--- trunk/tengDissertation/Introduction.tex	2006/04/06 04:07:57	2695
+++ trunk/tengDissertation/Introduction.tex	2006/04/07 22:05:48	2698
@@ -63,10 +63,12 @@ that if all forces are conservative, Energy $E = T + V
 \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 $E = T + V$ is
-conserved. All of these conserved quantities are important factors
-to determine the quality of numerical integration scheme for rigid
-body \cite{Dullweber1997}.
+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}
 
@@ -200,36 +202,19 @@ When studying Hamiltonian system, it is more convenien
 independent variables and it only works with 1st-order differential
 equations\cite{Marion90}.
 
-When studying Hamiltonian system, it is more convenient to use
-notation
+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}
-r = r(q,p)^T
-\end{equation}
-and to introduce a $2n \times 2n$ canonical structure matrix $J$,
-\begin{equation}
-J = \left( {\begin{array}{*{20}c}
-   0 & I  \\
-   { - I} & 0  \\
-\end{array}} \right)
-\label{introEquation:canonicalMatrix}
+\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}
-where $I$ is a $n \times n$ identity matrix and $J$ is a
-skew-symmetric matrix ($ J^T  =  - J $). Thus, Hamiltonian system
-can be rewritten as,
-\begin{equation}
-\frac{d}{{dt}}r = J\nabla _r H(r)
-\label{introEquation:compactHamiltonian}
-\end{equation}
 
-%\subsection{\label{introSection:canonicalTransformation}Canonical
-%Transformation}
-
-\section{\label{introSection:geometricIntegratos}Geometric Integrators}
-
-\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods}
-
-\subsection{\label{Construction of Symplectic Methods}}
-
 \section{\label{introSection:statisticalMechanics}Statistical
 Mechanics}
 
@@ -238,7 +223,7 @@ Statistical Mechanics concepts presented in this disse
 The following section will give a brief introduction to some of the
 Statistical Mechanics concepts presented in this dissertation.
 
-\subsection{\label{introSection::ensemble}Ensemble and Phase Space}
+\subsection{\label{introSection:ensemble}Ensemble and Phase Space}
 
 \subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
 
@@ -269,8 +254,153 @@ will be the best choice.
 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.
+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}
+   0 & 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}
+   0 & {\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. The flow of a Hamiltonian vector field
+on a symplectic manifold is a symplectomorphism. Let $\varphi$ be
+the flow of Hamiltonian vector field, $\varphi$ is a
+\emph{symplectic} flow if it satisfies,
+\begin{equation}
+d \varphi^T J d \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. As to the Poisson system,
+\begin{equation}
+d\varphi ^T Jd\varphi  = J \circ \varphi
+\end{equation}
+is the property must be preserved by the integrator. It is possible
+to construct a \emph{volume-preserving} flow for a source free($
+\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi  =
+1$. 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:splittingAndComposition}Splitting and Composition Methods}
+
 \section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
 As a special discipline of molecular modeling, Molecular dynamics
@@ -303,6 +433,208 @@ Applications of dynamics of rigid bodies.
 
 \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,
+\[
+H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
+{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
+w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
+1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
+\]
+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}
