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
properties:
\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}

\begin{equation}\label{eq:gofr}
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
\end{equation}

\begin{multline}\label{eq:gofrCosTheta}
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
\end{multline}

\begin{multline}\label{eq:gofrCosOmega}
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
\end{multline}

\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}

\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}
Revision:	668
Committed:	Wed Aug 6 18:35:50 2003 UTC (21 years, 1 month ago) by mmeineke
Content type:	application/x-tex
File size:	1724 byte(s)
Log Message:	added the main equations to the analysis section
#	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			properties:
7			\begin{itemize}
8			\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			\item $\langle \cos \omega \rangle_{\text{AB}}(r)$:
13			Eq. \ref{eq:cosOmegaOfR}
14			\end{itemize}
15	mmeineke	662
16	mmeineke	668	\begin{equation}\label{eq:gofr}
17			g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle
18			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
19			\delta( r - \|\mathbf{r}_{ij}\|) \rangle
20			\end{equation}
21	mmeineke	662
22	mmeineke	668	\begin{multline}\label{eq:gofrCosTheta}
23			g_{\text{AB}}(r, \cos \theta) = \\
24			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
25			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
26			\delta( \cos \theta - \cos \theta_{ij})
27			\delta( r - \|\mathbf{r}_{ij}\|) \rangle
28			\end{multline}
29
30			\begin{multline}\label{eq:gofrCosOmega}
31			g_{\text{AB}}(r, \cos \omega) = \\
32			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
33			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
34			\delta( \cos \omega - \cos \omega_{ij})
35			\delta( r - \|\mathbf{r}_{ij}\|) \rangle
36			\end{multline}
37
38			\begin{multline}\label{eq:gofrXYZ}
39			g_{\text{AB}}(x, y, z) = \\
40			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
41			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
42			\delta( x - x_{ij})
43			\delta( y - y_{ij})
44			\delta( z - z_{ij}) \rangle
45			\end{multline}
46
47			\begin{equation}\label{eq:cosOmegaOfR}
48			\langle \cos \omega \rangle_{\text{AB}}(r) =
49			\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}}
50			(\cos \omega_{ij}) \delta( r - \|\mathbf{r}_{ij}\|) \rangle
51			\end{equation}