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+\end{equation}
+A point mass interacting with other bodies moves with the
+acceleration along the direction of the force acting on it. Let
-<
+$F_ij$ be the force that particle $i$ exerts on particle $j$, and
-<
+$F_ji$ be the force that particle $j$ exerts on particle $i$.
->
+$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and
->
+$F_{ji}$ be the force that particle $j$ exerts on particle $i$.
+Newton’s third law states that
+\begin{equation}
-<
+F_ij = -F_ji
->
+F_{ij} = -F_{ji}
+ \label{introEquation:newtonThirdLaw}
+\end{equation}
+\subsubsection{\label{introSection:equationOfMotionLagrangian}The
+Equations of Motion in Lagrangian Mechanics}
-<
+for a holonomic system of $f$ degrees of freedom, the equations of
->
+For a holonomic 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 }} -
+}}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
+H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
+\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
-<
+q_i }}} \right) = 0}
-<
+\label{introEquation:conserveHalmitonian}
->
+q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian}
+\end{equation}
-–
+When studying Hamiltonian system, it is more convenient to use
-–
+notation
-–
+\begin{equation}
-–
+r = r(q,p)^T
-–
+\end{equation}
-–
+and to introduce a $2n \times 2n$ canonical structure matrix $J$,
-–
+\begin{equation}
-–
+J = \left( {\begin{array}{*{20}c}
-–
+& I  \\
-–
+   { - I} & 0  \\
-–
+\end{array}} \right)
-–
+\label{introEquation:canonicalMatrix}
-–
+\end{equation}
-–
+where $I$ is a $n \times n$ identity matrix and $J$ is a
-–
+skew-symmetric matrix ($ J^T  =  - J $). Thus, Hamiltonian system
-–
+can be rewritten as,
-–
+\begin{equation}
-–
+\frac{d}{{dt}}r = J\nabla _r H(r)
-–
+\label{introEquation:compactHamiltonian}
-–
+\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 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 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.
->
->
+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 of distribution for an ensemble with $f$ degrees
->
+of freedom is defined as,
->
+\begin{equation}
->
+\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
->
+\label{introEquation:densityDistribution}
->
+\end{equation}
->
+Governed by the principles of mechanics, the phase points change
->
+their value which would change the density at any time at phase
->
+space. Hence, the density of distribution is also to be taken as a
->
+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 are exploited, we can
->
+sufficiently approximate $\delta N$ without introducing
->
+discontinuity when we go from one region in the phase space to
->
+another. By integrating over the whole phase space,
->
+\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 Equation(\ref{introEquation:unitProbability}) and
->
+the knowledge of the system, it is possible to calculate the average
->
+value of any desired quantity which depends on the coordinates and
->
+momenta of the system. Even when the dynamics of the real system is
->
+complex, or stochastic, or even discontinuous, the average
->
+properties of the ensemble of possibilities as a whole may still
->
+remain well defined. For a classical system in thermal equilibrium
->
+with its environment, the ensemble average of a mechanical quantity,
->
+$\langle A(q , p) \rangle_t$, takes the form of an integral over the
->
+phase space of the system,
->
+\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}, partition function can
->
+be used to describe the statistical properties of a system in
->
+thermodynamic equilibrium.
->
->
+As an ensemble of systems, each of which is known to be thermally
->
+isolated and conserve energy, Microcanonical ensemble(NVE) has a
->
+partition function like,
->
+\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 experiment are carried out under constant pressure
->
+condition, isothermal-isobaric ensemble(NPT) play a very important
->
+role in molecular simulation. The isothermal-isobaric ensemble allow
->
+the system to exchange energy with a heat bath of temperature $T$
->
+and to change the volume as well. Its partition function is given as
->
+\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}
-+
-+
+The Liouville's theorem is the foundation on which statistical
-+
+mechanics rests. It describes the time evolution of phase space
-+
+distribution function. In order to calculate the rate of change of
-+
+$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we
-+
+consider the two faces perpendicular to the $q_1$ axis, which are
-+
+located at $q_1$ and $q_1 + \delta q_1$, the number of phase points
-+
+leaving the opposite face is given by the expression,
-+
+\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,
-+
+divining $ \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 particles in the system
-+
+is huge, we may be able to believe the system is stationary,
-+
+\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}Conservation of Phase Space}
-+
+Lets consider a region in the phase space,
-+
+\begin{equation}
-+
+\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
-+
+\end{equation}
-+
+If this region is small enough, the density $\rho$ can be regarded
-+
+as uniform over the whole phase space. Thus, the number of phase
-+
+points inside this region is given by,
-+
+\begin{equation}
-+
+\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
-+
+dp_1 } ..dp_f.
-+
+\end{equation}
-+
-+
+\begin{equation}
-+
+\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho
-+
+\frac{d}{{dt}}(\delta v) = 0.
-+
+\end{equation}
-+
+With the help of stationary assumption
-+
+(\ref{introEquation:stationary}), we obtain the principle of the
-+
+\emph{conservation of extension in phase space},
-+
+\begin{equation}
-+
+\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
-+
+...dq_f dp_1 } ..dp_f  = 0.
-+
+\label{introEquation:volumePreserving}
-+
+\end{equation}
-+
-+
+\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms}
-+
-+
+Liouville's theorem can be expresses in a variety of different forms
-+
+which are convenient within different contexts. For any two function
-+
+$F$ and $G$ of the coordinates and momenta of a system, the Poisson
-+
+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(
-+
+\ref{introEquation:motionHamiltonianCoordinate} ,
-+
+\ref{introEquation:motionHamiltonianMomentum} ) into
-+
+(\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
+ensemble average. It states that time average and average over the
+statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
+\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{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}.
+\section{\label{introSection:geometricIntegratos}Geometric Integrators}
+A variety of numerical integrators were proposed to simulate the
+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. According to Liouville's theorem, the
-<
+Hamiltonian \emph{flow} or \emph{symplectomorphism} generated by the
-<
+Hamiltonian vector filed preserves the volume form on the phase
-<
+space, which is the basis of classical statistical mechanics.
->
+classical mechanics.
-<
+\subsection{\label{introSection:exactFlow}The Exact Flow of ODE}
->
+\subsection{\label{introSection:ODE}Ordinary Differential Equations}
-<
+\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting}
->
+For a ordinary differential system defined as
->
+\begin{equation}
->
+\dot x = f(x)
->
+\end{equation}
->
+where $x = x(q,p)^T$, this system is canonical Hamiltonian, if
->
+\begin{equation}
->
+f(r) = J\nabla _x H(r).
->
+\end{equation}
->
+$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric
->
+matrix
->
+\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,
-+
+\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 $x$.
-+
-+
+\subsection{\label{introSection:exactFlow}Exact Flow}
-+
-+
+Let $x(t)$ be the exact solution of the ODE system,
-+
+\begin{equation}
-+
+\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}
-+
+\end{equation}
-+
+The exact flow(solution) $\varphi_\tau$ is defined by
-+
+\[
-+
+x(t+\tau) =\varphi_\tau(x(t))
-+
+\]
-+
+where $\tau$ is a fixed time step and $\varphi$ is a map from phase
-+
+space to itself. The flow has the continuous group property,
-+
+\begin{equation}
-+
+\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
-+
++ \tau _2 } .
-+
+\end{equation}
-+
+In particular,
-+
+\begin{equation}
-+
+\varphi _\tau   \circ \varphi _{ - \tau }  = I
-+
+\end{equation}
-+
+Therefore, the exact flow is self-adjoint,
-+
+\begin{equation}
-+
+\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
-+
+\end{equation}
-+
+The exact flow can also be written in terms of the of an operator,
-+
+\begin{equation}
-+
+\varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
-+
+}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
-+
+\label{introEquation:exponentialOperator}
-+
+\end{equation}
-+
-+
+In most cases, it is not easy to find the exact flow $\varphi_\tau$.
-+
+Instead, we use a approximate map, $\psi_\tau$, which is usually
-+
+called integrator. The order of an integrator $\psi_\tau$ is $p$, if
-+
+the Taylor series of $\psi_\tau$ agree to order $p$,
-+
+\begin{equation}
-+
+\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1})
-+
+\end{equation}
-+
-+
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
-+
-+
+The hidden geometric properties of ODE and its flow play important
-+
+roles in numerical studies. Many of them can be found in systems
-+
+which occur naturally in applications.
-+
-+
+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
-+
+\end{equation}
-+
+is the property must be preserved by the integrator.
-+
-+
+It is possible to construct a \emph{volume-preserving} flow for a
-+
+source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $
-+
+\det d\varphi  = 1$. One can show easily that a symplectic flow will
-+
+be volume-preserving.
-+
-+
+Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE}
-+
+will result in a new system,
-+
+\[
-+
+\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
-+
+\]
-+
+The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
-+
+In other words, the flow of this vector field is reversible if and
-+
+only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
-+
-+
+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{equation}
-+
+\begin{array}{c}
-+
+ \frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\
-+
+  = [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
-+
+ \end{array}
-+
+\label{introEquation:firstIntegral1}
-+
+\end{equation}
-+
+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 using one of four different methods.
-+
+\begin{enumerate}
-+
+\item Generating functions
-+
+\item Variational methods
-+
+\item Runge-Kutta methods
-+
+\item Splitting methods
-+
+\end{enumerate}
-+
-+
+Generating function tends to lead to methods which are cumbersome
-+
+and difficult to use. In dissipative systems, variational methods
-+
+can capture the decay of energy accurately. Since their
-+
+geometrically unstable nature against non-Hamiltonian perturbations,
-+
+ordinary implicit Runge-Kutta methods are not suitable for
-+
+Hamiltonian system. Recently, various high-order explicit
-+
+Runge--Kutta methods have been developed to overcome this
-+
+instability. However, due to computational penalty involved in
-+
+implementing the Runge-Kutta methods, they do not attract too much
-+
+attention from Molecular Dynamics community. Instead, splitting have
-+
+been widely accepted since they exploit natural decompositions of
-+
+the system\cite{Tuckerman92}.
-+
-+
+\subsubsection{\label{introSection:splittingMethod}Splitting Method}
-+
-+
+The main idea behind splitting methods is to decompose the discrete
-+
+$\varphi_h$ as a composition of simpler flows,
-+
+\begin{equation}
-+
+\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
-+
+\varphi _{h_n }
-+
+\label{introEquation:FlowDecomposition}
-+
+\end{equation}
-+
+where each of the sub-flow is chosen such that each represent a
-+
+simpler integration of the system.
-+
-+
+Suppose that a Hamiltonian system takes the form,
-+
+\[
-+
+H = H_1 + H_2.
-+
+\]
-+
+Here, $H_1$ and $H_2$ may represent different physical processes of
-+
+the system. For instance, they may relate to kinetic and potential
-+
+energy respectively, which is a natural decomposition of the
-+
+problem. If $H_1$ and $H_2$ can be integrated using exact flows
-+
+$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first
-+
+order is then given by the Lie-Trotter formula
-+
+\begin{equation}
-+
+\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
-+
+\label{introEquation:firstOrderSplitting}
-+
+\end{equation}
-+
+where $\varphi _h$ is the result of applying the corresponding
-+
+continuous $\varphi _i$ over a time $h$. By definition, as
-+
+$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
-+
+must follow that each operator $\varphi_i(t)$ is a symplectic map.
-+
+It is easy to show that any composition of symplectic flows yields a
-+
+symplectic map,
-+
+\begin{equation}
-+
+(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
-+
+'\phi ' = \phi '^T J\phi ' = J,
-+
+ \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(\ref{introEquation:firstOrderSplitting})
-+
+introduces local errors proportional to $h^2$, while Strang
-+
+splitting gives a second-order decomposition,
-+
+\begin{equation}
-+
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
-+
+_{1,h/2} , \label{introEquation:secondOrderSplitting}
-+
+\end{equation}
-+
+which has a local error proportional to $h^3$. Sprang splitting's
-+
+popularity in molecular simulation community attribute to its
-+
+symmetric property,
-+
+\begin{equation}
-+
+\varphi _h^{ - 1} = \varphi _{ - h}.
-+
+\label{introEquation:timeReversible}
-+
+\end{equation}
-+
-+
+\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method}
-+
+The classical equation for a system consisting of interacting
-+
+particles can be written in Hamiltonian form,
-+
+\[
-+
+H = T + V
-+
+\]
-+
+where $T$ is the kinetic energy and $V$ is the potential energy.
-+
+Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one
-+
+obtains the following:
-+
+\begin{align}
-+
+q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
-+
+    \frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
-+
+\label{introEquation:Lp10a} \\%
-+
+%
-+
+\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
-+
+    \biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
-+
+\label{introEquation:Lp10b}
-+
+\end{align}
-+
+where $F(t)$ is the force at time $t$. This integration scheme is
-+
+known as \emph{velocity verlet} which is
-+
+symplectic(\ref{introEquation:SymplecticFlowComposition}),
-+
+time-reversible(\ref{introEquation:timeReversible}) and
-+
+volume-preserving (\ref{introEquation:volumePreserving}). These
-+
+geometric properties attribute to its long-time stability and its
-+
+popularity in the community. However, the most commonly used
-+
+velocity verlet integration scheme is written as below,
-+
+\begin{align}
-+
+\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
-+
+    \dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
-+
+%
-+
+q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
-+
+    \label{introEquation:Lp9b}\\%
-+
+%
-+
+\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
-+
+    \frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c}
-+
+\end{align}
-+
+From the preceding splitting, one can see that the integration of
-+
+the equations of motion would follow:
-+
+\begin{enumerate}
-+
+\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
-+
-+
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
-+
-+
+\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move.
-+
-+
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
-+
+\end{enumerate}
-+
-+
+Simply switching the order of splitting and composing, a new
-+
+integrator, the \emph{position verlet} integrator, can be generated,
-+
+\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:positionVerlet1}
-+
+\end{align}
-+
-+
+\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
-+
-+
+Baker-Campbell-Hausdorff formula can be used to determine the local
-+
+error of splitting method in terms of commutator of the
-+
+operators(\ref{introEquation:exponentialOperator}) associated with
-+
+the sub-flow. For operators $hX$ and $hY$ which are associate to
-+
+$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have
-+
+\begin{equation}
-+
+\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 Baker-Campbell-Hausdorff formula to Sprang splitting, we
-+
+can obtain
-+
+\begin{eqnarray*}
-+
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2
-+
+[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
-+
+& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} +
-+
+\ldots )
-+
+\end{eqnarray*}
-+
+Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local
-+
+error of Spring splitting is proportional to $h^3$. The same
-+
+procedure can be applied to general splitting,  of the form
-+
+\begin{equation}
-+
+\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
-+
+} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
-+
+\end{equation}
-+
+Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher
-+
+order method. Yoshida proposed an elegant way to compose higher
-+
+order methods based on symmetric splitting. Given a symmetric second
-+
+order base method $ \varphi _h^{(2)} $, a fourth-order symmetric
-+
+method can be constructed by composing,
-+
+\[
-+
+\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
-+
+h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
-+
+\]
-+
+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 a special discipline of molecular modeling, Molecular dynamics
+\subsection{\label{introSec:mdInit}Initialization}
-+
+\subsection{\label{introSec:forceEvaluation}Force Evaluation}
-+
+\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
-<
+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.
->
+Rigid bodies are frequently involved in the modeling of different
->
+areas, from engineering, physics, to chemistry. For example,
->
+missiles and vehicle are usually modeled by rigid bodies.  The
->
+movement of the objects in 3D gaming engine or other physics
->
+simulator is governed by the rigid body dynamics. In molecular
->
+simulation, rigid body is used to simplify the model in
->
+protein-protein docking study{\cite{Gray03}}.
-<
+Applications of dynamics of rigid bodies.
->
+It is very important to develop stable and efficient methods to
->
+integrate the equations of motion of orientational degrees of
->
+freedom. Euler angles are the nature choice to describe the
->
+rotational degrees of freedom. However, due to its singularity, the
->
+numerical integration of corresponding equations of motion is very
->
+inefficient and inaccurate. Although an alternative integrator using
->
+different sets of Euler angles can overcome this difficulty\cite{},
->
+the computational penalty and the lost of angular momentum
->
+conservation still remain. A singularity free representation
->
+utilizing quaternions was developed by Evans in 1977. Unfortunately,
->
+this approach suffer from the nonseparable Hamiltonian resulted from
->
+quaternion representation, which prevents the symplectic algorithm
->
+to be utilized. Another different approach is to apply holonomic
->
+constraints to the atoms belonging to the rigid body. Each atom
->
+moves independently under the normal forces deriving from potential
->
+energy and constraint forces which are used to guarantee the
->
+rigidness. However, due to their iterative nature, SHAKE and Rattle
->
+algorithm converge very slowly when the number of constraint
->
+increases.
-<
+\subsection{\label{introSection:lieAlgebra}Lie Algebra}
->
+The break through in geometric literature suggests that, in order to
->
+develop a long-term integration scheme, one should preserve the
->
+symplectic structure of the flow. Introducing conjugate momentum to
->
+rotation matrix $A$ and re-formulating Hamiltonian's equation, a
->
+symplectic integrator, RSHAKE, was proposed to evolve the
->
+Hamiltonian system in a constraint manifold by iteratively
->
+satisfying the orthogonality constraint $A_t A = 1$. An alternative
->
+method using quaternion representation was developed by Omelyan.
->
+However, both of these methods are iterative and inefficient. In
->
+this section, we will present a symplectic Lie-Poisson integrator
->
+for rigid body developed by Dullweber and his
->
+coworkers\cite{Dullweber1997} in depth.
-<
+\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
->
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body}
->
+The motion of the rigid body is Hamiltonian with the Hamiltonian
->
+function
->
+\begin{equation}
->
+H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
->
+V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
->
+\label{introEquation:RBHamiltonian}
->
+\end{equation}
->
+Here, $q$ and $Q$  are the position and rotation matrix for the
->
+rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ , and
->
+$J$, a diagonal matrix, is defined by
->
+\[
->
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
->
+\]
->
+where $I_{ii}$ is the diagonal element of the inertia tensor. This
->
+constrained Hamiltonian equation subjects to a holonomic constraint,
->
+\begin{equation}
->
+Q^T Q = 1$, \label{introEquation:orthogonalConstraint}
->
+\end{equation}
->
+which is used to ensure rotation matrix's orthogonality.
->
+Differentiating \ref{introEquation:orthogonalConstraint} and using
->
+Equation \ref{introEquation:RBMotionMomentum}, one may obtain,
->
+\begin{equation}
->
+Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
->
+\label{introEquation:RBFirstOrderConstraint}
->
+\end{equation}
-<
+\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
->
+Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
->
+\ref{introEquation:motionHamiltonianMomentum}), one can write down
->
+the equations of motion,
->
+\[
->
+\begin{array}{c}
->
+ \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\
->
+ \frac{{dp}}{{dt}} =  - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
->
+ \frac{{dQ}}{{dt}} = PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
->
+ \frac{{dP}}{{dt}} =  - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\
->
+ \end{array}
->
+\]
-<
+%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
->
+In general, there are two ways to satisfy the holonomic constraints.
->
+We can use constraint force provided by lagrange multiplier on the
->
+normal manifold to keep the motion on constraint space. Or we can
->
+simply evolve the system in constraint manifold. The two method are
->
+proved to be equivalent. The holonomic constraint and equations of
->
+motions define a constraint manifold for rigid body
->
+\[
->
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
->
+\right\}.
->
+\]
-<
+\section{\label{introSection:correlationFunctions}Correlation Functions}
->
+Unfortunately, this constraint manifold is not the cotangent bundle
->
+$T_{\star}SO(3)$. However, it turns out that under symplectic
->
+transformation, the cotangent space and the phase space are
->
+diffeomorphic. Introducing
->
+\[
->
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
->
+\]
->
+the mechanical system subject to a holonomic constraint manifold $M$
->
+can be re-formulated as a Hamiltonian system on the cotangent space
->
+\[
->
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
->
+,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
->
+\]
-+
+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, angular momentum on body frame $\Pi  = Q^t P$ is introduced to
-+
+rewrite the equations of motion,
-+
+\begin{equation}
-+
+ \begin{array}{l}
-+
+ \mathop \Pi \limits^ \bullet   = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda  \\
-+
+ \mathop Q\limits^{{\rm{   }} \bullet }  = Q\Pi {\rm{ }}J^{ - 1}  \\
-+
+ \end{array}
-+
+ \label{introEqaution:RBMotionPI}
-+
+\end{equation}
-+
+, as well as holonomic constraints,
-+
+\[
-+
+\begin{array}{l}
-+
+ \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
-+
+ Q^T Q = 1 \\
-+
+ \end{array}
-+
+\]
-+
-+
+For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
-+
+so(3)^ \star$, the hat-map isomorphism,
-+
+\begin{equation}
-+
+v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
-+
+{\begin{array}{*{20}c}
-+
+& { - v_3 } & {v_2 }  \\
-+
+   {v_3 } & 0 & { - v_1 }  \\
-+
+   { - v_2 } & {v_1 } & 0  \\
-+
+\end{array}} \right),
-+
+\label{introEquation:hatmapIsomorphism}
-+
+\end{equation}
-+
+will let us associate the matrix products with traditional vector
-+
+operations
-+
+\[
-+
+\hat vu = v \times u
-+
+\]
-+
-+
+Using \ref{introEqaution:RBMotionPI}, one can construct a skew
-+
+matrix,
-+
+\begin{equation}
-+
+(\mathop \Pi \limits^ \bullet   - \mathop \Pi \limits^ \bullet  ^T
-+
+){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi  + \Pi J^{
-+
+- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]} -
-+
+(\Lambda  - \Lambda ^T ) . \label{introEquation:skewMatrixPI}
-+
+\end{equation}
-+
+Since $\Lambda$ is symmetric, the last term of Equation
-+
+\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
-+
+multiplier $\Lambda$ is absent from the equations of motion. This
-+
+unique property eliminate the requirement of iterations which can
-+
+not be avoided in other methods\cite{}.
-+
-+
+Applying hat-map isomorphism, we obtain the equation of motion for
-+
+angular momentum on body frame
-+
+\begin{equation}
-+
+\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
-+
+F_i (r,Q)} \right) \times X_i }.
-+
+\label{introEquation:bodyAngularMotion}
-+
+\end{equation}
-+
+In the same manner, the equation of motion for rotation matrix is
-+
+given by
-+
+\[
-+
+\dot Q = Qskew(I^{ - 1} \pi )
-+
+\]
-+
-+
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
-+
+Lie-Poisson Integrator for Free Rigid Body}
-+
-+
+If there is not external forces exerted on the rigid body, the only
-+
+contribution to the rotational is from the kinetic potential (the
-+
+first term of \ref{ introEquation:bodyAngularMotion}). The free
-+
+rigid body is an example of Lie-Poisson system with Hamiltonian
-+
+function
-+
+\begin{equation}
-+
+T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
-+
+\label{introEquation:rotationalKineticRB}
-+
+\end{equation}
-+
+where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
-+
+Lie-Poisson structure matrix,
-+
+\begin{equation}
-+
+J(\pi ) = \left( {\begin{array}{*{20}c}
-+
+& {\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
-+
+\[
-+
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
-+
+Q(0)e^{\Delta tR_1 }
-+
+\]
-+
+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 Cayley transformation,
-+
+\[
-+
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
-+
+)
-+
+\]
-+
-+
+The flow maps for $T_2^r$ and $T_2^r$ can be found in the same
-+
+manner.
-+
-+
+In order to construct a second-order symplectic method, we split the
-+
+angular kinetic Hamiltonian function can into five terms
-+
+\[
-+
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
-+
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
-+
+(\pi _1 )
-+
+\].
-+
+Concatenating flows corresponding to these five terms, we can obtain
-+
+an symplectic integrator,
-+
+\[
-+
+\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
-+
+\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
-+
+\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
-+
+_1 }.
-+
+\]
-+
-+
+The non-canonical Lie-Poisson bracket ${F, G}$ of two function
-+
+$F(\pi )$ and $G(\pi )$ is defined by
-+
+\[
-+
+\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
-+
+)
-+
+\]
-+
+If the Poisson bracket of a function $F$ with an arbitrary smooth
-+
+function $G$ is zero, $F$ is a \emph{Casimir}, which is the
-+
+conserved quantity in Poisson system. We can easily verify that the
-+
+norm of the angular momentum, $\parallel \pi
-+
+\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel
-+
+\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
-+
+then by the chain rule
-+
+\[
-+
+\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
-+
+}}{2})\pi
-+
+\]
-+
+Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
-+
+\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
-+
+Lie-Poisson integrator is found to be extremely efficient and stable
-+
+which can be explained by the fact the small angle approximation is
-+
+used and the norm of the angular momentum is conserved.
-+
-+
+\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
-+
+Splitting for Rigid Body}
-+
-+
+The Hamiltonian of rigid body can be separated in terms of kinetic
-+
+energy and potential energy,
-+
+\[
-+
+H = T(p,\pi ) + V(q,Q)
-+
+\]
-+
+The equations of motion corresponding to potential energy and
-+
+kinetic energy are listed in the below table,
-+
+\begin{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}
-+
+A second-order symplectic method is now obtained by the composition
-+
+of the flow maps,
-+
+\[
-+
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
-+
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
-+
+\]
-+
+Moreover, \varphi _{\Delta t/2,V} can be divided into two sub-flows
-+
+which corresponding to force and torque respectively,
-+
+\[
-+
+\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
-+
+_{\Delta t/2,\tau }.
-+
+\]
-+
+Since the associated operators of $\varphi _{\Delta t/2,F} $ and
-+
+$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition
-+
+order inside \varphi _{\Delta t/2,V} does not matter.
-+
-+
+Furthermore, kinetic potential can be separated to translational
-+
+kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$,
-+
+\begin{equation}
-+
+T(p,\pi ) =T^t (p) + T^r (\pi ).
-+
+\end{equation}
-+
+where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
-+
+defined by \ref{introEquation:rotationalKineticRB}. Therefore, the
-+
+corresponding flow maps are given by
-+
+\[
-+
+\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
-+
+_{\Delta t,T^r }.
-+
+\]
-+
+Finally, we obtain the overall symplectic flow maps for free moving
-+
+rigid body
-+
+\begin{equation}
-+
+\begin{array}{c}
-+
+ \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
-+
+  \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \\
-+
+  \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
-+
+ \end{array}
-+
+\label{introEquation:overallRBFlowMaps}
-+
+\end{equation}
-+
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
+\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
+\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}
->
+\label{introEquation:Lp5}
+\end{align}
+, and
+\begin{align}
+\subsection{\label{introSection:centersRigidBody}Centers of Rigid
+Body}
-+
-+
+\section{\label{introSection:correlationFunctions}Correlation Functions}

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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2697 by tim, Fri Apr 7 05:03:54 2006 UTC vs. Revision 2713 by tim, Sat Apr 15 01:08:18 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2697 by tim, Fri Apr 7 05:03:54 2006 UTC vs.
Revision 2713 by tim, Sat Apr 15 01:08:18 2006 UTC