| 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 |
< |
\begin{multline} |
| 38 |
< |
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 |
< |
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosTheta} |
| 43 |
< |
\end{multline} |
| 37 |
> |
\begin{equation} |
| 38 |
> |
g_{\text{AB}}(r, \cos \theta) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
| 39 |
> |
\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
| 40 |
> |
\delta( \cos \theta - \cos \theta_{ij}) |
| 41 |
> |
\delta( r - |\mathbf{r}_{ij}|) \rangle |
| 42 |
> |
\label{eq:gofrCosTheta} |
| 43 |
> |
\end{equation} |
| 44 |
|
Where |
| 45 |
|
\begin{equation*} |
| 46 |
|
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij} |
| 50 |
|
$\mathbf{r}_{ij}$. |
| 51 |
|
|
| 52 |
|
The second two dimensional histogram is of the form: |
| 53 |
< |
\begin{multline} |
| 54 |
< |
g_{\text{AB}}(r, \cos \omega) = \\ |
| 53 |
> |
\begin{equation} |
| 54 |
> |
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 |
|
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega} |
| 59 |
< |
\end{multline} |
| 59 |
> |
\end{equation} |
| 60 |
|
Here |
| 61 |
|
\begin{equation*} |
| 62 |
|
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}} |
| 66 |
|
|
| 67 |
|
The static analysis code is also cable of calculating a three |
| 68 |
|
dimensional pair correlation of the form: |
| 69 |
< |
\begin{multline}\label{eq:gofrXYZ} |
| 70 |
< |
g_{\text{AB}}(x, y, z) = \\ |
| 69 |
> |
\begin{equation}\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} |
| 76 |
> |
\end{equation} |
| 77 |
|
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
| 78 |
|
components respectively of vector $\mathbf{r}_{ij}$. |
| 79 |
|
|
| 132 |
|
multiple reads on the same file. |
| 133 |
|
|
| 134 |
|
\begin{figure} |
| 135 |
< |
\includegraphics[angle=-90,width=80mm]{dynamicPropsMem.eps} |
| 135 |
> |
\epsfxsize=6in |
| 136 |
> |
\epsfbox{dynamicPropsMem.eps} |
| 137 |
|
\caption{This diagram illustrates the dynamic memory allocation used by \texttt{dynamicProps}, which follows the scheme: $\sum^{N_{\text{memory blocks}}}_{i=1}[ \operatorname{self}(i) + \sum^{N_{\text{memory blocks}}}_{j>i} \operatorname{cross}(i,j)]$. The shaded region represents the self correlation of the memory block, and the open blocks are read one at a time and the cross correlations between blocks are calculated.} |
| 138 |
|
\label{fig:dynamicPropsMemory} |
| 139 |
|
\end{figure} |
