[ViewVC] Diff of: group/trunk/tengDissertation/Introduction.tex

Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs.
Revision 2695 by tim, Thu Apr 6 04:07:57 2006 UTC

+\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 $E = T + V$ is
-+
+conserved. All of these conserved quantities are important factors
-+
+to determine the quality of numerical integration scheme for rigid
-+
+body \cite{Dullweber1997}.
-+
-+
+\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}.
-+
-+
+When studying Hamiltonian system, it is more convenient to use
-+
+notation
-+
+\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}
-+
+& I  \\
-+
+   { - I} & 0  \\
-+
+\end{array}} \right)
-+
+\label{introEquation:canonicalMatrix}
-+
+\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}
-+
-+
+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.
-+
-+
+\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}
-<
+\section{\label{introSection:langevinDynamics}Langevin 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.
-<
+\section{\label{introSection:hydroynamics}Hydrodynamics}
->
+\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:generalizedLangevinDynamics}Generalized Langevin Dynamics}
-+
-+
+\subsection{\label{introSection:hydroynamics}Hydrodynamics}

Diff Legend

-–
+Removed lines
-+
+Added lines
-<
+Changed lines
->
+Changed lines

Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs. Revision 2695 by tim, Thu Apr 6 04:07:57 2006 UTC

Diff Legend

Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs.
Revision 2695 by tim, Thu Apr 6 04:07:57 2006 UTC