
+Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
->
+to the Strang splitting, we can obtain
->
+\begin{eqnarray*}
->
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\
->
+                                   &   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
->
+                                   &   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots
->
+                                   ).
->
+\end{eqnarray*}
->
+Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local
->
+error of Strang splitting is proportional to $h^3$. The same
->
+procedure can be applied to a general splitting of the form
->
+\begin{equation}
->
+\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
->
+} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
->
+\end{equation}
->
+A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher
->
+order methods. Yoshida proposed an elegant way to compose higher
->
+order methods based on symmetric splitting.\cite{Yoshida1990} Given
->
+a symmetric second order base method $ \varphi _h^{(2)} $, a
->
+fourth-order symmetric method can be constructed by composing,
->
+\[
->
+\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
->
+h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
->
+\]
->
+where $ \alpha  =  - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta
->
+= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric
->
+integrator $ \varphi _h^{(2n + 2)}$ can be composed by
->
+\begin{equation}
->
+\varphi _h^{(2n + 2)}  = \varphi _{\alpha h}^{(2n)}  \circ \varphi
->
+_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)},
->
+\end{equation}
->
+if the weights are chosen as
->
+\[
->
+\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
->
+\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
->
+\]
->
->
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
->
->
+As one of the principal tools of molecular modeling, Molecular
->
+dynamics has proven to be a powerful tool for studying the functions
->
+of biological systems, providing structural, thermodynamic and
->
+dynamical information. The basic idea of molecular dynamics is that
->
+macroscopic properties are related to microscopic behavior and
->
+microscopic behavior can be calculated from the trajectories in
->
+simulations. For instance, instantaneous temperature of a
->
+Hamiltonian system of $N$ particles can be measured by
->
+\[
->
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
->
+\]
->
+where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
->
+respectively, $f$ is the number of degrees of freedom, and $k_B$ is
->
+the Boltzman constant.
->
->
+A typical molecular dynamics run consists of three essential steps:
->
+\begin{enumerate}
->
+  \item Initialization
->
+    \begin{enumerate}
->
+    \item Preliminary preparation
->
+    \item Minimization
->
+    \item Heating
->
+    \item Equilibration
->
+    \end{enumerate}
->
+  \item Production
->
+  \item Analysis
->
+\end{enumerate}
->
+These three individual steps will be covered in the following
->
+sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
->
+initialization of a simulation. Sec.~\ref{introSection:production}
->
+ discusses issues of production runs.
->
+Sec.~\ref{introSection:Analysis} provides the theoretical tools for
->
+analysis of trajectories.
-<
+\section{\label{introSection:correlationFunctions}Correlation Functions}
->
+\subsection{\label{introSec:initialSystemSettings}Initialization}
->
->
+\subsubsection{\textbf{Preliminary preparation}}
->
->
+When selecting the starting structure of a molecule for molecular
->
+simulation, one may retrieve its Cartesian coordinates from public
->
+databases, such as RCSB Protein Data Bank \textit{etc}. Although
->
+thousands of crystal structures of molecules are discovered every
->
+year, many more remain unknown due to the difficulties of
->
+purification and crystallization. Even for molecules with known
->
+structures, some important information is missing. For example, a
->
+missing hydrogen atom which acts as donor in hydrogen bonding must
->
+be added. Moreover, in order to include electrostatic interactions,
->
+one may need to specify the partial charges for individual atoms.
->
+Under some circumstances, we may even need to prepare the system in
->
+a special configuration. For instance, when studying transport
->
+phenomenon in membrane systems, we may prepare the lipids in a
->
+bilayer structure instead of placing lipids randomly in solvent,
->
+since we are not interested in the slow self-aggregation process.
->
->
+\subsubsection{\textbf{Minimization}}
->
->
+It is quite possible that some of molecules in the system from
->
+preliminary preparation may be overlapping with each other. This
->
+close proximity leads to high initial potential energy which
->
+consequently jeopardizes any molecular dynamics simulations. To
->
+remove these steric overlaps, one typically performs energy
->
+minimization to find a more reasonable conformation. Several energy
->
+minimization methods have been developed to exploit the energy
->
+surface and to locate the local minimum. While converging slowly
->
+near the minimum, the steepest descent method is extremely robust when
->
+systems are strongly anharmonic. Thus, it is often used to refine
->
+structures from crystallographic data. Relying on the Hessian,
->
+advanced methods like Newton-Raphson converge rapidly to a local
->
+minimum, but become unstable if the energy surface is far from
->
+quadratic. Another factor that must be taken into account, when
->
+choosing energy minimization method, is the size of the system.
->
+Steepest descent and conjugate gradient can deal with models of any
->
+size. Because of the limits on computer memory to store the hessian
->
+matrix and the computing power needed to diagonalize these matrices,
->
+most Newton-Raphson methods can not be used with very large systems.
->
->
+\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 setting the temperature of the system to a final temperature
->
+at which the simulation will be conducted. In the heating phase, we
->
+should also keep the system from drifting or rotating as a whole. To
->
+do this, the net linear momentum and angular momentum of the system
->
+is shifted to zero after each resampling from the Maxwell -Boltzman
->
+distribution.
->
->
+\subsubsection{\textbf{Equilibration}}
->
->
+The purpose of equilibration is to allow the system to evolve
->
+spontaneously for a period of time and reach equilibrium. The
->
+procedure is continued until various statistical properties, such as
->
+temperature, pressure, energy, volume and other structural
->
+properties \textit{etc}, become independent of time. Strictly
->
+speaking, minimization and heating are not necessary, provided the
->
+equilibration process is long enough. However, these steps can serve
->
+as a mean to arrive at an equilibrated structure in an effective
->
+way.
->
->
+\subsection{\label{introSection:production}Production}
->
->
+The production run is the most important step of the simulation, in
->
+which the equilibrated structure is used as a starting point and the
->
+motions of the molecules are collected for later analysis. In order
->
+to capture the macroscopic properties of the system, the molecular
->
+dynamics simulation must be performed by sampling correctly and
->
+efficiently from the relevant thermodynamic ensemble.
->
->
+The most expensive part of a molecular dynamics simulation is the
->
+calculation of non-bonded forces, such as van der Waals force and
->
+Coulombic forces \textit{etc}. For a system of $N$ particles, the
->
+complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
->
+which makes large simulations prohibitive in the absence of any
->
+algorithmic tricks. A natural approach to avoid system size issues
->
+is to represent the bulk behavior by a finite number of the
->
+particles. However, this approach will suffer from surface effects
->
+at the edges of the simulation. To offset this, \textit{Periodic
->
+boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to
->
+simulate bulk properties with a relatively small number of
->
+particles. In this method, the simulation box is replicated
->
+throughout space to form an infinite lattice. During the simulation,
->
+when a particle moves in the primary cell, its image in other cells
->
+move in exactly the same direction with exactly the same
->
+orientation. Thus, as a particle leaves the primary cell, one of its
->
+images will enter through the opposite face.
->
+\begin{figure}
->
+\centering
->
+\includegraphics[width=\linewidth]{pbc.eps}
->
+\caption[An illustration of periodic boundary conditions]{A 2-D
->
+illustration of periodic boundary conditions. As one particle leaves
->
+the left of the simulation box, an image of it enters the right.}
->
+\label{introFig:pbc}
->
+\end{figure}
->
->
+%cutoff and minimum image convention
->
+Another important technique to improve the efficiency of force
->
+evaluation is to apply spherical cutoffs where particles farther
->
+than a predetermined distance are not included in the
->
+calculation.\cite{Frenkel1996} The use of a cutoff radius will cause
->
+a discontinuity in the potential energy curve. Fortunately, one can
->
+shift a simple radial potential to ensure the potential curve go
->
+smoothly to zero at the cutoff radius. The cutoff strategy works
->
+well for Lennard-Jones interaction because of its short range
->
+nature. However, simply truncating the electrostatic interaction
->
+with the use of cutoffs has been shown to lead to severe artifacts
->
+in simulations. The Ewald summation, in which the slowly decaying
->
+Coulomb potential is transformed into direct and reciprocal sums
->
+with rapid and absolute convergence, has proved to minimize the
->
+periodicity artifacts in liquid simulations. Taking advantage of
->
+fast Fourier transform (FFT) techniques for calculating discrete
->
+Fourier transforms, the particle mesh-based
->
+methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
->
+$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the
->
+\emph{fast multipole method}\cite{Greengard1987, Greengard1994},
->
+which treats Coulombic interactions exactly at short range, and
->
+approximate the potential at long range through multipolar
->
+expansion. In spite of their wide acceptance at the molecular
->
+simulation community, these two methods are difficult to implement
->
+correctly and efficiently. Instead, we use a damped and
->
+charge-neutralized Coulomb potential method developed by Wolf and
->
+his coworkers.\cite{Wolf1999} The shifted Coulomb potential for
->
+particle $i$ and particle $j$ at distance $r_{rj}$ is given by:
->
+\begin{equation}
->
+V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
->
+r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
->
+R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
->
+r_{ij})}{r_{ij}}\right\}, \label{introEquation:shiftedCoulomb}
->
+\end{equation}
->
+where $\alpha$ is the convergence parameter. Due to the lack of
->
+inherent periodicity and rapid convergence,this method is extremely
->
+efficient and easy to implement.
->
+\begin{figure}
->
+\centering
->
+\includegraphics[width=\linewidth]{shifted_coulomb.eps}
->
+\caption[An illustration of shifted Coulomb potential]{An
->
+illustration of shifted Coulomb potential.}
->
+\label{introFigure:shiftedCoulomb}
->
+\end{figure}
->
->
+%multiple time step
->
->
+\subsection{\label{introSection:Analysis} Analysis}
->
->
+Recently, advanced visualization techniques have been applied to
->
+monitor the motions of molecules. Although the dynamics of the
->
+system can be described qualitatively from animation, quantitative
->
+trajectory analysis is more useful. According to the principles of
->
+Statistical Mechanics in
->
+Sec.~\ref{introSection:statisticalMechanics}, one can compute
->
+thermodynamic properties, analyze fluctuations of structural
->
+parameters, and investigate time-dependent processes of the molecule
->
+from the trajectories.
->
->
+\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}
->
->
+Thermodynamic properties, which can be expressed in terms of some
->
+function of the coordinates and momenta of all particles in the
->
+system, can be directly computed from molecular dynamics. The usual
->
+way to measure the pressure is based on virial theorem of Clausius
->
+which states that the virial is equal to $-3Nk_BT$. For a system
->
+with forces between particles, the total virial, $W$, contains the
->
+contribution from external pressure and interaction between the
->
+particles:
->
+\[
->
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
->
+f_{ij} } } \right\rangle
->
+\]
->
+where $f_{ij}$ is the force between particle $i$ and $j$ at a
->
+distance $r_{ij}$. Thus, the expression for the pressure is given
->
+by:
->
+\begin{equation}
->
+P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
->
+< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
->
+\end{equation}
->
->
+\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}
->
->
+Structural Properties of a simple fluid can be described by a set of
->
+distribution functions. Among these functions,the \emph{pair
->
+distribution function}, also known as \emph{radial distribution
->
+function}, is of most fundamental importance to liquid theory.
->
+Experimentally, pair distribution functions can be gathered by
->
+Fourier transforming raw data from a series of neutron diffraction
->
+experiments and integrating over the surface
->
+factor.\cite{Powles1973} The experimental results can serve as a
->
+criterion to justify the correctness of a liquid model. Moreover,
->
+various equilibrium thermodynamic and structural properties can also
->
+be expressed in terms of the radial distribution
->
+function.\cite{Allen1987} The pair distribution functions $g(r)$
->
+gives the probability that a particle $i$ will be located at a
->
+distance $r$ from a another particle $j$ in the system
->
+\begin{equation}
->
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
->
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
->
+(r)}{\rho}.
->
+\end{equation}
->
+Note that the delta function can be replaced by a histogram in
->
+computer simulation. Peaks in $g(r)$ represent solvent shells, and
->
+the height of these peaks gradually decreases to 1 as the liquid of
->
+large distance approaches the bulk density.
->
->
->
+\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
->
+Properties}}
->
->
+Time-dependent properties are usually calculated using \emph{time
->
+correlation functions}, which correlate random variables $A$ and $B$
->
+at two different times,
->
+\begin{equation}
->
+C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
->
+\label{introEquation:timeCorrelationFunction}
->
+\end{equation}
->
+If $A$ and $B$ refer to same variable, this kind of correlation
->
+functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation
->
+function which is directly related to transport properties of
->
+molecular liquids:
->
+\begin{equation}
->
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
->
+\right\rangle } dt
->
+\end{equation}
->
+where $D$ is diffusion constant. Unlike the velocity autocorrelation
->
+function, which is averaged over time origins and over all the
->
+atoms, the dipole autocorrelation functions is calculated for the
->
+entire system. The dipole autocorrelation function is given by:
->
+\begin{equation}
->
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
->
+\right\rangle
->
+\end{equation}
->
+Here $u_{tot}$ is the net dipole of the entire system and is given
->
+by
->
+\begin{equation}
->
+u_{tot} (t) = \sum\limits_i {u_i (t)}.
->
+\end{equation}
->
+In principle, many time correlation functions can be related to
->
+Fourier transforms of the infrared, Raman, and inelastic neutron
->
+scattering spectra of molecular liquids. In practice, one can
->
+extract the IR spectrum from the intensity of the molecular dipole
->
+fluctuation at each frequency using the following relationship:
->
+\begin{equation}
->
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
->
+i2\pi vt} dt}.
->
+\end{equation}
->
->
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
->
->
+Rigid bodies are frequently involved in the modeling of different
->
+areas, including engineering, physics and chemistry. For example,
->
+missiles and vehicles are usually modeled by rigid bodies.  The
->
+movement of the objects in 3D gaming engines or other physics
->
+simulators is governed by rigid body dynamics. In molecular
->
+simulations, rigid bodies are used to simplify protein-protein
->
+docking studies.\cite{Gray2003}
->
->
+It is very important to develop stable and efficient methods to
->
+integrate the equations of motion for orientational degrees of
->
+freedom. Euler angles are the natural choice to describe the
->
+rotational degrees of freedom. However, due to $\frac {1}{sin
->
+\theta}$ singularities, the numerical integration of corresponding
->
+equations of these motion is very inefficient and inaccurate.
->
+Although an alternative integrator using multiple sets of Euler
->
+angles can overcome this difficulty\cite{Barojas1973}, the
->
+computational penalty and the loss of angular momentum conservation
->
+still remain. A singularity-free representation utilizing
->
+quaternions was developed by Evans in 1977.\cite{Evans1977}
->
+Unfortunately, this approach used a nonseparable Hamiltonian
->
+resulting from the quaternion representation, which prevented the
->
+symplectic algorithm from being utilized. Another different approach
->
+is to apply holonomic constraints to the atoms belonging to the
->
+rigid body. Each atom moves independently under the normal forces
->
+deriving from potential energy and constraint forces which are used
->
+to guarantee the rigidness. However, due to their iterative nature,
->
+the SHAKE and Rattle algorithms also converge very slowly when the
->
+number of constraints increases.\cite{Ryckaert1977, Andersen1983}
->
->
+A break-through in geometric literature suggests that, in order to
->
+develop a long-term integration scheme, one should preserve the
->
+symplectic structure of the propagator. By introducing a conjugate
->
+momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's
->
+equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was
->
+proposed to evolve the Hamiltonian system in a constraint manifold
->
+by iteratively satisfying the orthogonality constraint $Q^T Q = 1$.
->
+An alternative method using the quaternion representation was
->
+developed by Omelyan.\cite{Omelyan1998} However, both of these
->
+methods are iterative and inefficient. In this section, we descibe a
->
+symplectic Lie-Poisson integrator for rigid bodies developed by
->
+Dullweber and his coworkers\cite{Dullweber1997} in depth.
->
->
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
->
+The Hamiltonian of a rigid body is given by
->
+\begin{equation}
->
+H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
->
+V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
->
+\label{introEquation:RBHamiltonian}
->
+\end{equation}
->
+Here, $q$ and $Q$  are the position vector and rotation matrix for
->
+the rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ ,
->
+and $J$, a diagonal matrix, is defined by
->
+\[
->
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
->
+\]
->
+where $I_{ii}$ is the diagonal element of the inertia tensor. This
->
+constrained Hamiltonian equation is subjected to a holonomic
->
+constraint,
->
+\begin{equation}
->
+Q^T Q = 1, \label{introEquation:orthogonalConstraint}
->
+\end{equation}
->
+which is used to ensure the rotation matrix's unitarity. Using
->
+Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~
->
+\ref{introEquation:motionHamiltonianMomentum}, one can write down
->
+the equations of motion,
->
+\begin{eqnarray}
->
+ \frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\
->
+ \frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\
->
+ \frac{{dQ}}{{dt}} & = & PJ^{ - 1},  \label{introEquation:RBMotionRotation}\\
->
+ \frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
->
+\end{eqnarray}
->
+Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and
->
+using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain,
->
+\begin{equation}
->
+Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
->
+\label{introEquation:RBFirstOrderConstraint}
->
+\end{equation}
->
+In general, there are two ways to satisfy the holonomic constraints.
->
+We can use a constraint force provided by a Lagrange multiplier on
->
+the normal manifold to keep the motion on the constraint space. Or
->
+we can simply evolve the system on the constraint manifold. These
->
+two methods have been proved to be equivalent. The holonomic
->
+constraint and equations of motions define a constraint manifold for
->
+rigid bodies
->
+\[
->
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
->
+\right\}.
->
+\]
->
+Unfortunately, this constraint manifold is not $T^* SO(3)$ which is
->
+a symplectic manifold on Lie rotation group $SO(3)$. However, it
->
+turns out that under symplectic transformation, the cotangent space
->
+and the phase space are diffeomorphic. By introducing
->
+\[
->
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
->
+\]
->
+the mechanical system subjected to a holonomic constraint manifold $M$
->
+can be re-formulated as a Hamiltonian system on the cotangent space
->
+\[
->
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
->
+,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
->
+\]
->
+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
->
+\[
->
+V(q,Q) = V(Q X_0 + q).
->
+\]
->
+Hence, the force and torque are given by
->
+\[
->
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
->
+\]
->
+and
->
+\[
->
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
->
+\]
->
+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}
->
+ \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
->
+ \dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
->
+ \end{array}
->
+ \label{introEqaution:RBMotionPI}
->
+\end{equation}
->
+as well as holonomic constraints $\Pi J^{ - 1}  + J^{ - 1} \Pi ^t  =
->
+$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a
->
+matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism,
->
+\begin{equation}
->
+v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
->
+{\begin{array}{*{20}c}
->
+& { - v_3 } & {v_2 }  \\
->
+   {v_3 } & 0 & { - v_1 }  \\
->
+   { - v_2 } & {v_1 } & 0  \\
->
+\end{array}} \right),
->
+\label{introEquation:hatmapIsomorphism}
->
+\end{equation}
->
+will let us associate the matrix products with traditional vector
->
+operations
->
+\[
->
+\hat vu = v \times u.
->
+\]
->
+Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
->
+matrix,
->
+\begin{eqnarray}
->
+(\dot \Pi  - \dot \Pi ^T )&= &(\Pi  - \Pi ^T )(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
->
+& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  -
->
+(\Lambda  - \Lambda ^T ). \label{introEquation:skewMatrixPI}
->
+\end{eqnarray}
->
+Since $\Lambda$ is symmetric, the last term of
->
+Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
->
+Lagrange multiplier $\Lambda$ is absent from the equations of
->
+motion. This unique property eliminates the requirement of
->
+iterations which can not be avoided in other methods.\cite{Kol1997,
->
+Omelyan1998} Applying the hat-map isomorphism, we obtain the
->
+equation of motion for angular momentum in the body frame
->
+\begin{equation}
->
+\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
->
+F_i (r,Q)} \right) \times X_i }.
->
+\label{introEquation:bodyAngularMotion}
->
+\end{equation}
->
+In the same manner, the equation of motion for rotation matrix is
->
+given by
->
+\[
->
+\dot Q = Qskew(I^{ - 1} \pi ).
->
+\]
->
->
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
->
+Lie-Poisson Integrator for Free Rigid Bodies}
->
->
+If there are no external forces exerted on the rigid body, the only
->
+contribution to the rotational motion is from the kinetic energy
->
+(the first term of \ref{introEquation:bodyAngularMotion}). The free
->
+rigid body is an example of a Lie-Poisson system with Hamiltonian
->
+function
->
+\begin{equation}
->
+T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
->
+\label{introEquation:rotationalKineticRB}
->
+\end{equation}
->
+where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
->
+Lie-Poisson structure matrix,
->
+\begin{equation}
->
+J(\pi ) = \left( {\begin{array}{*{20}c}
->
+& {\pi _3 } & { - \pi _2 }  \\
->
+   { - \pi _3 } & 0 & {\pi _1 }  \\
->
+   {\pi _2 } & { - \pi _1 } & 0  \\
->
+\end{array}} \right).
->
+\end{equation}
->
+Thus, the dynamics of free rigid body is governed by
->
+\begin{equation}
->
+\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
->
+\end{equation}
->
+One may notice that each $T_i^r$ in
->
+Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly.
->
+For instance, the equations of motion due to $T_1^r$ are given by
->
+\begin{equation}
->
+\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
->
+\label{introEqaution:RBMotionSingleTerm}
->
+\end{equation}
->
+with
->
+\[ R_1  = \left( {\begin{array}{*{20}c}
->
+& 0 & 0  \\
->
+& 0 & {\pi _1 }  \\
->
+& { - \pi _1 } & 0  \\
->
+\end{array}} \right).
->
+\]
->
+The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is
->
+\[
->
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
->
+Q(0)e^{\Delta tR_1 }
->
+\]
->
+with
->
+\[
->
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
->
+& 0 & 0  \\
->
+& {\cos \theta _1 } & {\sin \theta _1 }  \\
->
+& { - \sin \theta _1 } & {\cos \theta _1 }  \\
->
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
->
+\]
->
+To reduce the cost of computing expensive functions in $e^{\Delta
->
+tR_1 }$, we can use the Cayley transformation to obtain a
->
+single-aixs propagator,
->
+\begin{eqnarray*}
->
+e^{\Delta tR_1 }  & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta
->
+tR_1 ) \\
->
+%
->
+& \approx & \left( \begin{array}{ccc}
->
+& 0 & 0 \\
->
+& \frac{1-\theta^2 / 4}{1 + \theta^2 / 4}  & -\frac{\theta}{1+
->
+\theta^2 / 4} \\
->
+& \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 +
->
+\theta^2 / 4}
->
+\end{array}
->
+\right).
->
+\end{eqnarray*}
->
+The propagators for $T_2^r$ and $T_3^r$ can be found in the same
->
+manner. In order to construct a second-order symplectic method, we
->
+split the angular kinetic Hamiltonian function into five terms
->
+\[
->
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
->
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
->
+(\pi _1 ).
->
+\]
->
+By concatenating the propagators corresponding to these five terms,
->
+we can obtain an symplectic integrator,
->
+\[
->
+\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
->
+\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
->
+\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
->
+_1 }.
->
+\]
->
+The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by
->
+\[
->
+\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
->
+).
->
+\]
->
+If the Poisson bracket of a function $F$ with an arbitrary smooth
->
+function $G$ is zero, $F$ is a \emph{Casimir}, which is the
->
+conserved quantity in Poisson system. We can easily verify that the
->
+norm of the angular momentum, $\parallel \pi
->
+\parallel$, is a \emph{Casimir}.\cite{McLachlan1993} Let $F(\pi ) = S(\frac{{\parallel
->
+\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
->
+then by the chain rule
->
+\[
->
+\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
->
+}}{2})\pi.
->
+\]
->
+Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
->
+\pi
->
+\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
->
+Lie-Poisson integrator is found to be both extremely efficient and
->
+stable. These properties can be explained by the fact the small
->
+angle approximation is used and the norm of the angular momentum is
->
+conserved.
->
->
+\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
->
+Splitting for Rigid Body}
->
->
+The Hamiltonian of rigid body can be separated in terms of kinetic
->
+energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations
->
+of motion corresponding to potential energy and kinetic energy are
->
+listed in Table~\ref{introTable:rbEquations}.
->
+\begin{table}
->
+\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
->
+\label{introTable:rbEquations}
->
+\begin{center}
->
+\begin{tabular}{|l|l|}
->
+  \hline
->
+  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
->
+  Potential & Kinetic \\
->
+  $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
->
+  $\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
->
+  $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
->
+  $ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
->
+  \hline
->
+\end{tabular}
->
+\end{center}
->
+\end{table}
->
+A second-order symplectic method is now obtained by the composition
->
+of the position and velocity propagators,
->
+\[
->
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
->
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
->
+\]
->
+Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
->
+sub-propagators which corresponding to force and torque
->
+respectively,
->
+\[
->
+\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
->
+_{\Delta t/2,\tau }.
->
+\]
->
+Since the associated operators of $\varphi _{\Delta t/2,F} $ and
->
+$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order
->
+inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the
->
+kinetic energy can be separated to translational kinetic term, $T^t
->
+(p)$, and rotational kinetic term, $T^r (\pi )$,
->
+\begin{equation}
->
+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 Eq.~\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{eqnarray}
->
+ \varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \notag\\
->
+  & & \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \notag\\
->
+  & & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .
->
+\label{introEquation:overallRBFlowMaps}
->
+\end{eqnarray}
->
->
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
->
+As an alternative to newtonian dynamics, Langevin dynamics, which
->
+mimics a simple heat bath with stochastic and dissipative forces,
->
+has been applied in a variety of studies. This section will review
->
+the theory of Langevin dynamics. A brief derivation of the generalized
->
+Langevin equation will be given first. Following that, we will
->
+discuss the physical meaning of the terms appearing in the equation.
->
->
+\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
->
->
+A harmonic bath model, in which an effective set of harmonic
->
+oscillators are used to mimic the effect of a linearly responding
->
+environment, has been widely used in quantum chemistry and
->
+statistical mechanics. One of the successful applications of
->
+Harmonic bath model is the derivation of the Generalized Langevin
->
+Dynamics (GLE). Consider a system, in which the degree of
->
+freedom $x$ is assumed to couple to the bath linearly, giving a
->
+Hamiltonian of the form
->
+\begin{equation}
->
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
->
+\label{introEquation:bathGLE}.
->
+\end{equation}
->
+Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated
->
+with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian,
->
+\[
->
+H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
->
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  x_\alpha ^2 }
->
+\right\}}
->
+\]
->
+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 Eq.~\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, we
->
+can retrieve the solutions of the original problems. Let $f(t)$ be a
->
+function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$
->
+is a new function defined as
->
+\[
->
+L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
->
+\]
->
+where  $p$ is real and  $L$ is called the Laplace Transform
->
+Operator. Below are some important properties of the Laplace transform
->
+\begin{eqnarray*}
->
+ L(x + y)  & = & L(x) + L(y) \\
->
+ L(ax)     & = & aL(x) \\
->
+ L(\dot x) & = & pL(x) - px(0) \\
->
+ L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
->
+ L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
->
+ \end{eqnarray*}
->
+Applying the Laplace transform to the bath coordinates, we obtain
->
+\begin{eqnarray*}
->
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) & = & - \omega _\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha  }}L(x), \\
->
+L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}. \\
->
+\end{eqnarray*}
->
+In the same way, the system coordinates become
->
+\begin{eqnarray*}
->
+ mL(\ddot x) & = &
->
+  - \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0) - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}  \\
->
+  & & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}.
->
+\end{eqnarray*}
->
+With the help of some relatively important inverse Laplace
->
+transformations:
->
+\[
->
+\begin{array}{c}
->
+ L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
->
+ L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
->
+ L(1) = \frac{1}{p} \\
->
+ \end{array}
->
+\]
->
+we obtain
->
+\begin{eqnarray*}
->
+m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
->
+\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
->
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
->
+_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
->
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
->
+x_\alpha (0) - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}}
->
+\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
->
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}\\
->
+%
->
+& = & -
->
+\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha
->
+= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha
->
+^2 }}} \right)\cos (\omega _\alpha
->
+t)\dot x(t - \tau )d} \tau }  \\
->
+& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
->
+x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha  }}}
->
+\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
->
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
->
+\end{eqnarray*}
->
+Introducing a \emph{dynamic friction kernel}
->
+\begin{equation}
->
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
->
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
->
+\label{introEquation:dynamicFrictionKernelDefinition}
->
+\end{equation}
->
+and \emph{a random force}
->
+\begin{equation}
->
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
->
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
->
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
->
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
->
+\label{introEquation:randomForceDefinition}
->
+\end{equation}
->
+the equation of motion can be rewritten as
->
+\begin{equation}
->
+m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
->
+(t)\dot x(t - \tau )d\tau }  + R(t)
->
+\label{introEuqation:GeneralizedLangevinDynamics}
->
+\end{equation}
->
+which is known as the \emph{generalized Langevin equation} (GLE).
->
->
+\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 $\dot x$, $\left\langle {x(t)R(t)}
->
+\right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
->
+.$ This property is what we expect from a truly random process. As
->
+long as the model chosen for $R(t)$ was a gaussian distribution in
->
+general, the stochastic nature of the GLE still remains.
->
+%dynamic friction kernel
->
+The convolution integral
->
+\[
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
->
+\]
->
+depends on the entire history of the evolution of $x$, which implies
->
+that the bath retains memory of previous motions. In other words,
->
+the bath requires a finite time to respond to change in the motion
->
+of the system. For a sluggish bath which responds slowly to changes
->
+in the system coordinate, we may regard $\xi(t)$ as a constant
->
+$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
->
+\[
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
->
+\]
->
+and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
->
+\[
->
+m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
->
+\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
->
+\]
->
+which can be used to describe the effect of dynamic caging in
->
+viscous solvents. The other extreme is the bath that responds
->
+infinitely quickly to motions in the system. Thus, $\xi (t)$ can be
->
+taken as a $delta$ function in time:
->
+\[
->
+\xi (t) = 2\xi _0 \delta (t).
->
+\]
->
+Hence, the convolution integral becomes
->
+\[
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
->
+{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
->
+\]
->
+and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
->
+\begin{equation}
->
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
->
+x(t) + R(t) \label{introEquation:LangevinEquation}
->
+\end{equation}
->
+which is known as the Langevin equation. The static friction
->
+coefficient $\xi _0$ can either be calculated from spectral density
->
+or be determined by Stokes' law for regular shaped particles. A
->
+brief review on calculating friction tensors for arbitrary shaped
->
+particles is given in Sec.~\ref{introSection:frictionTensor}.
->
->
+\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
->
->
+Defining a new set of coordinates
->
+\[
->
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
->
+^2 }}x(0),
->
+\]
->
+we can rewrite $R(t)$ as
->
+\[
->
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
->
+\]
->
+And since the $q$ coordinates are harmonic oscillators,
->
+\begin{eqnarray*}
->
+ \left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
->
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
->
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
->
+ \left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha  {\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle } }  \\
->
+  & = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
->
+  & = &kT\xi (t)
->
+\end{eqnarray*}
->
+Thus, we recover the \emph{second fluctuation dissipation theorem}
->
+\begin{equation}
->
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
->
+\label{introEquation:secondFluctuationDissipation},
->
+\end{equation}
->
+which acts as a constraint on the possible ways in which one can
->
+model the random force and friction kernel.

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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs. Revision 2941 by tim, Mon Jul 17 20:01:05 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs.
Revision 2941 by tim, Mon Jul 17 20:01:05 2006 UTC