| 993 |
|
|
| 994 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 995 |
|
|
| 996 |
+ |
Recently, advanced visualization technique are widely applied to |
| 997 |
+ |
monitor the motions of molecules. Although the dynamics of the |
| 998 |
+ |
system can be described qualitatively from animation, quantitative |
| 999 |
+ |
trajectory analysis are more appreciable. According to the |
| 1000 |
+ |
principles of Statistical Mechanics, |
| 1001 |
+ |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 1002 |
+ |
thermodynamics properties, analyze fluctuations of structural |
| 1003 |
+ |
parameters, and investigate time-dependent processes of the molecule |
| 1004 |
+ |
from the trajectories. |
| 1005 |
+ |
|
| 1006 |
+ |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
| 1007 |
+ |
|
| 1008 |
+ |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
| 1009 |
+ |
|
| 1010 |
+ |
Structural Properties of a simple fluid can be described by a set of |
| 1011 |
+ |
distribution functions. Among these functions,\emph{pair |
| 1012 |
+ |
distribution function}, also known as \emph{radial distribution |
| 1013 |
+ |
function}, are of most fundamental importance to liquid-state |
| 1014 |
+ |
theory. Pair distribution function can be gathered by Fourier |
| 1015 |
+ |
transforming raw data from a series of neutron diffraction |
| 1016 |
+ |
experiments and integrating over the surface factor \cite{Powles73}. |
| 1017 |
+ |
The experiment result can serve as a criterion to justify the |
| 1018 |
+ |
correctness of the theory. Moreover, various equilibrium |
| 1019 |
+ |
thermodynamic and structural properties can also be expressed in |
| 1020 |
+ |
terms of radial distribution function \cite{allen87:csl}. |
| 1021 |
+ |
|
| 1022 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
| 1023 |
+ |
particle $i$ will be located at a distance $r$ from a another |
| 1024 |
+ |
particle $j$ in the system |
| 1025 |
+ |
\[ |
| 1026 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1027 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
| 1028 |
+ |
\] |
| 1029 |
+ |
Note that the delta function can be replaced by a histogram in |
| 1030 |
+ |
computer simulation. Figure |
| 1031 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
| 1032 |
+ |
distribution function for the liquid argon system. The occurrence of |
| 1033 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
| 1034 |
+ |
to find particles at certain radial values than at others. This is a |
| 1035 |
+ |
result of the attractive interaction at such distances. Because of |
| 1036 |
+ |
the strong repulsive forces at short distance, the probability of |
| 1037 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
| 1038 |
+ |
other is essentially zero. |
| 1039 |
+ |
|
| 1040 |
+ |
%\begin{figure} |
| 1041 |
+ |
%\centering |
| 1042 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
| 1043 |
+ |
%\caption[Pair distribution function for the liquid argon |
| 1044 |
+ |
%]{Pair distribution function for the liquid argon} |
| 1045 |
+ |
%\label{introFigure:pairDistributionFunction} |
| 1046 |
+ |
%\end{figure} |
| 1047 |
+ |
|
| 1048 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
| 1049 |
+ |
Properties} |
| 1050 |
+ |
|
| 1051 |
+ |
Time-dependent properties are usually calculated using \emph{time |
| 1052 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
| 1053 |
+ |
at two different time |
| 1054 |
+ |
\begin{equation} |
| 1055 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
| 1056 |
+ |
\label{introEquation:timeCorrelationFunction} |
| 1057 |
+ |
\end{equation} |
| 1058 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1059 |
+ |
function is called \emph{auto correlation function}. One example of |
| 1060 |
+ |
auto correlation function is velocity auto-correlation function |
| 1061 |
+ |
which is directly related to transport properties of molecular |
| 1062 |
+ |
liquids. Another example is the calculation of the IR spectrum |
| 1063 |
+ |
through a Fourier transform of the dipole autocorrelation function. |
| 1064 |
+ |
|
| 1065 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1066 |
|
|
| 1067 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1923 |
|
\] |
| 1924 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1925 |
|
joining center of resistance $R$ and origin $O$. |
| 1857 |
– |
|
| 1858 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
