| 1 |
\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
| 2 |
|
| 3 |
\section{\label{introSection:classicalMechanics}Classical |
| 4 |
Mechanics} |
| 5 |
|
| 6 |
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 |
|
| 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 |
| 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 |
\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
| 85 |
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 |
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} |
| 98 |
\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 |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
| 112 |
\label{introEquation:halmitonianPrinciple2} |
| 113 |
\end{equation} |
| 114 |
|
| 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 |
| 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 |
\label{introEquation:eqMotionLagrangian} |
| 124 |
\end{equation} |
| 125 |
where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is |
| 126 |
generalized velocity. |
| 127 |
|
| 128 |
\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} |
| 129 |
|
| 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 |
The Lagrange equations of motion are then expressed by |
| 140 |
\begin{equation} |
| 141 |
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 |
\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 |
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 |
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 |
| 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 |
equations\cite{Marion90}. |
| 202 |
|
| 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 |
| 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 |
| 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. |
| 240 |
|
| 241 |
\subsection{\label{introSection::ensemble}Ensemble} |
| 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} |
| 272 |
|
| 273 |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 274 |
|
| 275 |
\subsection{\label{introSection:hydroynamics}Hydrodynamics} |
