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+\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
-–
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
-–
-–
+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.
-–
+\section{\label{introSection:classicalMechanics}Classical
+Mechanics}
+sufficient to predict the future behavior of the system.
+\subsection{\label{introSection:newtonian}Newtonian Mechanics}
-+
+The discovery of Newton's three laws of mechanics which govern the
-+
+motion of particles is the foundation of the classical mechanics.
-+
+Newton’s first law defines a class of inertial frames. Inertial
-+
+frames are reference frames where a particle not interacting with
-+
+other bodies will move with constant speed in the same direction.
-+
+With respect to inertial frames Newton’s second law has the form
-+
+\begin{equation}
-+
+F = \frac {dp}{dt} = \frac {mv}{dt}
-+
+\label{introEquation:newtonSecondLaw}
-+
+\end{equation}
-+
+A point mass interacting with other bodies moves with the
-+
+acceleration along the direction of the force acting on it. Let
-+
+$F_ij$ be the force that particle $i$ exerts on particle $j$, and
-+
+$F_ji$ be the force that particle $j$ exerts on particle $i$.
-+
+Newton’s third law states that
-+
+\begin{equation}
-+
+F_ij = -F_ji
-+
+ \label{introEquation:newtonThirdLaw}
-+
+\end{equation}
-+
+Conservation laws of Newtonian Mechanics play very important roles
-+
+in solving mechanics problems. The linear momentum of a particle is
-+
+conserved if it is free or it experiences no force. The second
-+
+conservation theorem concerns the angular momentum of a particle.
-+
+The angular momentum $L$ of a particle with respect to an origin
-+
+from which $r$ is measured is defined to be
-+
+\begin{equation}
-+
+L \equiv r \times p \label{introEquation:angularMomentumDefinition}
-+
+\end{equation}
-+
+The torque $\tau$ with respect to the same origin is defined to be
-+
+\begin{equation}
-+
+N \equiv r \times F \label{introEquation:torqueDefinition}
-+
+\end{equation}
-+
+Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
-+
+\[
-+
+\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times
-+
+\dot p)
-+
+\]
-+
+since
-+
+\[
-+
+\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0
-+
+\]
-+
+thus,
-+
+\begin{equation}
-+
+\dot L = r \times \dot p = N
-+
+\end{equation}
-+
+If there are no external torques acting on a body, the angular
-+
+momentum of it is conserved. The last conservation theorem state
-+
+that if all forces are conservative, Energy
-+
+\begin{equation}E = T + V \label{introEquation:energyConservation}
-+
+\end{equation}
-+
+ is conserved. All of these conserved quantities are
-+
+important factors to determine the quality of numerical integration
-+
+scheme for rigid body \cite{Dullweber1997}.
-+
+\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
+Newtonian Mechanics suffers from two important limitations: it
+which arise in attempts to apply Newton's equation to complex
+system, alternative procedures may be developed.
-<
+\subsection{\label{introSection:halmiltonPrinciple}Hamilton's
->
+\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
+Principle}
+Hamilton introduced the dynamical principle upon which it is
+The actual trajectory, along which a dynamical system may move from
+one point to another within a specified time, is derived by finding
+the path which minimizes the time integral of the difference between
-<
+the kinetic, $K$, and potential energies, $U$.
->
+the kinetic, $K$, and potential energies, $U$ \cite{tolman79}.
+\begin{equation}
+\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
+\label{introEquation:halmitonianPrinciple1}
+\label{introEquation:halmitonianPrinciple2}
+\end{equation}
-<
+\subsection{\label{introSection:equationOfMotionLagrangian}The
->
+\subsubsection{\label{introSection:equationOfMotionLagrangian}The
+Equations of Motion in Lagrangian Mechanics}
+for a holonomic system of $f$ degrees of freedom, the equations of
+Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
+Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
+equation of motion. Due to their symmetrical formula, they are also
-<
+known as the canonical equations of motions.
->
+known as the canonical equations of motions \cite{Goldstein01}.
+An important difference between Lagrangian approach and the
+Hamiltonian approach is that the Lagrangian is considered to be a
+appropriate for application to statistical mechanics and quantum
+mechanics, since it treats the coordinate and its time derivative as
+independent variables and it only works with 1st-order differential
-<
+equations.
->
+equations\cite{Marion90}.
-<
+\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
->
+In Newtonian Mechanics, a system described by conservative forces
->
+conserves the total energy \ref{introEquation:energyConservation}.
->
+It follows that Hamilton's equations of motion conserve the total
->
+Hamiltonian.
->
+\begin{equation}
->
+\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
->
+H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
->
+}}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
->
+H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
->
+\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
->
+q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian}
->
+\end{equation}
-–
+\subsection{\label{introSection:canonicalTransformation}Canonical
-–
+Transformation}
-–
+\section{\label{introSection:statisticalMechanics}Statistical
+Mechanics}
-<
+The thermodynamic behaviors and properties  of Molecular Dynamics
->
+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.
-<
+\subsection{\label{introSection::ensemble}Ensemble}
->
+\subsection{\label{introSection:ensemble}Ensemble and Phase Space}
+\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
-+
+Various thermodynamic properties can be calculated from Molecular
-+
+Dynamics simulation. By comparing experimental values with the
-+
+calculated properties, one can determine the accuracy of the
-+
+simulation and the quality of the underlying model. However, both of
-+
+experiment and computer simulation are usually performed during a
-+
+certain time interval and the measurements are averaged over a
-+
+period of them which is different from the average behavior of
-+
+many-body system in Statistical Mechanics. Fortunately, Ergodic
-+
+Hypothesis is proposed to make a connection between time average and
-+
+ensemble average. It states that time average and average over the
-+
+statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
-+
+\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
-+
+\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}.
-+
-+
+\section{\label{introSection:geometricIntegratos}Geometric Integrators}
-+
+A variety of numerical integrators were proposed to simulate the
-+
+motions. They usually begin with an initial conditionals and move
-+
+the objects in the direction governed by the differential equations.
-+
+However, most of them ignore the hidden physical law contained
-+
+within the equations. Since 1990, geometric integrators, which
-+
+preserve various phase-flow invariants such as symplectic structure,
-+
+volume and time reversal symmetry, are developed to address this
-+
+issue. The velocity verlet method, which happens to be a simple
-+
+example of symplectic integrator, continues to gain its popularity
-+
+in molecular dynamics community. This fact can be partly explained
-+
+by its geometric nature.
-+
-+
+\subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
-+
+A \emph{manifold} is an abstract mathematical space. It locally
-+
+looks like Euclidean space, but when viewed globally, it may have
-+
+more complicate structure. A good example of manifold is the surface
-+
+of Earth. It seems to be flat locally, but it is round if viewed as
-+
+a whole. A \emph{differentiable manifold} (also known as
-+
+\emph{smooth manifold}) is a manifold with an open cover in which
-+
+the covering neighborhoods are all smoothly isomorphic to one
-+
+another. In other words,it is possible to apply calculus on
-+
+\emph{differentiable manifold}. A \emph{symplectic manifold} is
-+
+defined as a pair $(M, \omega)$ which consisting of a
-+
+\emph{differentiable manifold} $M$ and a close, non-degenerated,
-+
+bilinear symplectic form, $\omega$. A symplectic form on a vector
-+
+space $V$ is a function $\omega(x, y)$ which satisfies
-+
+$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
-+
+\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
-+
+$\omega(x, x) = 0$. Cross product operation in vector field is an
-+
+example of symplectic form.
-+
-+
+One of the motivations to study \emph{symplectic manifold} in
-+
+Hamiltonian Mechanics is that a symplectic manifold can represent
-+
+all possible configurations of the system and the phase space of the
-+
+system can be described by it's cotangent bundle. Every symplectic
-+
+manifold is even dimensional. For instance, in Hamilton equations,
-+
+coordinate and momentum always appear in pairs.
-+
-+
+Let  $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map
-+
+\[
-+
+f : M \rightarrow N
-+
+\]
-+
+is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and
-+
+the \emph{pullback} of $\eta$ under f is equal to $\omega$.
-+
+Canonical transformation is an example of symplectomorphism in
-+
+classical mechanics.
-+
-+
+\subsection{\label{introSection:ODE}Ordinary Differential Equations}
-+
-+
+For a ordinary differential system defined as
-+
+\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 $x$.
-+
+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:geometricProperties}Geometric Properties}
-+
+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. 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}
-+
-+
+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,
-+
+\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  =
-+
+$. 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.
-+
-+
+\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\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,
-+
+\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,
-+
+\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}
-+
+Suppose that a Hamiltonian system has a form with $H = T + V$
-+
-+
-+
-+
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
-+
-+
+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.
-+
-+
+\subsection{\label{introSec:mdInit}Initialization}
-+
-+
+\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.
-+
-+
+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}
-<
+\subsection{\label{introSection:hydroynamics}Hydrodynamics}
->
+\begin{equation}
->
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
->
+\label{introEquation:bathGLE}
->
+\end{equation}
->
+where $H_B$ is harmonic bath Hamiltonian,
->
+\[
->
+H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
->
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
->
+\]
->
+and $\Delta U$ is bilinear system-bath coupling,
->
+\[
->
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
->
+\]
->
+Completing the square,
->
+\[
->
+H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
->
+{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
->
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
->
+w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
->
+}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
->
+\]
->
+and putting it back into Eq.~\ref{introEquation:bathGLE},
->
+\[
->
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
->
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
->
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
->
+w_\alpha ^2 }}x} \right)^2 } \right\}}
->
+\]
->
+where
->
+\[
->
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
->
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
->
+\]
->
+Since the first two terms of the new Hamiltonian depend only on the
->
+system coordinates, we can get the equations of motion for
->
+Generalized Langevin Dynamics by Hamilton's equations
->
+\ref{introEquation:motionHamiltonianCoordinate,
->
+introEquation:motionHamiltonianMomentum},
->
+\begin{align}
->
+\dot p &=  - \frac{{\partial H}}{{\partial x}}
->
+       &= m\ddot x
->
+       &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)}
->
+\label{introEq:Lp5}
->
+\end{align}
->
+, and
->
+\begin{align}
->
+\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
->
+                &= m\ddot x_\alpha
->
+                &= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
->
+\end{align}
->
->
+\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
->
->
+\[
->
+L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
->
+\]
->
->
+\[
->
+L(x + y) = L(x) + L(y)
->
+\]
->
->
+\[
->
+L(ax) = aL(x)
->
+\]
->
->
+\[
->
+L(\dot x) = pL(x) - px(0)
->
+\]
->
->
+\[
->
+L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
->
+\]
->
->
+\[
->
+L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
->
+\]
->
->
+Some relatively important transformation,
->
+\[
->
+L(\cos at) = \frac{p}{{p^2  + a^2 }}
->
+\]
->
->
+\[
->
+L(\sin at) = \frac{a}{{p^2  + a^2 }}
->
+\]
->
->
+\[
->
+L(1) = \frac{1}{p}
->
+\]
->
->
+First, the bath coordinates,
->
+\[
->
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
->
+_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
->
+}}L(x)
->
+\]
->
+\[
->
+L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
->
+px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
->
+\]
->
+Then, the system coordinates,
->
+\begin{align}
->
+mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
->
+\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{\frac{{g_\alpha
->
+}}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha
->
+(0)}}{{p^2  + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha
->
+}}\omega _\alpha ^2 L(x)} \right\}}
->
+%
->
+ &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
->
+ \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x)
->
+ - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0)
->
+ - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}
->
+\end{align}
->
+Then, the inverse transform,
->
->
+\begin{align}
->
+m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
->
+\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
->
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
->
+_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
->
+- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
->
+(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
->
+_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
->
+%
->
+&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
->
+{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
->
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
->
+t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
->
+{\left[ {g_\alpha  x_\alpha  (0) - \frac{{g_\alpha  }}{{m_\alpha
->
+\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
->
+\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
->
+(\omega _\alpha  t)} \right\}}
->
+\end{align}
->
->
+\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.
->
->
+\[
->
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
->
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
->
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
->
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
->
+\]
->
+The random forces depend only on initial conditions.
->
->
+\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
->
+So we can define a new set of coordinates,
->
+\[
->
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
->
+^2 }}x(0)
->
+\]
->
+This makes
->
+\[
->
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
->
+\]
->
+And since the $q$ coordinates are harmonic oscillators,
->
+\[
->
+\begin{array}{l}
->
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
->
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
->
+ \end{array}
->
+\]
->
->
+\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}
->
->
+\begin{equation}
->
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
->
+\label{introEquation:secondFluctuationDissipation}
->
+\end{equation}
->
->
+\section{\label{introSection:hydroynamics}Hydrodynamics}
->
->
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
->
+\subsection{\label{introSection:analyticalApproach}Analytical
->
+Approach}
->
->
+\subsection{\label{introSection:approximationApproach}Approximation
->
+Approach}
->
->
+\subsection{\label{introSection:centersRigidBody}Centers of Rigid
->
+Body}

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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2693 by tim, Wed Apr 5 03:44:32 2006 UTC vs. Revision 2699 by tim, Mon Apr 10 05:35:55 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2693 by tim, Wed Apr 5 03:44:32 2006 UTC vs.
Revision 2699 by tim, Mon Apr 10 05:35:55 2006 UTC