--- trunk/tengDissertation/Introduction.tex	2006/04/10 05:35:55	2699
+++ trunk/tengDissertation/Introduction.tex	2006/04/11 03:38:09	2700
@@ -117,7 +117,7 @@ for a holonomic system of $f$ degrees of freedom, the 
 \subsubsection{\label{introSection:equationOfMotionLagrangian}The
 Equations of Motion in Lagrangian Mechanics}
 
-for a holonomic system of $f$ degrees of freedom, the equations of
+For a holonomic system of $f$ degrees of freedom, the equations of
 motion in the Lagrangian form is
 \begin{equation}
 \frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
@@ -221,10 +221,217 @@ Statistical Mechanics concepts presented in this disse
 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.
+Statistical Mechanics concepts and theorem presented in this
+dissertation.
 
-\subsection{\label{introSection:ensemble}Ensemble and Phase Space}
+\subsection{\label{introSection:ensemble}Phase Space and Ensemble}
+
+Mathematically, phase space is the space which represents all
+possible states. Each possible state of the system corresponds to
+one unique point in the phase space. For mechanical systems, the
+phase space usually consists of all possible values of position and
+momentum variables. Consider a dynamic system in a cartesian space,
+where each of the $6f$ coordinates and momenta is assigned to one of
+$6f$ mutually orthogonal axes, the phase space of this system is a
+$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 ,
+\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and
+momenta is a phase space vector.
+
+A microscopic state or microstate of a classical system is
+specification of the complete phase space vector of a system at any
+instant in time. An ensemble is defined as a collection of systems
+sharing one or more macroscopic characteristics but each being in a
+unique microstate. The complete ensemble is specified by giving all
+systems or microstates consistent with the common macroscopic
+characteristics of the ensemble. Although the state of each
+individual system in the ensemble could be precisely described at
+any instance in time by a suitable phase space vector, when using
+ensembles for statistical purposes, there is no need to maintain
+distinctions between individual systems, since the numbers of
+systems at any time in the different states which correspond to
+different regions of the phase space are more interesting. Moreover,
+in the point of view of statistical mechanics, one would prefer to
+use ensembles containing a large enough population of separate
+members so that the numbers of systems in such different states can
+be regarded as changing continuously as we traverse different
+regions of the phase space. The condition of an ensemble at any time
+can be regarded as appropriately specified by the density $\rho$
+with which representative points are distributed over the phase
+space. The density of distribution for an ensemble with $f$ degrees
+of freedom is defined as,
+\begin{equation}
+\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
+\label{introEquation:densityDistribution}
+\end{equation}
+Governed by the principles of mechanics, the phase points change
+their value which would change the density at any time at phase
+space. Hence, the density of distribution is also to be taken as a
+function of the time.
+
+The number of systems $\delta N$ at time $t$ can be determined by,
+\begin{equation}
+\delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
+\label{introEquation:deltaN}
+\end{equation}
+Assuming a large enough population of systems are exploited, we can
+sufficiently approximate $\delta N$ without introducing
+discontinuity when we go from one region in the phase space to
+another. By integrating over the whole phase space,
+\begin{equation}
+N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
+\label{introEquation:totalNumberSystem}
+\end{equation}
+gives us an expression for the total number of the systems. Hence,
+the probability per unit in the phase space can be obtained by,
+\begin{equation}
+\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int
+{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
+\label{introEquation:unitProbability}
+\end{equation}
+With the help of Equation(\ref{introEquation:unitProbability}) and
+the knowledge of the system, it is possible to calculate the average
+value of any desired quantity which depends on the coordinates and
+momenta of the system. Even when the dynamics of the real system is
+complex, or stochastic, or even discontinuous, the average
+properties of the ensemble of possibilities as a whole may still
+remain well defined. For a classical system in thermal equilibrium
+with its environment, the ensemble average of a mechanical quantity,
+$\langle A(q , p) \rangle_t$, takes the form of an integral over the
+phase space of the system,
+\begin{equation}
+\langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
+(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}
+\label{introEquation:ensembelAverage}
+\end{equation}
+
+There are several different types of ensembles with different
+statistical characteristics. As a function of macroscopic
+parameters, such as temperature \textit{etc}, partition function can
+be used to describe the statistical properties of a system in
+thermodynamic equilibrium.
+
+As an ensemble of systems, each of which is known to be thermally
+isolated and conserve energy, Microcanonical ensemble(NVE) has a
+partition function like,
+\begin{equation}
+\Omega (N,V,E) = e^{\beta TS}
+\label{introEqaution: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
+of the total energy amongst the possible dynamical states is given
+by the partition function,
+\begin{equation}
+\Omega (N,V,T) = e^{ - \beta A}
+\label{introEquation:NVTPartition}
+\end{equation}
+Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
+TS$. Since most experiment are carried out under constant pressure
+condition, isothermal-isobaric ensemble(NPT) play a very important
+role in molecular simulation. The isothermal-isobaric ensemble allow
+the system to exchange energy with a heat bath of temperature $T$
+and to change the volume as well. Its partition function is given as
+\begin{equation}
+\Delta (N,P,T) =  - e^{\beta G}.
+ \label{introEquation:NPTPartition}
+\end{equation}
+Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy.
+
+\subsection{\label{introSection:liouville}Liouville's theorem}
+
+The Liouville's theorem is the foundation on which statistical
+mechanics rests. It describes the time evolution of phase space
+distribution function. In order to calculate the rate of change of
+$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we
+consider the two faces perpendicular to the $q_1$ axis, which are
+located at $q_1$ and $q_1 + \delta q_1$, the number of phase points
+leaving the opposite face is given by the expression,
+\begin{equation}
+\left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
+\right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
+}}\delta q_1 } \right)\delta q_2  \ldots \delta q_f \delta p_1
+\ldots \delta p_f .
+\end{equation}
+Summing all over the phase space, we obtain
+\begin{equation}
+\frac{{d(\delta N)}}{{dt}} =  - \sum\limits_{i = 1}^f {\left[ {\rho
+\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} +
+\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left(
+{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  + \frac{{\partial
+\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1
+\ldots \delta q_f \delta p_1  \ldots \delta p_f .
+\end{equation}
+Differentiating the equations of motion in Hamiltonian formalism
+(\ref{introEquation:motionHamiltonianCoordinate},
+\ref{introEquation:motionHamiltonianMomentum}), we can show,
+\begin{equation}
+\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }}
++ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
+\end{equation}
+which cancels the first terms of the right hand side. Furthermore,
+divining $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
+p_f $ in both sides, we can write out Liouville's theorem in a
+simple form,
+\begin{equation}
+\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f
+{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  +
+\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
+\label{introEquation:liouvilleTheorem}
+\end{equation}
 
+Liouville's theorem states that the distribution function is
+constant along any trajectory in phase space. In classical
+statistical mechanics, since the number of particles in the system
+is huge, we may be able to believe the system is stationary,
+\begin{equation}
+\frac{{\partial \rho }}{{\partial t}} = 0.
+\label{introEquation:stationary}
+\end{equation}
+In such stationary system, the density of distribution $\rho$ can be
+connected to the Hamiltonian $H$ through Maxwell-Boltzmann
+distribution,
+\begin{equation}
+\rho  \propto e^{ - \beta H}
+\label{introEquation:densityAndHamiltonian}
+\end{equation}
+
+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
+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 }} -
+\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial
+q_i }}} \right)}.
+\label{introEquation:poissonBracket}
+\end{equation}
+Substituting equations of motion in Hamiltonian formalism(
+\ref{introEquation:motionHamiltonianCoordinate} ,
+\ref{introEquation:motionHamiltonianMomentum} ) into
+(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's
+theorem using Poisson bracket notion,
+\begin{equation}
+\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
+{\rho ,H} \right\}.
+\label{introEquation:liouvilleTheromInPoissin}
+\end{equation}
+Moreover, the Liouville operator is defined as
+\begin{equation}
+iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial
+p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial
+H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)}
+\label{introEquation:liouvilleOperator}
+\end{equation}
+In terms of Liouville operator, Liouville's equation can also be
+expressed as
+\begin{equation}
+\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - iL\rho
+\label{introEquation:liouvilleTheoremInOperator}
+\end{equation}
+
+
 \subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
 
 Various thermodynamic properties can be calculated from Molecular
@@ -239,22 +446,23 @@ statistical ensemble are identical \cite{Frenkel1996, 
 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
+\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
+\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
+{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp
 \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}.
+where $\langle  A(q , p) \rangle_t$ 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
@@ -442,8 +650,6 @@ Suppose that a Hamiltonian system has a form with $H =
  \label{introEquation:SymplecticFlowComposition}
 \end{equation}
 Suppose that a Hamiltonian system has a form with $H = T + V$
-
-
 
 \section{\label{introSection:molecularDynamics}Molecular Dynamics}
 
