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+\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
-<
+\section{\label{introSection:classicalMechanics}Classical Mechanics}
->
+\section{\label{introSection:classicalMechanics}Classical
->
+Mechanics}
-<
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
->
+Closely related to Classical Mechanics, Molecular Dynamics
->
+simulations are carried out by integrating the equations of motion
->
+for a given system of particles. There are three fundamental ideas
->
+behind classical mechanics. Firstly, One can determine the state of
->
+a mechanical system at any time of interest; Secondly, all the
->
+mechanical properties of the system at that time can be determined
->
+by combining the knowledge of the properties of the system with the
->
+specification of this state; Finally, the specification of the state
->
+when further combine with the laws of mechanics will also be
->
+sufficient to predict the future behavior of the system.
-<
+\section{\label{introSection:statisticalMechanics}Statistical Mechanics}
->
+\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
-+
+describes their motion in special cartesian coordinate systems.
-+
+Another limitation of Newtonian mechanics becomes obvious when we
-+
+try to describe systems with large numbers of particles. It becomes
-+
+very difficult to predict the properties of the system by carrying
-+
+out calculations involving the each individual interaction between
-+
+all the particles, even if we know all of the details of the
-+
+interaction. In order to overcome some of the practical difficulties
-+
+which arise in attempts to apply Newton's equation to complex
-+
+system, alternative procedures may be developed.
-+
-+
+\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
-+
+Principle}
-+
-+
+Hamilton introduced the dynamical principle upon which it is
-+
+possible to base all of mechanics and, indeed, most of classical
-+
+physics. Hamilton's Principle may be stated as follow,
-+
-+
+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, $U$ \cite{tolman79}.
-+
+\begin{equation}
-+
+\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
-+
+\label{introEquation:halmitonianPrinciple1}
-+
+\end{equation}
-+
-+
+For simple mechanical systems, where the forces acting on the
-+
+different part are derivable from a potential and the velocities are
-+
+small compared with that of light, the Lagrangian function $L$ can
-+
+be define as the difference between the kinetic energy of the system
-+
+and its potential energy,
-+
+\begin{equation}
-+
+L \equiv K - U = L(q_i ,\dot q_i ) ,
-+
+\label{introEquation:lagrangianDef}
-+
+\end{equation}
-+
+then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
-+
+\begin{equation}
-+
+\delta \int_{t_1 }^{t_2 } {L dt = 0} ,
-+
+\label{introEquation:halmitonianPrinciple2}
-+
+\end{equation}
-+
-+
+\subsubsection{\label{introSection:equationOfMotionLagrangian}The
-+
+Equations of Motion in Lagrangian Mechanics}
-+
-+
+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 }} -
-+
+\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
-+
+\label{introEquation:eqMotionLagrangian}
-+
+\end{equation}
-+
+where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
-+
+generalized velocity.
-+
-+
+\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
-+
-+
+Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
-+
+introduced by William Rowan Hamilton in 1833 as a re-formulation of
-+
+classical mechanics. If the potential energy of a system is
-+
+independent of generalized velocities, the generalized momenta can
-+
+be defined as
-+
+\begin{equation}
-+
+p_i = \frac{\partial L}{\partial \dot q_i}
-+
+\label{introEquation:generalizedMomenta}
-+
+\end{equation}
-+
+The Lagrange equations of motion are then expressed by
-+
+\begin{equation}
-+
+p_i  = \frac{{\partial L}}{{\partial q_i }}
-+
+\label{introEquation:generalizedMomentaDot}
-+
+\end{equation}
-+
-+
+With the help of the generalized momenta, we may now define a new
-+
+quantity $H$ by the equation
-+
+\begin{equation}
-+
+H = \sum\limits_k {p_k \dot q_k }  - L ,
-+
+\label{introEquation:hamiltonianDefByLagrangian}
-+
+\end{equation}
-+
+where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
-+
+$L$ is the Lagrangian function for the system.
-+
-+
+Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
-+
+one can obtain
-+
+\begin{equation}
-+
+dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
-+
+\frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
-+
+L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
-+
+L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
-+
+\end{equation}
-+
+Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
-+
+second and fourth terms in the parentheses cancel. Therefore,
-+
+Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
-+
+\begin{equation}
-+
+dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
-+
+\right)}  - \frac{{\partial L}}{{\partial t}}dt
-+
+\label{introEquation:diffHamiltonian2}
-+
+\end{equation}
-+
+By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
-+
+find
-+
+\begin{equation}
-+
+\frac{{\partial H}}{{\partial p_k }} = q_k
-+
+\label{introEquation:motionHamiltonianCoordinate}
-+
+\end{equation}
-+
+\begin{equation}
-+
+\frac{{\partial H}}{{\partial q_k }} =  - p_k
-+
+\label{introEquation:motionHamiltonianMomentum}
-+
+\end{equation}
-+
+and
-+
+\begin{equation}
-+
+\frac{{\partial H}}{{\partial t}} =  - \frac{{\partial L}}{{\partial
-+
+t}}
-+
+\label{introEquation:motionHamiltonianTime}
-+
+\end{equation}
-+
-+
+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 \cite{Goldstein01}.
-+
-+
+An important difference between Lagrangian approach and the
-+
+Hamiltonian approach is that the Lagrangian is considered to be a
-+
+function of the generalized velocities $\dot q_i$ and the
-+
+generalized coordinates $q_i$, while the Hamiltonian is considered
-+
+to be a function of the generalized momenta $p_i$ and the conjugate
-+
+generalized coordinate $q_i$. Hamiltonian Mechanics is more
-+
+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\cite{Marion90}.
-+
-+
+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}
-+
-+
+\section{\label{introSection:statisticalMechanics}Statistical
-+
+Mechanics}
-+
-+
+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
-+
+dissertation.
-+
-+
+\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
-+
+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(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 $\langle  A(q , p) \rangle_t$ 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
-+
+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}
-+
-+
+\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}
-<
+\section{\label{introSection:correlationFunctions}Correlation Functions}
->
+\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 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs. Revision 2700 by tim, Tue Apr 11 03:38:09 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs.
Revision 2700 by tim, Tue Apr 11 03:38:09 2006 UTC