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\section{Analysis Code} |
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We ran some numbers through some functions. Did that a couple of times |
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in quick succession to obtain some lovely graphs.\cite{allen87:csl} |
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\subsection{Static Property Analysis} |
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The static properties of the trajectories are analyzed with the |
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program staticProps. The code is capable of calculating the following |
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pair correlations between species A and B: |
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\begin{itemize} |
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\item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr} |
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\item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta} |
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\item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega} |
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\item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ} |
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\item $\langle \cos \omega \rangle_{\text{AB}}(r)$: |
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Eq.~\ref{eq:cosOmegaOfR} |
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\end{itemize} |
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The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows: |
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\begin{equation} |
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g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %% |
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\sum_{i \in \text{A}} \sum_{j \in \text{B}} %% |
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\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofr} |
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\end{equation} |
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Where $\mathbf{r}_{ij}$ is the vector |
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\begin{equation*} |
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\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag |
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\end{equation*} |
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and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over |
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the expected pair density at a given $r$. |
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The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and |
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$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two |
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dimensional histograms. Both use $r$ for the primary axis then a |
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$\cos$ for the secondary axis ($\cos \theta$ for |
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Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for |
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Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to |
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correlate alignment on directional entities. $g_{\text{AB}}(r, \cos |
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\theta)$ is defined as follows: |
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\begin{multline} |
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g_{\text{AB}}(r, \cos \theta) = \\ |
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\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
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\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
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\delta( \cos \theta - \cos \theta_{ij}) |
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\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosTheta} |
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\end{multline} |
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Where |
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\begin{equation*} |
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\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij} |
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\end{equation*} |
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Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$ |
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and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector |
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$\mathbf{r}_{ij}$. |
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The second two dimensional histogram is of the form: |
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\begin{multline} |
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g_{\text{AB}}(r, \cos \omega) = \\ |
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\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
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\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
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\delta( \cos \omega - \cos \omega_{ij}) |
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\delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega} |
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\end{multline} |
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Here |
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\begin{equation*} |
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\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}} |
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\end{equation*} |
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Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit |
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directional vectors of species $i$ and $j$. |
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The static analysis code is also cable of calculating a three |
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dimensional pair correlation of the form: |
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\begin{multline}\label{eq:gofrXYZ} |
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g_{\text{AB}}(x, y, z) = \\ |
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\frac{V}{N_{\text{A}}N_{\text{B}}}\langle |
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\sum_{i \in \text{A}} \sum_{j \in \text{B}} |
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\delta( x - x_{ij}) |
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\delta( y - y_{ij}) |
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\delta( z - z_{ij}) \rangle |
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\end{multline} |
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Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$ |
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components respectively of vector $\mathbf{r}_{ij}$. |
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The final pair correlation is similar to |
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Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega |
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\rangle_{\text{AB}}(r)$ is calculated in the following way: |
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\begin{equation}\label{eq:cosOmegaOfR} |
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\langle \cos \omega \rangle_{\text{AB}}(r) = |
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\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}} |
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(\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle |
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\end{equation} |
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Here $\cos \omega_{ij}$ is defined in the same way as in |
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Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair |
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correlation that gives the average correlation of two directional |
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entities as a function of their distance from each other. |
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\subsection{Dynamic Property Analysis} |
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The dynamic properties of a trajectory are calculated with the program |
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dynamicProps. |
