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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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properties: |
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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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\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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\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 |
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\label{eq:gofrCosTheta} |
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\end{multline} |
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\begin{multline}\label{eq:gofrCosOmega} |
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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 |
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\end{multline} |
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|
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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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\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} |
