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 |
– |
|
3 |
|
\section{\label{introSection:classicalMechanics}Classical |
4 |
|
Mechanics} |
5 |
|
|
15 |
|
sufficient to predict the future behavior of the system. |
16 |
|
|
17 |
|
\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
18 |
+ |
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 |
|
|
38 |
+ |
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 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
72 |
|
|
73 |
|
Newtonian Mechanics suffers from two important limitations: it |
81 |
|
which arise in attempts to apply Newton's equation to complex |
82 |
|
system, alternative procedures may be developed. |
83 |
|
|
84 |
< |
\subsection{\label{introSection:halmiltonPrinciple}Hamilton's |
84 |
> |
\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
85 |
|
Principle} |
86 |
|
|
87 |
|
Hamilton introduced the dynamical principle upon which it is |
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 |
< |
the kinetic, $K$, and potential energies, $U$. |
94 |
> |
the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
95 |
|
\begin{equation} |
96 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
97 |
|
\label{introEquation:halmitonianPrinciple1} |
112 |
|
\label{introEquation:halmitonianPrinciple2} |
113 |
|
\end{equation} |
114 |
|
|
115 |
< |
\subsection{\label{introSection:equationOfMotionLagrangian}The |
115 |
> |
\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
116 |
|
Equations of Motion in Lagrangian Mechanics} |
117 |
|
|
118 |
|
for a holonomic system of $f$ degrees of freedom, the equations of |
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 |
< |
known as the canonical equations of motions. |
190 |
> |
known as the canonical equations of motions \cite{Goldstein01}. |
191 |
|
|
192 |
|
An important difference between Lagrangian approach and the |
193 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
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 |
< |
equations. |
201 |
> |
equations\cite{Marion90}. |
202 |
|
|
203 |
< |
\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
203 |
> |
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 |
> |
Thus, Hamiltonian system can be rewritten as, |
217 |
> |
\begin{equation} |
218 |
> |
\frac{d}{{dt}}r = J\nabla _r H(r) |
219 |
> |
\label{introEquation:compactHamiltonian} |
220 |
> |
\end{equation} |
221 |
> |
where $I$ is an identity matrix and $J$ is a skew-symmetrix matrix |
222 |
> |
($ J^T = - J $). |
223 |
|
|
224 |
< |
\subsection{\label{introSection:canonicalTransformation}Canonical |
224 |
> |
%\subsection{\label{introSection:canonicalTransformation}Canonical |
225 |
|
Transformation} |
226 |
|
|
227 |
+ |
\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 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
234 |
|
Mechanics} |
235 |
|
|
236 |
< |
The thermodynamic behaviors and properties of Molecular Dynamics |
236 |
> |
The thermodynamic behaviors and properties of Molecular Dynamics |
237 |
|
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. |
242 |
|
|
243 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
244 |
|
|
245 |
+ |
|
246 |
+ |
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
247 |
+ |
|
248 |
+ |
As a special discipline of molecular modeling, Molecular dynamics |
249 |
+ |
has proven to be a powerful tool for studying the functions of |
250 |
+ |
biological systems, providing structural, thermodynamic and |
251 |
+ |
dynamical information. |
252 |
+ |
|
253 |
+ |
\subsection{\label{introSec:mdInit}Initialization} |
254 |
+ |
|
255 |
+ |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
256 |
+ |
|
257 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
258 |
|
|
259 |
+ |
A rigid body is a body in which the distance between any two given |
260 |
+ |
points of a rigid body remains constant regardless of external |
261 |
+ |
forces exerted on it. A rigid body therefore conserves its shape |
262 |
+ |
during its motion. |
263 |
+ |
|
264 |
+ |
Applications of dynamics of rigid bodies. |
265 |
+ |
|
266 |
+ |
|
267 |
+ |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
268 |
+ |
|
269 |
|
\section{\label{introSection:correlationFunctions}Correlation Functions} |
270 |
|
|
271 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |