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\end{array}} \right) |
214 |
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\label{introEquation:canonicalMatrix} |
215 |
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\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 |
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\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} |