| 3 |
|
\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: |
| 6 |
> |
pair correlations between species A and B: |
| 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} |
| 13 |
|
Eq.~\ref{eq:cosOmegaOfR} |
| 14 |
|
\end{itemize} |
| 15 |
|
|
| 16 |
+ |
The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows: |
| 17 |
|
\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 |
|
\end{equation} |
| 22 |
+ |
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 |
|
|
| 29 |
+ |
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 |
|
\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 |
| 28 |
< |
\label{eq:gofrCosTheta} |
| 42 |
> |
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosTheta} |
| 43 |
|
\end{multline} |
| 44 |
+ |
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 |
|
|
| 52 |
< |
\begin{multline}\label{eq:gofrCosOmega} |
| 52 |
> |
The second two dimensional histogram is of the form: |
| 53 |
> |
\begin{multline} |
| 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 |
| 58 |
> |
\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega} |
| 59 |
|
\end{multline} |
| 60 |
+ |
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 |
|
|
| 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) = \\ |
| 71 |
|
\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
| 74 |
|
\delta( y - y_{ij}) |
| 75 |
|
\delta( z - z_{ij}) \rangle |
| 76 |
|
\end{multline} |
| 77 |
+ |
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
| 78 |
+ |
components respectively of vector $\mathbf{r}_{ij}$. |
| 79 |
|
|
| 80 |
+ |
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 |
|
\begin{equation}\label{eq:cosOmegaOfR} |
| 84 |
< |
\langle \cos \omega \rangle_{\text{AB}}(r) = |
| 84 |
> |
\langle \cos \omega \rangle_{\text{AB}}(r) = |
| 85 |
|
\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 |
+ |
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. |
