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\section{\label{sec:analysis}Trajectory Analysis} |
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\subsection{\label{subSec:staticProps}Static Property Analysis} |
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The static properties of the trajectories are analyzed with the |
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program \texttt{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{equation} |
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g_{\text{AB}}(r, \cos \theta) = \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{equation} |
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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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|
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The second two dimensional histogram is of the form: |
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\begin{equation} |
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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{equation} |
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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{equation}\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{equation} |
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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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All static properties are calculated on a frame by frame basis. The |
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trajectory is read a single frame at a time, and the appropriate |
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calculations are done on each frame. Once one frame is finished, the |
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next frame is read in, and a running average of the property being |
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calculated is accumulated in each frame. The program allows for the |
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user to specify more than one property be calculated in single run, |
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preventing the need to read a file multiple times. |
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\subsection{\label{dynamicProps}Dynamic Property Analysis} |
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The dynamic properties of a trajectory are calculated with the program |
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\texttt{dynamicProps}. The program will calculate the following properties: |
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\begin{gather} |
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\langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle \label{eq:rms}\\ |
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\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\ |
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\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr} |
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\end{gather} |
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Eq.~\ref{eq:rms} is the root mean square displacement |
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function. Eq.~\ref{eq:velCorr} and Eq.~\ref{eq:angularVelCorr} are the |
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velocity and angular velocity correlation functions respectively. The |
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latter is only applicable to directional species in the simulation. |
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The \texttt{dynamicProps} program handles he file in a manner different from |
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\texttt{staticProps}. As the properties calculated by this program are time |
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dependent, multiple frames must be read in simultaneously by the |
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program. For small trajectories this is no problem, and the entire |
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trajectory is read into memory. However, for long trajectories of |
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large systems, the files can be quite large. In order to accommodate |
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large files, \texttt{dynamicProps} adopts a scheme whereby two blocks of memory |
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are allocated to read in several frames each. |
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In this two block scheme, the correlation functions are first |
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calculated within each memory block, then the cross correlations |
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between the frames contained within the two blocks are |
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calculated. Once completed, the memory blocks are incremented, and the |
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process is repeated. A diagram illustrating the process is shown in |
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Fig.~\ref{fig:dynamicPropsMemory}. As was the case with \texttt{staticProps}, |
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multiple properties may be calculated in a single run to avoid |
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multiple reads on the same file. |
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\begin{figure} |
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\epsfxsize=6in |
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\epsfbox{dynamicPropsMem.eps} |
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\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.} |
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\label{fig:dynamicPropsMemory} |
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\end{figure} |
