--- trunk/tengDissertation/Introduction.tex	2006/04/05 21:00:19	2694
+++ trunk/tengDissertation/Introduction.tex	2006/04/06 04:07:57	2695
@@ -213,16 +213,16 @@ Thus, Hamiltonian system can be rewritten as,
 \end{array}} \right)
 \label{introEquation:canonicalMatrix}
 \end{equation}
-Thus, Hamiltonian system can be rewritten as,
+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}
-where $I$ is an identity matrix and $J$ is a skew-symmetrix matrix
-($ J^T  =  - J $).
 
 %\subsection{\label{introSection:canonicalTransformation}Canonical
-Transformation}
+%Transformation}
 
 \section{\label{introSection:geometricIntegratos}Geometric Integrators}
 
@@ -238,10 +238,38 @@ 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}
+\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.
 
 \section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
@@ -263,7 +291,12 @@ Applications of dynamics of rigid bodies.
 
 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}