trunk/oopsePaper/analysis.tex

\section{Analysis Code}

\subsection{Static Property Analysis}
The static properties of the trajectories are analyzed with the
program staticProps. The code is capable of calculating the following
pair correlations between species A and B:
\begin{itemize}
        \item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr}
        \item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta}
        \item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega}
        \item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ}
        \item $\langle \cos \omega \rangle_{\text{AB}}(r)$: 
                Eq.~\ref{eq:cosOmegaOfR}
\end{itemize}

The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows:
\begin{equation}
g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %%
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} %%
        \delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofr}
\end{equation}
Where $\mathbf{r}_{ij}$ is the vector
\begin{equation*}
\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag
\end{equation*}
and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over
the expected pair density at a given $r$.

The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and
$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two
dimensional histograms. Both use $r$ for the primary axis then a
$\cos$ for the secondary axis ($\cos \theta$ for
Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for
Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to
correlate alignment on directional entities. $g_{\text{AB}}(r, \cos
\theta)$ is defined as follows:
\begin{multline}
g_{\text{AB}}(r, \cos \theta) = \\
        \frac{V}{N_{\text{A}}N_{\text{B}}}\langle 
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} 
        \delta( \cos \theta - \cos \theta_{ij})
        \delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosTheta}
\end{multline}
Where
\begin{equation*}
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij}
\end{equation*}
Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$
and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector
$\mathbf{r}_{ij}$.

The second two dimensional histogram is of the form:
\begin{multline}
g_{\text{AB}}(r, \cos \omega) = \\
        \frac{V}{N_{\text{A}}N_{\text{B}}}\langle 
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} 
        \delta( \cos \omega - \cos \omega_{ij})
        \delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega}
\end{multline}
Here
\begin{equation*}
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}}
\end{equation*}
Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit
directional vectors of species $i$ and $j$.

The static analysis code is also cable of calculating a three
dimensional pair correlation of the form:
\begin{multline}\label{eq:gofrXYZ}
g_{\text{AB}}(x, y, z) = \\
        \frac{V}{N_{\text{A}}N_{\text{B}}}\langle 
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} 
        \delta( x - x_{ij})
        \delta( y - y_{ij})
        \delta( z - z_{ij}) \rangle
\end{multline}
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$
components respectively of vector $\mathbf{r}_{ij}$.

The final pair correlation is similar to
Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega
\rangle_{\text{AB}}(r)$ is calculated in the following way:
\begin{equation}\label{eq:cosOmegaOfR}
\langle \cos \omega \rangle_{\text{AB}}(r)  = 
        \langle \sum_{i \in \text{A}} \sum_{j \in \text{B}}
        (\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle
\end{equation}
Here $\cos \omega_{ij}$ is defined in the same way as in
Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair
correlation that gives the average correlation of two directional
entities as a function of their distance from each other.

\subsection{Dynamic Property Analysis}
The dynamic properties of a trajectory are calculated with the program
dynamicProps.
Revision:	695
Committed:	Wed Aug 13 21:22:44 2003 UTC (21 years, 1 month ago) by mmeineke
Content type:	application/x-tex
File size:	3802 byte(s)
Log Message:	added text to the analysis code.
#	User	Rev	Content
1	mmeineke	662	\section{Analysis Code}
2
3	mmeineke	668	\subsection{Static Property Analysis}
4			The static properties of the trajectories are analyzed with the
5			program staticProps. The code is capable of calculating the following
6	mmeineke	695	pair correlations between species A and B:
7	mmeineke	668	\begin{itemize}
8	mmeineke	691	\item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr}
9			\item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta}
10			\item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega}
11			\item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ}
12	mmeineke	668	\item $\langle \cos \omega \rangle_{\text{AB}}(r)$:
13	mmeineke	691	Eq.~\ref{eq:cosOmegaOfR}
14	mmeineke	668	\end{itemize}
15	mmeineke	662
16	mmeineke	695	The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows:
17	mmeineke	691	\begin{equation}
18			g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %%
19			\sum_{i \in \text{A}} \sum_{j \in \text{B}} %%
20			\delta( r - \|\mathbf{r}_{ij}\|) \rangle \label{eq:gofr}
21	mmeineke	668	\end{equation}
22	mmeineke	695	Where $\mathbf{r}_{ij}$ is the vector
23			\begin{equation*}
24			\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag
25			\end{equation*}
26			and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over
27			the expected pair density at a given $r$.
28	mmeineke	662
29	mmeineke	695	The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and
30			$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two
31			dimensional histograms. Both use $r$ for the primary axis then a
32			$\cos$ for the secondary axis ($\cos \theta$ for
33			Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for
34			Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to
35			correlate alignment on directional entities. $g_{\text{AB}}(r, \cos
36			\theta)$ is defined as follows:
37	mmeineke	691	\begin{multline}
38	mmeineke	668	g_{\text{AB}}(r, \cos \theta) = \\
39			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
40			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
41			\delta( \cos \theta - \cos \theta_{ij})
42	mmeineke	695	\delta( r - \|\mathbf{r}_{ij}\|) \rangle \label{eq:gofrCosTheta}
43	mmeineke	668	\end{multline}
44	mmeineke	695	Where
45			\begin{equation*}
46			\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij}
47			\end{equation*}
48			Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$
49			and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector
50			$\mathbf{r}_{ij}$.
51	mmeineke	668
52	mmeineke	695	The second two dimensional histogram is of the form:
53			\begin{multline}
54	mmeineke	668	g_{\text{AB}}(r, \cos \omega) = \\
55			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
56			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
57			\delta( \cos \omega - \cos \omega_{ij})
58	mmeineke	695	\delta( r - \|\mathbf{r}_{ij}\|) \rangle \label{eq:gofrCosOmega}
59	mmeineke	668	\end{multline}
60	mmeineke	695	Here
61			\begin{equation*}
62			\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}}
63			\end{equation*}
64			Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit
65			directional vectors of species $i$ and $j$.
66	mmeineke	668
67	mmeineke	695	The static analysis code is also cable of calculating a three
68			dimensional pair correlation of the form:
69	mmeineke	668	\begin{multline}\label{eq:gofrXYZ}
70			g_{\text{AB}}(x, y, z) = \\
71			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
72			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
73			\delta( x - x_{ij})
74			\delta( y - y_{ij})
75			\delta( z - z_{ij}) \rangle
76			\end{multline}
77	mmeineke	695	Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$
78			components respectively of vector $\mathbf{r}_{ij}$.
79	mmeineke	668
80	mmeineke	695	The final pair correlation is similar to
81			Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega
82			\rangle_{\text{AB}}(r)$ is calculated in the following way:
83	mmeineke	668	\begin{equation}\label{eq:cosOmegaOfR}
84	mmeineke	695	\langle \cos \omega \rangle_{\text{AB}}(r) =
85	mmeineke	668	\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}}
86			(\cos \omega_{ij}) \delta( r - \|\mathbf{r}_{ij}\|) \rangle
87			\end{equation}
88	mmeineke	695	Here $\cos \omega_{ij}$ is defined in the same way as in
89			Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair
90			correlation that gives the average correlation of two directional
91			entities as a function of their distance from each other.
92
93			\subsection{Dynamic Property Analysis}
94			The dynamic properties of a trajectory are calculated with the program
95			dynamicProps.