| 213 |
|
\end{array}} \right) |
| 214 |
|
\label{introEquation:canonicalMatrix} |
| 215 |
|
\end{equation} |
| 216 |
< |
Thus, Hamiltonian system can be rewritten as, |
| 216 |
> |
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 |
|
\begin{equation} |
| 220 |
|
\frac{d}{{dt}}r = J\nabla _r H(r) |
| 221 |
|
\label{introEquation:compactHamiltonian} |
| 222 |
|
\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} |
| 225 |
> |
%Transformation} |
| 226 |
|
|
| 227 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 228 |
|
|
| 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} |
| 241 |
> |
\subsection{\label{introSection::ensemble}Ensemble and Phase Space} |
| 242 |
|
|
| 243 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
| 244 |
|
|
| 245 |
+ |
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 |
|
|
| 274 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 275 |
|
|
| 291 |
|
|
| 292 |
|
Applications of dynamics of rigid bodies. |
| 293 |
|
|
| 294 |
+ |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 295 |
|
|
| 296 |
+ |
\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 |
|
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
| 301 |
|
|
| 302 |
|
\section{\label{introSection:correlationFunctions}Correlation Functions} |
