trunk/tengDissertation/Introduction.tex

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

\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}
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}
   0 & 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}

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: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	2693	\section{\label{introSection:classicalMechanics}Classical
4			Mechanics}
5	tim	2685
6	tim	2692	Closely related to Classical Mechanics, Molecular Dynamics
7			simulations are carried out by integrating the equations of motion
8			for a given system of particles. There are three fundamental ideas
9			behind classical mechanics. Firstly, One can determine the state of
10			a mechanical system at any time of interest; Secondly, all the
11			mechanical properties of the system at that time can be determined
12			by combining the knowledge of the properties of the system with the
13			specification of this state; Finally, the specification of the state
14			when further combine with the laws of mechanics will also be
15			sufficient to predict the future behavior of the system.
16	tim	2685
17	tim	2693	\subsection{\label{introSection:newtonian}Newtonian Mechanics}
18	tim	2694	The discovery of Newton's three laws of mechanics which govern the
19			motion of particles is the foundation of the classical mechanics.
20			Newton’s first law defines a class of inertial frames. Inertial
21			frames are reference frames where a particle not interacting with
22			other bodies will move with constant speed in the same direction.
23			With respect to inertial frames Newton’s second law has the form
24			\begin{equation}
25			F = \frac {dp}{dt} = \frac {mv}{dt}
26			\label{introEquation:newtonSecondLaw}
27			\end{equation}
28			A point mass interacting with other bodies moves with the
29			acceleration along the direction of the force acting on it. Let
30			$F_ij$ be the force that particle $i$ exerts on particle $j$, and
31			$F_ji$ be the force that particle $j$ exerts on particle $i$.
32			Newton’s third law states that
33			\begin{equation}
34			F_ij = -F_ji
35			\label{introEquation:newtonThirdLaw}
36			\end{equation}
37	tim	2692
38	tim	2694	Conservation laws of Newtonian Mechanics play very important roles
39			in solving mechanics problems. The linear momentum of a particle is
40			conserved if it is free or it experiences no force. The second
41			conservation theorem concerns the angular momentum of a particle.
42			The angular momentum $L$ of a particle with respect to an origin
43			from which $r$ is measured is defined to be
44			\begin{equation}
45			L \equiv r \times p \label{introEquation:angularMomentumDefinition}
46			\end{equation}
47			The torque $\tau$ with respect to the same origin is defined to be
48			\begin{equation}
49			N \equiv r \times F \label{introEquation:torqueDefinition}
50			\end{equation}
51			Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
52			\[
53			\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times
54			\dot p)
55			\]
56			since
57			\[
58			\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0
59			\]
60			thus,
61			\begin{equation}
62			\dot L = r \times \dot p = N
63			\end{equation}
64			If there are no external torques acting on a body, the angular
65			momentum of it is conserved. The last conservation theorem state
66			that if all forces are conservative, Energy $E = T + V$ is
67			conserved. All of these conserved quantities are important factors
68			to determine the quality of numerical integration scheme for rigid
69			body \cite{Dullweber1997}.
70
71	tim	2693	\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
72	tim	2692
73			Newtonian Mechanics suffers from two important limitations: it
74			describes their motion in special cartesian coordinate systems.
75			Another limitation of Newtonian mechanics becomes obvious when we
76			try to describe systems with large numbers of particles. It becomes
77			very difficult to predict the properties of the system by carrying
78			out calculations involving the each individual interaction between
79			all the particles, even if we know all of the details of the
80			interaction. In order to overcome some of the practical difficulties
81			which arise in attempts to apply Newton's equation to complex
82			system, alternative procedures may be developed.
83
84	tim	2694	\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
85	tim	2692	Principle}
86
87			Hamilton introduced the dynamical principle upon which it is
88			possible to base all of mechanics and, indeed, most of classical
89			physics. Hamilton's Principle may be stated as follow,
90
91			The actual trajectory, along which a dynamical system may move from
92			one point to another within a specified time, is derived by finding
93			the path which minimizes the time integral of the difference between
94	tim	2694	the kinetic, $K$, and potential energies, $U$ \cite{tolman79}.
95	tim	2692	\begin{equation}
96			\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
97	tim	2693	\label{introEquation:halmitonianPrinciple1}
98	tim	2692	\end{equation}
99
100			For simple mechanical systems, where the forces acting on the
101			different part are derivable from a potential and the velocities are
102			small compared with that of light, the Lagrangian function $L$ can
103			be define as the difference between the kinetic energy of the system
104			and its potential energy,
105			\begin{equation}
106			L \equiv K - U = L(q_i ,\dot q_i ) ,
107			\label{introEquation:lagrangianDef}
108			\end{equation}
109			then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
110			\begin{equation}
111	tim	2693	\delta \int_{t_1 }^{t_2 } {L dt = 0} ,
112			\label{introEquation:halmitonianPrinciple2}
113	tim	2692	\end{equation}
114
115	tim	2694	\subsubsection{\label{introSection:equationOfMotionLagrangian}The
116	tim	2692	Equations of Motion in Lagrangian Mechanics}
117
118			for a holonomic system of $f$ degrees of freedom, the equations of
119			motion in the Lagrangian form is
120			\begin{equation}
121			\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
122			\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
123	tim	2693	\label{introEquation:eqMotionLagrangian}
124	tim	2692	\end{equation}
125			where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
126			generalized velocity.
127
128	tim	2693	\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}
129	tim	2692
130			Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
131			introduced by William Rowan Hamilton in 1833 as a re-formulation of
132			classical mechanics. If the potential energy of a system is
133			independent of generalized velocities, the generalized momenta can
134			be defined as
135			\begin{equation}
136			p_i = \frac{\partial L}{\partial \dot q_i}
137			\label{introEquation:generalizedMomenta}
138			\end{equation}
139	tim	2693	The Lagrange equations of motion are then expressed by
140	tim	2692	\begin{equation}
141	tim	2693	p_i = \frac{{\partial L}}{{\partial q_i }}
142			\label{introEquation:generalizedMomentaDot}
143			\end{equation}
144
145			With the help of the generalized momenta, we may now define a new
146			quantity $H$ by the equation
147			\begin{equation}
148			H = \sum\limits_k {p_k \dot q_k } - L ,
149	tim	2692	\label{introEquation:hamiltonianDefByLagrangian}
150			\end{equation}
151			where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and
152			$L$ is the Lagrangian function for the system.
153
154	tim	2693	Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
155			one can obtain
156			\begin{equation}
157			dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k -
158			\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial
159			L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial
160			L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
161			\end{equation}
162			Making use of Eq.~\ref{introEquation:generalizedMomenta}, the
163			second and fourth terms in the parentheses cancel. Therefore,
164			Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
165			\begin{equation}
166			dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k }
167			\right)} - \frac{{\partial L}}{{\partial t}}dt
168			\label{introEquation:diffHamiltonian2}
169			\end{equation}
170			By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
171			find
172			\begin{equation}
173			\frac{{\partial H}}{{\partial p_k }} = q_k
174			\label{introEquation:motionHamiltonianCoordinate}
175			\end{equation}
176			\begin{equation}
177			\frac{{\partial H}}{{\partial q_k }} = - p_k
178			\label{introEquation:motionHamiltonianMomentum}
179			\end{equation}
180			and
181			\begin{equation}
182			\frac{{\partial H}}{{\partial t}} = - \frac{{\partial L}}{{\partial
183			t}}
184			\label{introEquation:motionHamiltonianTime}
185			\end{equation}
186
187			Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
188			Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
189			equation of motion. Due to their symmetrical formula, they are also
190	tim	2694	known as the canonical equations of motions \cite{Goldstein01}.
191	tim	2693
192	tim	2692	An important difference between Lagrangian approach and the
193			Hamiltonian approach is that the Lagrangian is considered to be a
194			function of the generalized velocities $\dot q_i$ and the
195			generalized coordinates $q_i$, while the Hamiltonian is considered
196			to be a function of the generalized momenta $p_i$ and the conjugate
197			generalized coordinate $q_i$. Hamiltonian Mechanics is more
198			appropriate for application to statistical mechanics and quantum
199			mechanics, since it treats the coordinate and its time derivative as
200			independent variables and it only works with 1st-order differential
201	tim	2694	equations\cite{Marion90}.
202	tim	2692
203	tim	2694	When studying Hamiltonian system, it is more convenient to use
204			notation
205			\begin{equation}
206			r = r(q,p)^T
207			\end{equation}
208			and to introduce a $2n \times 2n$ canonical structure matrix $J$,
209			\begin{equation}
210			J = \left( {\begin{array}{*{20}c}
211			0 & I \\
212			{ - I} & 0 \\
213			\end{array}} \right)
214			\label{introEquation:canonicalMatrix}
215			\end{equation}
216	tim	2695	where $I$ is a $n \times n$ identity matrix and $J$ is a
217			skew-symmetric matrix ($ J^T = - J $). Thus, Hamiltonian system
218			can be rewritten as,
219	tim	2694	\begin{equation}
220			\frac{d}{{dt}}r = J\nabla _r H(r)
221			\label{introEquation:compactHamiltonian}
222			\end{equation}
223	tim	2692
224	tim	2694	%\subsection{\label{introSection:canonicalTransformation}Canonical
225	tim	2695	%Transformation}
226	tim	2692
227	tim	2694	\section{\label{introSection:geometricIntegratos}Geometric Integrators}
228
229			\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods}
230
231			\subsection{\label{Construction of Symplectic Methods}}
232
233	tim	2693	\section{\label{introSection:statisticalMechanics}Statistical
234			Mechanics}
235	tim	2692
236	tim	2694	The thermodynamic behaviors and properties of Molecular Dynamics
237	tim	2692	simulation are governed by the principle of Statistical Mechanics.
238			The following section will give a brief introduction to some of the
239			Statistical Mechanics concepts presented in this dissertation.
240
241	tim	2695	\subsection{\label{introSection::ensemble}Ensemble and Phase Space}
242	tim	2692
243	tim	2693	\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
244	tim	2692
245	tim	2695	Various thermodynamic properties can be calculated from Molecular
246			Dynamics simulation. By comparing experimental values with the
247			calculated properties, one can determine the accuracy of the
248			simulation and the quality of the underlying model. However, both of
249			experiment and computer simulation are usually performed during a
250			certain time interval and the measurements are averaged over a
251			period of them which is different from the average behavior of
252			many-body system in Statistical Mechanics. Fortunately, Ergodic
253			Hypothesis is proposed to make a connection between time average and
254			ensemble average. It states that time average and average over the
255			statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
256			\begin{equation}
257			\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty }
258			\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma
259			{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq
260			\end{equation}
261			where $\langle A \rangle_t$ is an equilibrium value of a physical
262			quantity and $\rho (p(t), q(t))$ is the equilibrium distribution
263			function. If an observation is averaged over a sufficiently long
264			time (longer than relaxation time), all accessible microstates in
265			phase space are assumed to be equally probed, giving a properly
266			weighted statistical average. This allows the researcher freedom of
267			choice when deciding how best to measure a given observable. In case
268			an ensemble averaged approach sounds most reasonable, the Monte
269			Carlo techniques\cite{metropolis:1949} can be utilized. Or if the
270			system lends itself to a time averaging approach, the Molecular
271			Dynamics techniques in Sec.~\ref{introSection:molecularDynamics}
272			will be the best choice.
273	tim	2694
274			\section{\label{introSection:molecularDynamics}Molecular Dynamics}
275
276			As a special discipline of molecular modeling, Molecular dynamics
277			has proven to be a powerful tool for studying the functions of
278			biological systems, providing structural, thermodynamic and
279			dynamical information.
280
281			\subsection{\label{introSec:mdInit}Initialization}
282
283			\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
284
285	tim	2693	\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
286	tim	2692
287	tim	2694	A rigid body is a body in which the distance between any two given
288			points of a rigid body remains constant regardless of external
289			forces exerted on it. A rigid body therefore conserves its shape
290			during its motion.
291
292			Applications of dynamics of rigid bodies.
293
294	tim	2695	\subsection{\label{introSection:lieAlgebra}Lie Algebra}
295	tim	2694
296	tim	2695	\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
297
298			\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
299
300	tim	2694	%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
301
302	tim	2693	\section{\label{introSection:correlationFunctions}Correlation Functions}
303	tim	2692
304	tim	2685	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
305
306	tim	2692	\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
307	tim	2685
308	tim	2692	\subsection{\label{introSection:hydroynamics}Hydrodynamics}