--- trunk/tengDissertation/Introduction.tex	2006/06/30 02:45:29	2912
+++ trunk/tengDissertation/Introduction.tex	2006/07/17 15:28:44	2938
@@ -208,7 +208,7 @@ Statistical Mechanics concepts and theorem presented i
 The thermodynamic behaviors and properties of Molecular Dynamics
 simulation are governed by the principle of Statistical Mechanics.
 The following section will give a brief introduction to some of the
-Statistical Mechanics concepts and theorem presented in this
+Statistical Mechanics concepts and theorems presented in this
 dissertation.
 
 \subsection{\label{introSection:ensemble}Phase Space and Ensemble}
@@ -372,7 +372,7 @@ bracket ${F, G}$ is defined as
 Liouville's theorem can be expressed in a variety of different forms
 which are convenient within different contexts. For any two function
 $F$ and $G$ of the coordinates and momenta of a system, the Poisson
-bracket ${F, G}$ is defined as
+bracket $\{F,G\}$ is defined as
 \begin{equation}
 \left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial
 F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} -
@@ -439,7 +439,7 @@ initial conditionals and move the objects in the direc
 \section{\label{introSection:geometricIntegratos}Geometric Integrators}
 A variety of numerical integrators have been proposed to simulate
 the motions of atoms in MD simulation. They usually begin with
-initial conditionals and move the objects in the direction governed
+initial conditions and move the objects in the direction governed
 by the differential equations. However, most of them ignore the
 hidden physical laws contained within the equations. Since 1990,
 geometric integrators, which preserve various phase-flow invariants
@@ -459,7 +459,7 @@ defined as a pair $(M, \omega)$ which consists of a
 \emph{smooth manifold}) is a manifold on which it is possible to
 apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is
 defined as a pair $(M, \omega)$ which consists of a
-\emph{differentiable manifold} $M$ and a close, non-degenerated,
+\emph{differentiable manifold} $M$ and a close, non-degenerate,
 bilinear symplectic form, $\omega$. A symplectic form on a vector
 space $V$ is a function $\omega(x, y)$ which satisfies
 $\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
@@ -479,7 +479,7 @@ where $x = x(q,p)^T$, this system is a canonical Hamil
 \begin{equation}
 \dot x = f(x)
 \end{equation}
-where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if
+where $x = x(q,p)$, this system is a canonical Hamiltonian, if
 $f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian
 function and $J$ is the skew-symmetric matrix
 \begin{equation}
@@ -527,7 +527,7 @@ The exact propagator can also be written in terms of o
 \begin{equation}
 \varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
 \end{equation}
-The exact propagator can also be written in terms of operator,
+The exact propagator can also be written as an operator,
 \begin{equation}
 \varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
 }{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
@@ -729,7 +729,7 @@ the equations of motion would follow:
 
 \item Use the half step velocities to move positions one whole step, $\Delta t$.
 
-\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move.
+\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move.
 
 \item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
 \end{enumerate}
@@ -868,7 +868,7 @@ near the minimum, steepest descent method is extremely
 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
+near the minimum, the steepest descent method is extremely robust when
 systems are strongly anharmonic. Thus, it is often used to refine
 structures from crystallographic data. Relying on the Hessian,
 advanced methods like Newton-Raphson converge rapidly to a local
@@ -887,7 +887,7 @@ at which the simulation will be conducted. In heating 
 temperature. Beginning at a lower temperature and gradually
 increasing the temperature by assigning larger random velocities, we
 end up setting the temperature of the system to a final temperature
-at which the simulation will be conducted. In heating phase, we
+at which the simulation will be conducted. In the heating phase, we
 should also keep the system from drifting or rotating as a whole. To
 do this, the net linear momentum and angular momentum of the system
 is shifted to zero after each resampling from the Maxwell -Boltzman
@@ -954,8 +954,8 @@ periodicity artifacts in liquid simulations. Taking th
 in simulations. The Ewald summation, in which the slowly decaying
 Coulomb potential is transformed into direct and reciprocal sums
 with rapid and absolute convergence, has proved to minimize the
-periodicity artifacts in liquid simulations. Taking the advantages
-of the fast Fourier transform (FFT) for calculating discrete Fourier
+periodicity artifacts in liquid simulations. Taking advantage
+of fast Fourier transform (FFT) techniques for calculating discrete Fourier
 transforms, the particle mesh-based
 methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
 $O(N^{3/2})$ to $O(N logN)$. An alternative approach is the
@@ -1059,41 +1059,40 @@ functions are called \emph{autocorrelation functions}.
 \label{introEquation:timeCorrelationFunction}
 \end{equation}
 If $A$ and $B$ refer to same variable, this kind of correlation
-functions are called \emph{autocorrelation functions}. One example
-of auto correlation function is the velocity auto-correlation
+functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation
 function which is directly related to transport properties of
 molecular liquids:
-\[
+\begin{equation}
 D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
 \right\rangle } dt
-\]
+\end{equation}
 where $D$ is diffusion constant. Unlike the velocity autocorrelation
 function, which is averaged over time origins and over all the
 atoms, the dipole autocorrelation functions is calculated for the
 entire system. The dipole autocorrelation function is given by:
-\[
+\begin{equation}
 c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
 \right\rangle
-\]
+\end{equation}
 Here $u_{tot}$ is the net dipole of the entire system and is given
 by
-\[
+\begin{equation}
 u_{tot} (t) = \sum\limits_i {u_i (t)}.
-\]
+\end{equation}
 In principle, many time correlation functions can be related to
 Fourier transforms of the infrared, Raman, and inelastic neutron
 scattering spectra of molecular liquids. In practice, one can
 extract the IR spectrum from the intensity of the molecular dipole
 fluctuation at each frequency using the following relationship:
-\[
+\begin{equation}
 \hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
 i2\pi vt} dt}.
-\]
+\end{equation}
 
 \section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
 
 Rigid bodies are frequently involved in the modeling of different
-areas, from engineering, physics, to chemistry. For example,
+areas, including engineering, physics and chemistry. For example,
 missiles and vehicles are usually modeled by rigid bodies.  The
 movement of the objects in 3D gaming engines or other physics
 simulators is governed by rigid body dynamics. In molecular
@@ -1135,8 +1134,7 @@ The motion of a rigid body is Hamiltonian with the Ham
 Dullweber and his coworkers\cite{Dullweber1997} in depth.
 
 \subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
-The motion of a rigid body is Hamiltonian with the Hamiltonian
-function
+The Hamiltonian of a rigid body is given by 
 \begin{equation}
 H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
 V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
@@ -1348,8 +1346,7 @@ The non-canonical Lie-Poisson bracket ${F, G}$ of two 
 \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
+The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by
 \[
 \{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
 ).
@@ -1358,7 +1355,7 @@ norm of the angular momentum, $\parallel \pi
 function $G$ is zero, $F$ is a \emph{Casimir}, which is the
 conserved quantity in Poisson system. We can easily verify that the
 norm of the angular momentum, $\parallel \pi
-\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let$ F(\pi ) = S(\frac{{\parallel
+\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let $F(\pi ) = S(\frac{{\parallel
 \pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
 then by the chain rule
 \[
@@ -1379,7 +1376,7 @@ listed in Table~\ref{introTable:rbEquations} 
 The Hamiltonian of rigid body can be separated in terms of kinetic
 energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations
 of motion corresponding to potential energy and kinetic energy are
-listed in Table~\ref{introTable:rbEquations} 
+listed in Table~\ref{introTable:rbEquations}. 
 \begin{table}
 \caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
 \label{introTable:rbEquations}
@@ -1437,11 +1434,9 @@ the theory of Langevin dynamics. A brief derivation of
 As an alternative to newtonian dynamics, Langevin dynamics, which
 mimics a simple heat bath with stochastic and dissipative forces,
 has been applied in a variety of studies. This section will review
-the theory of Langevin dynamics. A brief derivation of generalized
+the theory of Langevin dynamics. A brief derivation of the generalized
 Langevin equation will be given first. Following that, we will
-discuss the physical meaning of the terms appearing in the equation
-as well as the calculation of friction tensor from hydrodynamics
-theory.
+discuss the physical meaning of the terms appearing in the equation.
 
 \subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
 
@@ -1450,7 +1445,7 @@ Dynamics (GLE). Lets consider a system, in which the d
 environment, has been widely used in quantum chemistry and
 statistical mechanics. One of the successful applications of
 Harmonic bath model is the derivation of the Generalized Langevin
-Dynamics (GLE). Lets consider a system, in which the degree of
+Dynamics (GLE). Consider a system, in which the degree of
 freedom $x$ is assumed to couple to the bath linearly, giving a
 Hamiltonian of the form
 \begin{equation}
@@ -1461,7 +1456,7 @@ H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_
 with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian,
 \[
 H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  \omega _\alpha ^2 }
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  x_\alpha ^2 }
 \right\}}
 \]
 where the index $\alpha$ runs over all the bath degrees of freedom,
@@ -1514,7 +1509,7 @@ Operator. Below are some important properties of Lapla
 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
+Operator. Below are some important properties of the Laplace transform
 \begin{eqnarray*}
  L(x + y)  & = & L(x) + L(y) \\
  L(ax)     & = & aL(x) \\
@@ -1583,14 +1578,14 @@ which is known as the \emph{generalized Langevin equat
 (t)\dot x(t - \tau )d\tau }  + R(t)
 \label{introEuqation:GeneralizedLangevinDynamics}
 \end{equation}
-which is known as the \emph{generalized Langevin equation}.
+which is known as the \emph{generalized Langevin equation} (GLE).
 
 \subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
 
 One may notice that $R(t)$ depends only on initial conditions, which
 implies it is completely deterministic within the context of a
 harmonic bath. However, it is easy to verify that $R(t)$ is totally
-uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)}
+uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)}
 \right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
 0.$ This property is what we expect from a truly random process. As
 long as the model chosen for $R(t)$ was a gaussian distribution in
@@ -1619,7 +1614,7 @@ taken as a $delta$ function in time:
 infinitely quickly to motions in the system. Thus, $\xi (t)$ can be
 taken as a $delta$ function in time:
 \[
-\xi (t) = 2\xi _0 \delta (t)
+\xi (t) = 2\xi _0 \delta (t).
 \]
 Hence, the convolution integral becomes
 \[
@@ -1644,7 +1639,7 @@ we can rewrite $R(T)$ as
 q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
 ^2 }}x(0),
 \]
-we can rewrite $R(T)$ as
+we can rewrite $R(t)$ as
 \[
 R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
 \]
