trunk/tengDissertation/Introduction.tex

\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}

\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{introSection:classicalMechanics}Classical Mechanics}

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.

\subsubsection{\label{introSection:newtonian}Newtonian Mechanics}

\subsubsection{\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.

\subsubsubsection{\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, $U$.
\begin{equation}
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
\lable{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 } {K dt = 0} ,
\lable{introEquation:halmitonianPrinciple2}
\end{equation}

\subsubsubsection{\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
\lable{introEquation:eqMotionLagrangian}
\end{equation}
where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
generalized velocity.

\subsubsection{\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}
With the help of these momenta, we may now define a new quantity $H$
by the equation
\begin{equation}
H = p_1 \dot q_1  +  \ldots  + p_f \dot q_f  - 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.

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.


\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}

\subsection{\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.

\subsubsection{\label{introSection::ensemble}Ensemble}

\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}

\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}

\subsection{\label{introSection:correlationFunctions}Correlation Functions}

\section{\label{introSection:langevinDynamics}Langevin Dynamics}

\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}

\subsection{\label{introSection:hydroynamics}Hydrodynamics}
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#	User	Rev	Content
1	tim	2685	\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
2
3	tim	2692	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
4	tim	2685
5	tim	2692	As a special discipline of molecular modeling, Molecular dynamics
6			has proven to be a powerful tool for studying the functions of
7			biological systems, providing structural, thermodynamic and
8			dynamical information.
9	tim	2685
10	tim	2692	\subsection{\label{introSection:classicalMechanics}Classical Mechanics}
11	tim	2685
12	tim	2692	Closely related to Classical Mechanics, Molecular Dynamics
13			simulations are carried out by integrating the equations of motion
14			for a given system of particles. There are three fundamental ideas
15			behind classical mechanics. Firstly, One can determine the state of
16			a mechanical system at any time of interest; Secondly, all the
17			mechanical properties of the system at that time can be determined
18			by combining the knowledge of the properties of the system with the
19			specification of this state; Finally, the specification of the state
20			when further combine with the laws of mechanics will also be
21			sufficient to predict the future behavior of the system.
22	tim	2685
23	tim	2692	\subsubsection{\label{introSection:newtonian}Newtonian Mechanics}
24
25			\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics}
26
27			Newtonian Mechanics suffers from two important limitations: it
28			describes their motion in special cartesian coordinate systems.
29			Another limitation of Newtonian mechanics becomes obvious when we
30			try to describe systems with large numbers of particles. It becomes
31			very difficult to predict the properties of the system by carrying
32			out calculations involving the each individual interaction between
33			all the particles, even if we know all of the details of the
34			interaction. In order to overcome some of the practical difficulties
35			which arise in attempts to apply Newton's equation to complex
36			system, alternative procedures may be developed.
37
38			\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
39			Principle}
40
41			Hamilton introduced the dynamical principle upon which it is
42			possible to base all of mechanics and, indeed, most of classical
43			physics. Hamilton's Principle may be stated as follow,
44
45			The actual trajectory, along which a dynamical system may move from
46			one point to another within a specified time, is derived by finding
47			the path which minimizes the time integral of the difference between
48			the kinetic, $K$, and potential energies, $U$.
49			\begin{equation}
50			\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
51			\lable{introEquation:halmitonianPrinciple1}
52			\end{equation}
53
54			For simple mechanical systems, where the forces acting on the
55			different part are derivable from a potential and the velocities are
56			small compared with that of light, the Lagrangian function $L$ can
57			be define as the difference between the kinetic energy of the system
58			and its potential energy,
59			\begin{equation}
60			L \equiv K - U = L(q_i ,\dot q_i ) ,
61			\label{introEquation:lagrangianDef}
62			\end{equation}
63			then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
64			\begin{equation}
65			\delta \int_{t_1 }^{t_2 } {K dt = 0} ,
66			\lable{introEquation:halmitonianPrinciple2}
67			\end{equation}
68
69			\subsubsubsection{\label{introSection:equationOfMotionLagrangian}The
70			Equations of Motion in Lagrangian Mechanics}
71
72			for a holonomic system of $f$ degrees of freedom, the equations of
73			motion in the Lagrangian form is
74			\begin{equation}
75			\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
76			\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
77			\lable{introEquation:eqMotionLagrangian}
78			\end{equation}
79			where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
80			generalized velocity.
81
82			\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
83
84			Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
85			introduced by William Rowan Hamilton in 1833 as a re-formulation of
86			classical mechanics. If the potential energy of a system is
87			independent of generalized velocities, the generalized momenta can
88			be defined as
89			\begin{equation}
90			p_i = \frac{\partial L}{\partial \dot q_i}
91			\label{introEquation:generalizedMomenta}
92			\end{equation}
93			With the help of these momenta, we may now define a new quantity $H$
94			by the equation
95			\begin{equation}
96			H = p_1 \dot q_1 + \ldots + p_f \dot q_f - L,
97			\label{introEquation:hamiltonianDefByLagrangian}
98			\end{equation}
99			where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and
100			$L$ is the Lagrangian function for the system.
101
102			An important difference between Lagrangian approach and the
103			Hamiltonian approach is that the Lagrangian is considered to be a
104			function of the generalized velocities $\dot q_i$ and the
105			generalized coordinates $q_i$, while the Hamiltonian is considered
106			to be a function of the generalized momenta $p_i$ and the conjugate
107			generalized coordinate $q_i$. Hamiltonian Mechanics is more
108			appropriate for application to statistical mechanics and quantum
109			mechanics, since it treats the coordinate and its time derivative as
110			independent variables and it only works with 1st-order differential
111			equations.
112
113
114			\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation}
115
116			\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics}
117
118			The thermodynamic behaviors and properties of Molecular Dynamics
119			simulation are governed by the principle of Statistical Mechanics.
120			The following section will give a brief introduction to some of the
121			Statistical Mechanics concepts presented in this dissertation.
122
123			\subsubsection{\label{introSection::ensemble}Ensemble}
124
125			\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis}
126
127			\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
128
129			\subsection{\label{introSection:correlationFunctions}Correlation Functions}
130
131	tim	2685	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
132
133	tim	2692	\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
134	tim	2685
135	tim	2692	\subsection{\label{introSection:hydroynamics}Hydrodynamics}