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.

\section{\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.

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

\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.

\subsection{\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} ,
\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}

\subsection{\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.

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.

\subsection{\label{introSection:poissonBrackets}Poisson Brackets}

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

\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}

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

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

\section{\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	2693	\section{\label{introSection:classicalMechanics}Classical
11			Mechanics}
12	tim	2685
13	tim	2692	Closely related to Classical Mechanics, Molecular Dynamics
14			simulations are carried out by integrating the equations of motion
15			for a given system of particles. There are three fundamental ideas
16			behind classical mechanics. Firstly, One can determine the state of
17			a mechanical system at any time of interest; Secondly, all the
18			mechanical properties of the system at that time can be determined
19			by combining the knowledge of the properties of the system with the
20			specification of this state; Finally, the specification of the state
21			when further combine with the laws of mechanics will also be
22			sufficient to predict the future behavior of the system.
23	tim	2685
24	tim	2693	\subsection{\label{introSection:newtonian}Newtonian Mechanics}
25	tim	2692
26	tim	2693	\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
27	tim	2692
28			Newtonian Mechanics suffers from two important limitations: it
29			describes their motion in special cartesian coordinate systems.
30			Another limitation of Newtonian mechanics becomes obvious when we
31			try to describe systems with large numbers of particles. It becomes
32			very difficult to predict the properties of the system by carrying
33			out calculations involving the each individual interaction between
34			all the particles, even if we know all of the details of the
35			interaction. In order to overcome some of the practical difficulties
36			which arise in attempts to apply Newton's equation to complex
37			system, alternative procedures may be developed.
38
39	tim	2693	\subsection{\label{introSection:halmiltonPrinciple}Hamilton's
40	tim	2692	Principle}
41
42			Hamilton introduced the dynamical principle upon which it is
43			possible to base all of mechanics and, indeed, most of classical
44			physics. Hamilton's Principle may be stated as follow,
45
46			The actual trajectory, along which a dynamical system may move from
47			one point to another within a specified time, is derived by finding
48			the path which minimizes the time integral of the difference between
49			the kinetic, $K$, and potential energies, $U$.
50			\begin{equation}
51			\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
52	tim	2693	\label{introEquation:halmitonianPrinciple1}
53	tim	2692	\end{equation}
54
55			For simple mechanical systems, where the forces acting on the
56			different part are derivable from a potential and the velocities are
57			small compared with that of light, the Lagrangian function $L$ can
58			be define as the difference between the kinetic energy of the system
59			and its potential energy,
60			\begin{equation}
61			L \equiv K - U = L(q_i ,\dot q_i ) ,
62			\label{introEquation:lagrangianDef}
63			\end{equation}
64			then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
65			\begin{equation}
66	tim	2693	\delta \int_{t_1 }^{t_2 } {L dt = 0} ,
67			\label{introEquation:halmitonianPrinciple2}
68	tim	2692	\end{equation}
69
70	tim	2693	\subsection{\label{introSection:equationOfMotionLagrangian}The
71	tim	2692	Equations of Motion in Lagrangian Mechanics}
72
73			for a holonomic system of $f$ degrees of freedom, the equations of
74			motion in the Lagrangian form is
75			\begin{equation}
76			\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
77			\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
78	tim	2693	\label{introEquation:eqMotionLagrangian}
79	tim	2692	\end{equation}
80			where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
81			generalized velocity.
82
83	tim	2693	\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
84	tim	2692
85			Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
86			introduced by William Rowan Hamilton in 1833 as a re-formulation of
87			classical mechanics. If the potential energy of a system is
88			independent of generalized velocities, the generalized momenta can
89			be defined as
90			\begin{equation}
91			p_i = \frac{\partial L}{\partial \dot q_i}
92			\label{introEquation:generalizedMomenta}
93			\end{equation}
94	tim	2693	The Lagrange equations of motion are then expressed by
95	tim	2692	\begin{equation}
96	tim	2693	p_i = \frac{{\partial L}}{{\partial q_i }}
97			\label{introEquation:generalizedMomentaDot}
98			\end{equation}
99
100			With the help of the generalized momenta, we may now define a new
101			quantity $H$ by the equation
102			\begin{equation}
103			H = \sum\limits_k {p_k \dot q_k } - L ,
104	tim	2692	\label{introEquation:hamiltonianDefByLagrangian}
105			\end{equation}
106			where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and
107			$L$ is the Lagrangian function for the system.
108
109	tim	2693	Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
110			one can obtain
111			\begin{equation}
112			dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k -
113			\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial
114			L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial
115			L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
116			\end{equation}
117			Making use of Eq.~\ref{introEquation:generalizedMomenta}, the
118			second and fourth terms in the parentheses cancel. Therefore,
119			Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
120			\begin{equation}
121			dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k }
122			\right)} - \frac{{\partial L}}{{\partial t}}dt
123			\label{introEquation:diffHamiltonian2}
124			\end{equation}
125			By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
126			find
127			\begin{equation}
128			\frac{{\partial H}}{{\partial p_k }} = q_k
129			\label{introEquation:motionHamiltonianCoordinate}
130			\end{equation}
131			\begin{equation}
132			\frac{{\partial H}}{{\partial q_k }} = - p_k
133			\label{introEquation:motionHamiltonianMomentum}
134			\end{equation}
135			and
136			\begin{equation}
137			\frac{{\partial H}}{{\partial t}} = - \frac{{\partial L}}{{\partial
138			t}}
139			\label{introEquation:motionHamiltonianTime}
140			\end{equation}
141
142			Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
143			Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
144			equation of motion. Due to their symmetrical formula, they are also
145			known as the canonical equations of motions.
146
147	tim	2692	An important difference between Lagrangian approach and the
148			Hamiltonian approach is that the Lagrangian is considered to be a
149			function of the generalized velocities $\dot q_i$ and the
150			generalized coordinates $q_i$, while the Hamiltonian is considered
151			to be a function of the generalized momenta $p_i$ and the conjugate
152			generalized coordinate $q_i$. Hamiltonian Mechanics is more
153			appropriate for application to statistical mechanics and quantum
154			mechanics, since it treats the coordinate and its time derivative as
155			independent variables and it only works with 1st-order differential
156			equations.
157
158	tim	2693	\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
159	tim	2692
160	tim	2693	\subsection{\label{introSection:canonicalTransformation}Canonical
161			Transformation}
162	tim	2692
163	tim	2693	\section{\label{introSection:statisticalMechanics}Statistical
164			Mechanics}
165	tim	2692
166			The thermodynamic behaviors and properties of Molecular Dynamics
167			simulation are governed by the principle of Statistical Mechanics.
168			The following section will give a brief introduction to some of the
169			Statistical Mechanics concepts presented in this dissertation.
170
171	tim	2693	\subsection{\label{introSection::ensemble}Ensemble}
172	tim	2692
173	tim	2693	\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
174	tim	2692
175	tim	2693	\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
176	tim	2692
177	tim	2693	\section{\label{introSection:correlationFunctions}Correlation Functions}
178	tim	2692
179	tim	2685	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
180
181	tim	2692	\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
182	tim	2685
183	tim	2692	\subsection{\label{introSection:hydroynamics}Hydrodynamics}