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}
Revision:	2693
Committed:	Wed Apr 5 03:44:32 2006 UTC (19 years, 7 months ago) by tim
Content type:	application/x-tex
File size:	7490 byte(s)
Log Message:	minor fix
#	Content
1	\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
2
3	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
4
5	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
10	\section{\label{introSection:classicalMechanics}Classical
11	Mechanics}
12
13	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
24	\subsection{\label{introSection:newtonian}Newtonian Mechanics}
25
26	\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
27
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	\subsection{\label{introSection:halmiltonPrinciple}Hamilton's
40	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	\label{introEquation:halmitonianPrinciple1}
53	\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	\delta \int_{t_1 }^{t_2 } {L dt = 0} ,
67	\label{introEquation:halmitonianPrinciple2}
68	\end{equation}
69
70	\subsection{\label{introSection:equationOfMotionLagrangian}The
71	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	\label{introEquation:eqMotionLagrangian}
79	\end{equation}
80	where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
81	generalized velocity.
82
83	\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
84
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	The Lagrange equations of motion are then expressed by
95	\begin{equation}
96	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	\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	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	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	\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
159
160	\subsection{\label{introSection:canonicalTransformation}Canonical
161	Transformation}
162
163	\section{\label{introSection:statisticalMechanics}Statistical
164	Mechanics}
165
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	\subsection{\label{introSection::ensemble}Ensemble}
172
173	\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
174
175	\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
176
177	\section{\label{introSection:correlationFunctions}Correlation Functions}
178
179	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
180
181	\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
182
183	\subsection{\label{introSection:hydroynamics}Hydrodynamics}