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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}
+isolated and conserve energy, Microcanonical ensemble(NVE) has a
+partition function like,
+\begin{equation}
-<
+\Omega (N,V,E) = e^{\beta TS}
-<
+\label{introEqaution:NVEPartition}.
->
+\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
+\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
+\label{introEquation:liouvilleTheoremInOperator}
+\end{equation}
-–
+\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
+Various thermodynamic properties can be calculated from Molecular
+\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
+\end{equation}
+The most obvious change being that matrix $J$ now depends on $x$.
-–
+The free rigid body is an example of Poisson system (actually a
-–
+Lie-Poisson system) with Hamiltonian function of angular kinetic
-–
+energy.
-–
+\begin{equation}
-–
+J(\pi ) = \left( {\begin{array}{*{20}c}
-–
+& {\pi _3 } & { - \pi _2 }  \\
-–
+   { - \pi _3 } & 0 & {\pi _1 }  \\
-–
+   {\pi _2 } & { - \pi _1 } & 0  \\
-–
+\end{array}} \right)
-–
+\end{equation}
-<
+\begin{equation}
-<
+H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2
-<
+}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right)
-<
+\end{equation}
->
+\subsection{\label{introSection:exactFlow}Exact Flow}
-–
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
+Let $x(t)$ be the exact solution of the ODE system,
+\begin{equation}
+\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}
+x(t+\tau) =\varphi_\tau(x(t))
+\]
+where $\tau$ is a fixed time step and $\varphi$ is a map from phase
-<
+space to itself. In most cases, it is not easy to find the exact
-<
+flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$,
-<
+which is usually called integrator. The order of an integrator
-<
+$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
-<
+order $p$,
->
+space to itself. The flow has the continuous group property,
+\begin{equation}
-+
+\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. Let $\varphi$ be the flow of Hamiltonian
-<
+vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies,
->
+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.
->
+{\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
+field on a symplectic manifold can be shown to be a
+symplectomorphism. As to the Poisson system,
+\begin{equation}
-<
+'\varphi ^T J '\varphi  = J \circ \varphi
->
+{\varphi '}^T J \varphi ' = J \circ \varphi
+\end{equation}
-<
+is the property must be preserved by the integrator. It is possible
-<
+to construct a \emph{volume-preserving} flow for a source free($
-<
+\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi  =
-<
+$. Changing the variables $y = h(x)$ in a
-<
+ODE\ref{introEquation:ODE} will result in a new system,
->
+is the property 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 $. When
-<
+designing any numerical methods, one should always try to preserve
-<
+the structural properties of the original ODE and its flow.
->
+only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
-+
+A \emph{first integral}, or conserved quantity of a general
-+
+differential function is a function $ G:R^{2d}  \to R^d $ which is
-+
+constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
-+
+\[
-+
+\frac{{dG(x(t))}}{{dt}} = 0.
-+
+\]
-+
+Using chain rule, one may obtain,
-+
+\[
-+
+\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G,
-+
+\]
-+
+which is the condition for conserving \emph{first integral}. For a
-+
+canonical Hamiltonian system, the time evolution of an arbitrary
-+
+smooth function $G$ is given by,
-+
+\begin{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
+\end{enumerate}
+Generating function tends to lead to methods which are cumbersome
-<
+and difficult to use\cite{}. In dissipative systems, variational
-<
+methods can capture the decay of energy accurately\cite{}. Since
-<
+their geometrically unstable nature against non-Hamiltonian
-<
+perturbations, ordinary implicit Runge-Kutta methods are not
-<
+suitable for Hamiltonian system. Recently, various high-order
-<
+explicit Runge--Kutta methods have been developed to overcome this
-<
+instability \cite{}. However, due to computational penalty involved
-<
+in implementing the Runge-Kutta methods, they do not attract too
-<
+much attention from Molecular Dynamics community. Instead, splitting
-<
+have been widely accepted since they exploit natural decompositions
-<
+of the system\cite{Tuckerman92}. The main idea behind splitting
-<
+methods is to decompose the discrete $\varphi_h$ as a composition of
-<
+simpler flows,
->
+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. Let $\phi$ and $\psi$ both be
-<
+symplectic maps, it is easy to show that any composition of
-<
+symplectic flows yields a symplectic map,
->
+simpler integration of the system.
->
->
+Suppose that a Hamiltonian system takes the form,
->
+\[
->
+H = H_1 + H_2.
->
+\]
->
+Here, $H_1$ and $H_2$ may represent different physical processes of
->
+the system. For instance, they may relate to kinetic and potential
->
+energy respectively, which is a natural decomposition of the
->
+problem. If $H_1$ and $H_2$ can be integrated using exact flows
->
+$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first
->
+order is then given by the Lie-Trotter formula
+\begin{equation}
-+
+\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
-+
+\label{introEquation:firstOrderSplitting}
-+
+\end{equation}
-+
+where $\varphi _h$ is the result of applying the corresponding
-+
+continuous $\varphi _i$ over a time $h$. By definition, as
-+
+$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
-+
+must follow that each operator $\varphi_i(t)$ is a symplectic map.
-+
+It is easy to show that any composition of symplectic flows yields a
-+
+symplectic map,
-+
+\begin{equation}
+(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
-<
+'\phi ' = \phi '^T J\phi ' = J.
->
+'\phi ' = \phi '^T J\phi ' = J,
+ \label{introEquation:SymplecticFlowComposition}
+\end{equation}
-<
+Suppose that a Hamiltonian system has a form with $H = T + V$
->
+where $\phi$ and $\psi$ both are symplectic maps. Thus operator
->
+splitting in this context automatically generates a symplectic map.
->
->
+The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting})
->
+introduces local errors proportional to $h^2$, while Strang
->
+splitting gives a second-order decomposition,
->
+\begin{equation}
->
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
->
+_{1,h/2} , \label{introEquation:secondOrderSplitting}
->
+\end{equation}
->
+which has a local error proportional to $h^3$. 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}
-<
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
->
+\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.
-<
+As a special discipline of molecular modeling, Molecular dynamics
-<
+has proven to be a powerful tool for studying the functions of
-<
+biological systems, providing structural, thermodynamic and
-<
+dynamical information.
->
+\item Use the half step velocities to move positions one whole step, $\Delta t$.
-<
+\subsection{\label{introSec:mdInit}Initialization}
->
+\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move.
-<
+\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
->
+\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
->
+\end{enumerate}
-<
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
->
+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:positionVerlet2}
->
+\end{align}
-<
+A rigid body is a body in which the distance between any two given
-<
+points of a rigid body remains constant regardless of external
-<
+forces exerted on it. A rigid body therefore conserves its shape
-<
+during its motion.
->
+\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
-<
+Applications of dynamics of rigid bodies.
-<
-<
+\subsection{\label{introSection:lieAlgebra}Lie Algebra}
-<
-<
+\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
-<
-<
+\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
-<
-<
+%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
-<
-<
+\section{\label{introSection:correlationFunctions}Correlation Functions}
-<
-<
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
-<
-<
+\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
-<
-<
+\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
-<
->
+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}
-<
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
-<
+\label{introEquation:bathGLE}
->
+\exp (hX + hY) = \exp (hZ)
+\end{equation}
-<
+where $H_B$ is harmonic bath Hamiltonian,
->
+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
+\[
-<
+H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-<
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
->
+[X,Y] = XY - YX .
+\]
-<
+and $\Delta U$ is bilinear system-bath coupling,
->
+Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we
->
+can obtain
->
+\begin{equation}
->
+\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{equation}
->
+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,
+\[
-<
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
->
+\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
->
+h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
+\]
-<
+Completing the square,
->
+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
+\[
-<
+H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
-<
+{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-<
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-<
+w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
-<
+}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
->
+\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
->
+\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
+\]
-–
+and putting it back into Eq.~\ref{introEquation:bathGLE},
-–
+\[
-–
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-–
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-–
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-–
+w_\alpha ^2 }}x} \right)^2 } \right\}}
-–
+\]
-–
+where
-–
+\[
-–
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
-–
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
-–
+\]
-–
+Since the first two terms of the new Hamiltonian depend only on the
-–
+system coordinates, we can get the equations of motion for
-–
+Generalized Langevin Dynamics by Hamilton's equations
-–
+\ref{introEquation:motionHamiltonianCoordinate,
-–
+introEquation:motionHamiltonianMomentum},
-–
+\begin{align}
-–
+\dot p &=  - \frac{{\partial H}}{{\partial x}}
-–
+       &= m\ddot x
-–
+       &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)}
-–
+\label{introEq:Lp5}
-–
+\end{align}
-–
+, and
-–
+\begin{align}
-–
+\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
-–
+                &= m\ddot x_\alpha
-–
+                &= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
-–
+\end{align}
-<
+\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
->
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
-<
+\[
-<
+L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
->
+As one of the principal tools of molecular modeling, Molecular
->
+dynamics has proven to be a powerful tool for studying the functions
->
+of biological systems, providing structural, thermodynamic and
->
+dynamical information. The basic idea of molecular dynamics is that
->
+macroscopic properties are related to microscopic behavior and
->
+microscopic behavior can be calculated from the trajectories in
->
+simulations. For instance, instantaneous temperature of an
->
+Hamiltonian system of $N$ particle can be measured by
->
+\[
->
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
+\]
-+
+where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
-+
+respectively, $f$ is the number of degrees of freedom, and $k_B$ is
-+
+the boltzman constant.
-+
+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{introSec:production} will
-+
+discusses issues in production run. Sec.~\ref{introSection:Analysis}
-+
+provides the theoretical tools for trajectory analysis.
-+
-+
+\subsection{\label{introSec:initialSystemSettings}Initialization}
-+
-+
+\subsubsection{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 the molecule with known
-+
+structure, some important information is missing. For example, the
-+
+missing hydrogen atom which acts as donor in hydrogen bonding must
-+
+be added. Moreover, in order to include electrostatic interaction,
-+
+one may need to specify the partial charges for individual atoms.
-+
+Under some circumstances, we may even need to prepare the system in
-+
+a special setup. For instance, when studying transport phenomenon in
-+
+membrane system, we may prepare the lipids in bilayer structure
-+
+instead of placing lipids randomly in solvent, since we are not
-+
+interested in self-aggregation and it takes a long time to happen.
-+
-+
+\subsubsection{Minimization}
-+
-+
+It is quite possible that some of molecules in the system from
-+
+preliminary preparation may be overlapped with each other. This
-+
+close proximity leads to high potential energy which consequently
-+
+jeopardizes any molecular dynamics simulations. To remove these
-+
+steric overlaps, one typically performs energy minimization to find
-+
+a more reasonable conformation. Several energy minimization methods
-+
+have been developed to exploit the energy surface and to locate the
-+
+local minimum. While converging slowly near the minimum, steepest
-+
+descent method is extremely robust when systems are far from
-+
+harmonic. Thus, it is often used to refine structure from
-+
+crystallographic data. Relied on the gradient or hessian, advanced
-+
+methods like conjugate gradient and Newton-Raphson converge rapidly
-+
+to a local minimum, while become unstable if the energy surface is
-+
+far from quadratic. Another factor 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 limit of computation power to calculate hessian
-+
+matrix and insufficient storage capacity to store them, most
-+
+Newton-Raphson methods can not be used with very large models.
-+
-+
+\subsubsection{Heating}
-+
-+
+Typically, Heating is performed by assigning random velocities
-+
+according to a Gaussian distribution for a temperature. Beginning at
-+
+a lower temperature and gradually increasing the temperature by
-+
+assigning greater random velocities, we end up with setting the
-+
+temperature of the system to a final temperature at which the
-+
+simulation will be conducted. In heating phase, we should also keep
-+
+the system from drifting or rotating as a whole. Equivalently, the
-+
+net linear momentum and angular momentum of the system should be
-+
+shifted to zero.
-+
-+
+\subsubsection{Equilibration}
-+
-+
+The purpose of equilibration is to allow the system to evolve
-+
+spontaneously for a period of time and reach equilibrium. The
-+
+procedure is continued until various statistical properties, such as
-+
+temperature, pressure, energy, volume and other structural
-+
+properties \textit{etc}, become independent of time. Strictly
-+
+speaking, minimization and heating are not necessary, provided the
-+
+equilibration process is long enough. However, these steps can serve
-+
+as a means to arrive at an equilibrated structure in an effective
-+
+way.
-+
-+
+\subsection{\label{introSection:production}Production}
-+
-+
+Production run is the most important steps 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 in correct and efficient way.
-+
-+
+The most expensive part of a molecular dynamics simulation is the
-+
+calculation of non-bonded forces, such as van der Waals force and
-+
+Coulombic forces \textit{etc}. For a system of $N$ particles, the
-+
+complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
-+
+which making large simulations prohibitive in the absence of any
-+
+computation saving techniques.
-+
-+
+A natural approach to avoid system size issue is to represent the
-+
+bulk behavior by a finite number of the particles. However, this
-+
+approach will suffer from the surface effect. To offset this,
-+
+\textit{Periodic boundary condition} is developed to simulate bulk
-+
+properties with a relatively small number of particles. In this
-+
+method, the simulation box is replicated throughout space to form an
-+
+infinite lattice. During the simulation, when a particle moves in
-+
+the primary cell, its image in other cells move in exactly the same
-+
+direction with exactly the same orientation. Thus, as a particle
-+
+leaves the primary cell, one of its images will enter through the
-+
+opposite face.
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pbcFig.eps}
-+
+%\caption[An illustration of periodic boundary conditions]{A 2-D
-+
+%illustration of periodic boundary conditions. As one particle leaves
-+
+%the right of the simulation box, an image of it enters the left.}
-+
+%\label{introFig:pbc}
-+
+%\end{figure}
-+
-+
+%cutoff and minimum image convention
-+
+Another important technique to improve the efficiency of force
-+
+evaluation is to apply cutoff where particles farther than a
-+
+predetermined distance, are not included in the calculation
-+
+\cite{Frenkel1996}. The use of a cutoff radius will cause a
-+
+discontinuity in the potential energy curve
-+
+(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the
-+
+potential to ensure the potential curve go smoothly to zero at the
-+
+cutoff radius. Cutoff strategy works pretty well for Lennard-Jones
-+
+interaction because of its short range nature. However, simply
-+
+truncating the electrostatic interaction with the use of cutoff has
-+
+been shown to lead to severe artifacts in simulations. Ewald
-+
+summation, in which the slowly conditionally convergent Coulomb
-+
+potential is transformed into direct and reciprocal sums with rapid
-+
+and absolute convergence, has proved to minimize the periodicity
-+
+artifacts in liquid simulations. Taking the advantages of the fast
-+
+Fourier transform (FFT) for calculating discrete Fourier transforms,
-+
+the particle mesh-based methods are accelerated from $O(N^{3/2})$ to
-+
+$O(N logN)$. An alternative approach is \emph{fast multipole
-+
+method}, which treats Coulombic interaction exactly at short range,
-+
+and approximate the potential at long range through multipolar
-+
+expansion. In spite of their wide acceptances at the molecular
-+
+simulation community, these two methods are hard to be implemented
-+
+correctly and efficiently. Instead, we use a damped and
-+
+charge-neutralized Coulomb potential method developed by Wolf and
-+
+his coworkers. 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]{pbcFig.eps}
-+
+%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.}
-+
+%\label{introFigure:shiftedCoulomb}
-+
+%\end{figure}
-+
-+
+%multiple time step
-+
-+
+\subsection{\label{introSection:Analysis} Analysis}
-+
-+
+Recently, advanced visualization technique are widely applied to
-+
+monitor the motions of molecules. Although the dynamics of the
-+
+system can be described qualitatively from animation, quantitative
-+
+trajectory analysis are more appreciable. According to the
-+
+principles of Statistical Mechanics,
-+
+Sec.~\ref{introSection:statisticalMechanics}, one can compute
-+
+thermodynamics properties, analyze fluctuations of structural
-+
+parameters, and investigate time-dependent processes of the molecule
-+
+from the trajectories.
-+
-+
+\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties}
-+
-+
+Thermodynamics properties, which can be expressed in terms of some
-+
+function of the coordinates and momenta of all particles in the
-+
+system, can be directly computed from molecular dynamics. The usual
-+
+way to measure the pressure is based on virial theorem of Clausius
-+
+which states that the virial is equal to $-3Nk_BT$. For a system
-+
+with forces between particles, the total virial, $W$, contains the
-+
+contribution from external pressure and interaction between the
-+
+particles:
+\[
-<
+L(x + y) = L(x) + L(y)
->
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
->
+f_{ij} } } \right\rangle
+\]
-+
+where $f_{ij}$ is the force between particle $i$ and $j$ at a
-+
+distance $r_{ij}$. Thus, the expression for the pressure is given
-+
+by:
-+
+\begin{equation}
-+
+P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
-+
+< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
-+
+\end{equation}
-+
+\subsubsection{\label{introSection:structuralProperties}Structural Properties}
-+
-+
+Structural Properties of a simple fluid can be described by a set of
-+
+distribution functions. Among these functions,\emph{pair
-+
+distribution function}, also known as \emph{radial distribution
-+
+function}, is of most fundamental importance to liquid-state theory.
-+
+Pair distribution function can be gathered by Fourier transforming
-+
+raw data from a series of neutron diffraction experiments and
-+
+integrating over the surface factor \cite{Powles73}. The experiment
-+
+result can serve as a criterion to justify the correctness of the
-+
+theory. Moreover, various equilibrium thermodynamic and structural
-+
+properties can also be expressed in terms of radial distribution
-+
+function \cite{allen87:csl}.
-+
-+
+A 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
+\[
-<
+L(ax) = aL(x)
->
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
->
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle.
+\]
-+
+Note that the delta function can be replaced by a histogram in
-+
+computer simulation. Figure
-+
+\ref{introFigure:pairDistributionFunction} shows a typical pair
-+
+distribution function for the liquid argon system. The occurrence of
-+
+several peaks in the plot of $g(r)$ suggests that it is more likely
-+
+to find particles at certain radial values than at others. This is a
-+
+result of the attractive interaction at such distances. Because of
-+
+the strong repulsive forces at short distance, the probability of
-+
+locating particles at distances less than about 2.5{\AA} from each
-+
+other is essentially zero.
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pdf.eps}
-+
+%\caption[Pair distribution function for the liquid argon
-+
+%]{Pair distribution function for the liquid argon}
-+
+%\label{introFigure:pairDistributionFunction}
-+
+%\end{figure}
-+
-+
+\subsubsection{\label{introSection:timeDependentProperties}Time-dependent
-+
+Properties}
-+
-+
+Time-dependent properties are usually calculated using \emph{time
-+
+correlation function}, which correlates random variables $A$ and $B$
-+
+at two different time
-+
+\begin{equation}
-+
+C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
-+
+\label{introEquation:timeCorrelationFunction}
-+
+\end{equation}
-+
+If $A$ and $B$ refer to same variable, this kind of correlation
-+
+function is called \emph{auto correlation function}. One example of
-+
+auto correlation function is velocity auto-correlation function
-+
+which is directly related to transport properties of molecular
-+
+liquids:
+\[
-<
+L(\dot x) = pL(x) - px(0)
->
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
->
+\right\rangle } dt
+\]
-+
+where $D$ is diffusion constant. Unlike velocity autocorrelation
-+
+function which is averaging over time origins and over all the
-+
+atoms, dipole autocorrelation are calculated for the entire system.
-+
+The dipole autocorrelation function is given by:
-+
+\[
-+
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
-+
+\right\rangle
-+
+\]
-+
+Here $u_{tot}$ is the net dipole of the entire system and is given
-+
+by
-+
+\[
-+
+u_{tot} (t) = \sum\limits_i {u_i (t)}
-+
+\]
-+
+In principle, many time correlation functions can be related with
-+
+Fourier transforms of the infrared, Raman, and inelastic neutron
-+
+scattering spectra of molecular liquids. In practice, one can
-+
+extract the IR spectrum from the intensity of dipole fluctuation at
-+
+each frequency using the following relationship:
-+
+\[
-+
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
-+
+i2\pi vt} dt}
-+
+\]
-+
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
-+
-+
+Rigid bodies are frequently involved in the modeling of different
-+
+areas, from engineering, physics, to chemistry. For example,
-+
+missiles and vehicle are usually modeled by rigid bodies.  The
-+
+movement of the objects in 3D gaming engine or other physics
-+
+simulator is governed by the rigid body dynamics. In molecular
-+
+simulation, rigid body is used to simplify the model in
-+
+protein-protein docking study{\cite{Gray03}}.
-+
-+
+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.
-+
-+
+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 $Q$ 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 $Q_T Q = 1$. 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: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
+\[
-<
+L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
->
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
+\]
-+
+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}
-+
+Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
-+
+\ref{introEquation:motionHamiltonianMomentum}), one can write down
-+
+the equations of motion,
+\[
-<
+L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
->
+\begin{array}{c}
->
+ \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\
->
+ \frac{{dp}}{{dt}} =  - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
->
+ \frac{{dQ}}{{dt}} = PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
->
+ \frac{{dP}}{{dt}} =  - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\
->
+ \end{array}
+\]
-<
+Some relatively important transformation,
->
+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
+\[
-<
+L(\cos at) = \frac{p}{{p^2  + a^2 }}
->
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
->
+\right\}.
+\]
-+
-+
+Unfortunately, this constraint manifold is not the cotangent bundle
-+
+$T_{\star}SO(3)$. However, it turns out that under symplectic
-+
+transformation, the cotangent space and the phase space are
-+
+diffeomorphic. 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,
+\[
-<
+L(\sin at) = \frac{a}{{p^2  + a^2 }}
->
+\begin{array}{l}
->
+ \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
->
+ Q^T Q = 1 \\
->
+ \end{array}
+\]
-+
+For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
-+
+so(3)^ \star$, the hat-map isomorphism,
-+
+\begin{equation}
-+
+v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
-+
+{\begin{array}{*{20}c}
-+
+& { - v_3 } & {v_2 }  \\
-+
+   {v_3 } & 0 & { - v_1 }  \\
-+
+   { - v_2 } & {v_1 } & 0  \\
-+
+\end{array}} \right),
-+
+\label{introEquation:hatmapIsomorphism}
-+
+\end{equation}
-+
+will let us associate the matrix products with traditional vector
-+
+operations
+\[
-<
+L(1) = \frac{1}{p}
->
+\hat vu = v \times u
+\]
-<
+First, the bath coordinates,
->
+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
+\[
-<
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
-<
+_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
-<
+}}L(x)
->
+\dot Q = Qskew(I^{ - 1} \pi )
+\]
-+
-+
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
-+
+Lie-Poisson Integrator for Free Rigid Body}
-+
-+
+If there 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
+\[
-<
+L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
-<
+px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
->
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
->
+Q(0)e^{\Delta tR_1 }
+\]
-<
+Then, the system coordinates,
-<
+\begin{align}
-<
+mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
-<
+\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{\frac{{g_\alpha
-<
+}}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha
-<
+(0)}}{{p^2  + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha
-<
+}}\omega _\alpha ^2 L(x)} \right\}}
-<
+%
-<
+ &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
-<
+ \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x)
-<
+ - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0)
-<
+ - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}
-<
+\end{align}
-<
+Then, the inverse transform,
->
+with
->
+\[
->
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
->
+& 0 & 0  \\
->
+& {\cos \theta _1 } & {\sin \theta _1 }  \\
->
+& { - \sin \theta _1 } & {\cos \theta _1 }  \\
->
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
->
+\]
->
+To reduce the cost of computing expensive functions in $e^{\Delta
->
+tR_1 }$, we can use Cayley transformation,
->
+\[
->
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
->
+)
->
+\]
->
+The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
->
+manner.
-+
+In order to construct a second-order symplectic method, we split the
-+
+angular kinetic Hamiltonian function can into five terms
-+
+\[
-+
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
-+
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
-+
+(\pi _1 )
-+
+\].
-+
+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}
-+
+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 simulation. A brief derivation of
-+
+generalized Langevin equation will be given first. Follow that, we
-+
+will discuss the physical meaning of the terms appearing in the
-+
+equation as well as the calculation of friction tensor from
-+
+hydrodynamics theory.
-+
-+
+\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
-+
-+
+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 Deriving Generalized
-+
+Langevin Dynamics. Lets consider a system, in which the degree of
-+
+freedom $x$ is assumed to couple to the bath linearly, giving a
-+
+Hamiltonian of the form
-+
+\begin{equation}
-+
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
-+
+\label{introEquation:bathGLE}.
-+
+\end{equation}
-+
+Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated
-+
+with this degree of freedom, $H_B$ is harmonic bath Hamiltonian,
-+
+\[
-+
+H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-+
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  \omega _\alpha ^2 }
-+
+\right\}}
-+
+\]
-+
+where the index $\alpha$ runs over all the bath degrees of freedom,
-+
+$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
-+
+the harmonic bath masses, and $\Delta U$ is 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 and the
-+
+coordinate $x$. Introducing
-+
+\[
-+
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
-+
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
-+
+\] and combining the last two terms in Equation
-+
+\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath
-+
+Hamiltonian as
-+
+\[
-+
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-+
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-+
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-+
+w_\alpha ^2 }}x} \right)^2 } \right\}}
-+
+\]
-+
+Since the first two terms of the new Hamiltonian depend only on the
-+
+system coordinates, we can get the equations of motion for
-+
+Generalized Langevin Dynamics by Hamilton's equations
-+
+\ref{introEquation:motionHamiltonianCoordinate,
-+
+introEquation:motionHamiltonianMomentum},
-+
+\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, 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 applying inverse Laplace transform, also known
-+
+as the Bromwich integral, we can retrieve the solutions of the
-+
+original problems.
-+
-+
+Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace
-+
+transform of f(t) is a new function defined as
-+
+\[
-+
+L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
-+
+\]
-+
+where  $p$ is real and  $L$ is called the Laplace Transform
-+
+Operator. Below are some important properties of Laplace transform
-+
+\begin{equation}
-+
+\begin{array}{c}
-+
+ 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{array}
-+
+\end{equation}
-+
-+
+Applying Laplace transform to the bath coordinates, we obtain
-+
+\[
-+
+\begin{array}{c}
-+
+ p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega _\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) \\
-+
+ L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }} \\
-+
+ \end{array}
-+
+\]
-+
+By the same way, the system coordinates become
-+
+\[
-+
+\begin{array}{c}
-+
+ mL(\ddot x) =  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
-+
+  - \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0) - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}  \\
-+
+ \end{array}
-+
+\]
-+
-+
+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{align}
+m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
+\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
+(\omega _\alpha  t)} \right\}}
+\end{align}
-+
+Introducing a \emph{dynamic friction kernel}
+\begin{equation}
-–
+m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
-–
+(t)\dot x(t - \tau )d\tau }  + R(t)
-–
+\label{introEuqation:GeneralizedLangevinDynamics}
-–
+\end{equation}
-–
+%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
-–
+%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
-–
+\[
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
-<
+\]
-<
+For an infinite harmonic bath, we can use the spectral density and
-<
+an integral over frequencies.
-<
-<
+\[
->
+\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)
->
+(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}.
->
->
+\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}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 $\dot x$,
->
+\[
->
+\begin{array}{l}
->
+ \left\langle {x(t)R(t)} \right\rangle  = 0, \\
->
+ \left\langle {\dot x(t)R(t)} \right\rangle  = 0. \\
->
+ \end{array}
+\]
-<
+The random forces depend only on initial conditions.
->
+This property is what we expect from a truly random process. As long
->
+as the model, which is gaussian distribution in general, chosen for
->
+$R(t)$ is a truly random process, 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 Equation \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 dynamic caging effect. 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 Equation \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
-+
+briefly review on calculating friction tensor for arbitrary shaped
-+
+particles is given in Sec.~\ref{introSection:frictionTensor}.
-+
+\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
-<
+So we can define a new set of coordinates,
->
->
+Defining a new set of coordinates,
+\[
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
+^2 }}x(0)
-<
+\]
-<
+This makes
->
+\],
->
+we can rewrite $R(T)$ as
+\[
-<
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
->
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
+\]
+And since the $q$ coordinates are harmonic oscillators,
+\[
-<
+\begin{array}{l}
->
+\begin{array}{c}
->
+ \left\langle {q_\alpha ^2 } \right\rangle  = \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
-+
+ \left\langle {R(t)R(0)} \right\rangle  = \sum\limits_\alpha  {\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle } }  \\
-+
+  = \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
-+
+  = kT\xi (t) \\
+ \end{array}
+\]
-<
-<
+\begin{align}
-<
+\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
-<
+{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
-<
+(t)q_\beta  (0)} \right\rangle } }
-<
+%
-<
+&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
-<
+\right\rangle \cos (\omega _\alpha  t)}
-<
+%
-<
+&= kT\xi (t)
-<
+\end{align}
-<
->
+Thus, we recover the \emph{second fluctuation dissipation theorem}
+\begin{equation}
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-<
+\label{introEquation:secondFluctuationDissipation}
->
+\label{introEquation:secondFluctuationDissipation}.
+\end{equation}
-+
+In effect, it acts as a constraint on the possible ways in which one
-+
+can model the random force and friction kernel.
-–
+\section{\label{introSection:hydroynamics}Hydrodynamics}
-–
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
-<
+\subsection{\label{introSection:analyticalApproach}Analytical
-<
+Approach}
->
+Theoretically, the friction kernel can be determined using velocity
->
+autocorrelation function. However, this approach become impractical
->
+when the system become more and more complicate. Instead, various
->
+approaches based on hydrodynamics have been developed to calculate
->
+the friction coefficients. The friction effect is isotropic in
->
+Equation, \zeta can be taken as a scalar. In general, friction
->
+tensor \Xi is a $6\times 6$ matrix given by
->
+\[
->
+\Xi  = \left( {\begin{array}{*{20}c}
->
+   {\Xi _{}^{tt} } & {\Xi _{}^{rt} }  \\
->
+   {\Xi _{}^{tr} } & {\Xi _{}^{rr} }  \\
->
+\end{array}} \right).
->
+\]
->
+Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction
->
+tensor and rotational resistance (friction) tensor respectively,
->
+while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $
->
+{\Xi^{rt} }$ is rotation-translation coupling tensor. When a
->
+particle moves in a fluid, it may experience friction force or
->
+torque along the opposite direction of the velocity or angular
->
+velocity,
->
+\[
->
+\left( \begin{array}{l}
->
+ F_R  \\
->
+ \tau _R  \\
->
+ \end{array} \right) =  - \left( {\begin{array}{*{20}c}
->
+   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
->
+   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
->
+\end{array}} \right)\left( \begin{array}{l}
->
+ v \\
->
+ w \\
->
+ \end{array} \right)
->
+\]
->
+where $F_r$ is the friction force and $\tau _R$ is the friction
->
+toque.
-<
+\subsection{\label{introSection:approximationApproach}Approximation
-<
+Approach}
->
+\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape}
-<
+\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-<
+Body}
->
+For a spherical particle, the translational and rotational friction
->
+constant can be calculated from Stoke's law,
->
+\[
->
+\Xi ^{tt}  = \left( {\begin{array}{*{20}c}
->
+   {6\pi \eta R} & 0 & 0  \\
->
+& {6\pi \eta R} & 0  \\
->
+& 0 & {6\pi \eta R}  \\
->
+\end{array}} \right)
->
+\]
->
+and
->
+\[
->
+\Xi ^{rr}  = \left( {\begin{array}{*{20}c}
->
+   {8\pi \eta R^3 } & 0 & 0  \\
->
+& {8\pi \eta R^3 } & 0  \\
->
+& 0 & {8\pi \eta R^3 }  \\
->
+\end{array}} \right)
->
+\]
->
+where $\eta$ is the viscosity of the solvent and $R$ is the
->
+hydrodynamics radius.
->
->
+Other non-spherical shape, such as cylinder and ellipsoid
->
+\textit{etc}, are widely used as reference for developing new
->
+hydrodynamics theory, because their properties can be calculated
->
+exactly. In 1936, Perrin extended Stokes's law to general ellipsoid,
->
+also called a triaxial ellipsoid, which is given in Cartesian
->
+coordinates by
->
+\[
->
+\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2
->
+}} = 1
->
+\]
->
+where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately,
->
+due to the complexity of the elliptic integral, only the ellipsoid
->
+with the restriction of two axes having to be equal, \textit{i.e.}
->
+prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved
->
+exactly. Introducing an elliptic integral parameter $S$ for prolate,
->
+\[
->
+S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
->
+} }}{b},
->
+\]
->
+and oblate,
->
+\[
->
+S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
->
+}}{a}
->
+\],
->
+one can write down the translational and rotational resistance
->
+tensors
->
+\[
->
+\begin{array}{l}
->
+ \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
->
+ \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
->
+ \end{array},
->
+\]
->
+and
->
+\[
->
+\begin{array}{l}
->
+ \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
->
+ \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
->
+ \end{array}.
->
+\]
->
->
+\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape}
->
->
+Unlike spherical and other regular shaped molecules, there is not
->
+analytical solution for friction tensor of any arbitrary shaped
->
+rigid molecules. The ellipsoid of revolution model and general
->
+triaxial ellipsoid model have been used to approximate the
->
+hydrodynamic properties of rigid bodies. However, since the mapping
->
+from all possible ellipsoidal space, $r$-space, to all possible
->
+combination of rotational diffusion coefficients, $D$-space is not
->
+unique\cite{Wegener79} as well as the intrinsic coupling between
->
+translational and rotational motion of rigid body\cite{}, general
->
+ellipsoid is not always suitable for modeling arbitrarily shaped
->
+rigid molecule. A number of studies have been devoted to determine
->
+the friction tensor for irregularly shaped rigid bodies using more
->
+advanced method\cite{} where the molecule of interest was modeled by
->
+combinations of spheres(beads)\cite{} and the hydrodynamics
->
+properties of the molecule can be calculated using the hydrodynamic
->
+interaction tensor. Let us consider a rigid assembly of $N$ beads
->
+immersed in a continuous medium. Due to hydrodynamics interaction,
->
+the ``net'' velocity of $i$th bead, $v'_i$ is different than its
->
+unperturbed velocity $v_i$,
->
+\[
->
+v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
->
+\]
->
+where $F_i$ is the frictional force, and $T_{ij}$ is the
->
+hydrodynamic interaction tensor. The friction force of $i$th bead is
->
+proportional to its ``net'' velocity
->
+\begin{equation}
->
+F_i  = \zeta _i v_i  - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }.
->
+\label{introEquation:tensorExpression}
->
+\end{equation}
->
+This equation is the basis for deriving the hydrodynamic tensor. In
->
+, Oseen and Burgers gave a simple solution to Equation
->
+\ref{introEquation:tensorExpression}
->
+\begin{equation}
->
+T_{ij}  = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij}
->
+R_{ij}^T }}{{R_{ij}^2 }}} \right).
->
+\label{introEquation:oseenTensor}
->
+\end{equation}
->
+Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
->
+A second order expression for element of different size was
->
+introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de
->
+la Torre and Bloomfield,
->
+\begin{equation}
->
+T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
->
+\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
->
+_i^2  + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} -
->
+\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
->
+\label{introEquation:RPTensorNonOverlapped}
->
+\end{equation}
->
+Both of the Equation \ref{introEquation:oseenTensor} and Equation
->
+\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij}
->
+\ge \sigma _i  + \sigma _j$. An alternative expression for
->
+overlapping beads with the same radius, $\sigma$, is given by
->
+\begin{equation}
->
+T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
->
+\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I +
->
+\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
->
+\label{introEquation:RPTensorOverlapped}
->
+\end{equation}
->
->
+To calculate the resistance tensor at an arbitrary origin $O$, we
->
+construct a $3N \times 3N$ matrix consisting of $N \times N$
->
+$B_{ij}$ blocks
->
+\begin{equation}
->
+B = \left( {\begin{array}{*{20}c}
->
+   {B_{11} } &  \ldots  & {B_{1N} }  \\
->
+    \vdots  &  \ddots  &  \vdots   \\
->
+   {B_{N1} } &  \cdots  & {B_{NN} }  \\
->
+\end{array}} \right),
->
+\end{equation}
->
+where $B_{ij}$ is given by
->
+\[
->
+B_{ij}  = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij}
->
+)T_{ij}
->
+\]
->
+where $\delta _{ij}$ is Kronecker delta function. Inverting matrix
->
+$B$, we obtain
->
->
+\[
->
+C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
->
+   {C_{11} } &  \ldots  & {C_{1N} }  \\
->
+    \vdots  &  \ddots  &  \vdots   \\
->
+   {C_{N1} } &  \cdots  & {C_{NN} }  \\
->
+\end{array}} \right)
->
+\]
->
+, which can be partitioned into $N \times N$ $3 \times 3$ block
->
+$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$
->
+\[
->
+U_i  = \left( {\begin{array}{*{20}c}
->
+& { - z_i } & {y_i }  \\
->
+   {z_i } & 0 & { - x_i }  \\
->
+   { - y_i } & {x_i } & 0  \\
->
+\end{array}} \right)
->
+\]
->
+where $x_i$, $y_i$, $z_i$ are the components of the vector joining
->
+bead $i$ and origin $O$. Hence, the elements of resistance tensor at
->
+arbitrary origin $O$ can be written as
->
+\begin{equation}
->
+\begin{array}{l}
->
+ \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\
->
+ \Xi _{}^{tr}  = \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
->
+ \Xi _{}^{rr}  =  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j  \\
->
+ \end{array}
->
+\label{introEquation:ResistanceTensorArbitraryOrigin}
->
+\end{equation}
->
->
+The resistance tensor depends on the origin to which they refer. The
->
+proper location for applying friction force is the center of
->
+resistance (reaction), at which the trace of rotational resistance
->
+tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of
->
+resistance is defined as an unique point of the rigid body at which
->
+the translation-rotation coupling tensor are symmetric,
->
+\begin{equation}
->
+\Xi^{tr}  = \left( {\Xi^{tr} } \right)^T
->
+\label{introEquation:definitionCR}
->
+\end{equation}
->
+Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin},
->
+we can easily find out that the translational resistance tensor is
->
+origin independent, while the rotational resistance tensor and
->
+translation-rotation coupling resistance tensor depend on the
->
+origin. Given resistance tensor at an arbitrary origin $O$, and a
->
+vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can
->
+obtain the resistance tensor at $P$ by
->
+\begin{equation}
->
+\begin{array}{l}
->
+ \Xi _P^{tt}  = \Xi _O^{tt}  \\
->
+ \Xi _P^{tr}  = \Xi _P^{rt}  = \Xi _O^{tr}  - U_{OP} \Xi _O^{tt}  \\
->
+ \Xi _P^{rr}  = \Xi _O^{rr}  - U_{OP} \Xi _O^{tt} U_{OP}  + \Xi _O^{tr} U_{OP}  - U_{OP} \Xi _O^{tr} ^{^T }  \\
->
+ \end{array}
->
+ \label{introEquation:resistanceTensorTransformation}
->
+\end{equation}
->
+where
->
+\[
->
+U_{OP}  = \left( {\begin{array}{*{20}c}
->
+& { - z_{OP} } & {y_{OP} }  \\
->
+   {z_i } & 0 & { - x_{OP} }  \\
->
+   { - y_{OP} } & {x_{OP} } & 0  \\
->
+\end{array}} \right)
->
+\]
->
+Using Equations \ref{introEquation:definitionCR} and
->
+\ref{introEquation:resistanceTensorTransformation}, one can locate
->
+the position of center of resistance,
->
+\[
->
+\left( \begin{array}{l}
->
+ x_{OR}  \\
->
+ y_{OR}  \\
->
+ z_{OR}  \\
->
+ \end{array} \right) = \left( {\begin{array}{*{20}c}
->
+   {(\Xi _O^{rr} )_{yy}  + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} }  \\
->
+   { - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz}  + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} }  \\
->
+   { - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx}  + (\Xi _O^{rr} )_{yy} }  \\
->
+\end{array}} \right)^{ - 1} \left( \begin{array}{l}
->
+ (\Xi _O^{tr} )_{yz}  - (\Xi _O^{tr} )_{zy}  \\
->
+ (\Xi _O^{tr} )_{zx}  - (\Xi _O^{tr} )_{xz}  \\
->
+ (\Xi _O^{tr} )_{xy}  - (\Xi _O^{tr} )_{yx}  \\
->
+ \end{array} \right).
->
+\]
->
+where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
->
+joining center of resistance $R$ and origin $O$.

Diff Legend

-–
+Removed lines
-+
+Added lines
-<
+Changed lines
->
+Changed lines

Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2700 by tim, Tue Apr 11 03:38:09 2006 UTC vs. Revision 2726 by tim, Fri Apr 21 15:37:53 2006 UTC

Diff Legend

Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2700 by tim, Tue Apr 11 03:38:09 2006 UTC vs.
Revision 2726 by tim, Fri Apr 21 15:37:53 2006 UTC