trunk/oopsePaper/analysis.tex

\section{\label{sec:analysis}Analysis Code}

\subsection{\label{subSec:staticProbs}Static Property Analysis}
The static properties of the trajectories are analyzed with the
program \texttt{staticProps}. The code is capable of calculating the following
pair correlations between species A and B:
\begin{itemize}
        \item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr}
        \item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta}
        \item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega}
        \item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ}
        \item $\langle \cos \omega \rangle_{\text{AB}}(r)$: 
                Eq.~\ref{eq:cosOmegaOfR}
\end{itemize}

The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows:
\begin{equation}
g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %%
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} %%
        \delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofr}
\end{equation}
Where $\mathbf{r}_{ij}$ is the vector
\begin{equation*}
\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag
\end{equation*}
and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over
the expected pair density at a given $r$.

The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and
$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two
dimensional histograms. Both use $r$ for the primary axis then a
$\cos$ for the secondary axis ($\cos \theta$ for
Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for
Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to
correlate alignment on directional entities. $g_{\text{AB}}(r, \cos
\theta)$ is defined as follows:
\begin{multline}
g_{\text{AB}}(r, \cos \theta) = \\
        \frac{V}{N_{\text{A}}N_{\text{B}}}\langle 
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} 
        \delta( \cos \theta - \cos \theta_{ij})
        \delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosTheta}
\end{multline}
Where
\begin{equation*}
\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij}
\end{equation*}
Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$
and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector
$\mathbf{r}_{ij}$.

The second two dimensional histogram is of the form:
\begin{multline}
g_{\text{AB}}(r, \cos \omega) = \\
        \frac{V}{N_{\text{A}}N_{\text{B}}}\langle 
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} 
        \delta( \cos \omega - \cos \omega_{ij})
        \delta( r - |\mathbf{r}_{ij}|) \rangle \label{eq:gofrCosOmega}
\end{multline}
Here
\begin{equation*}
\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}}
\end{equation*}
Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit
directional vectors of species $i$ and $j$.

The static analysis code is also cable of calculating a three
dimensional pair correlation of the form:
\begin{multline}\label{eq:gofrXYZ}
g_{\text{AB}}(x, y, z) = \\
        \frac{V}{N_{\text{A}}N_{\text{B}}}\langle 
        \sum_{i \in \text{A}} \sum_{j \in \text{B}} 
        \delta( x - x_{ij})
        \delta( y - y_{ij})
        \delta( z - z_{ij}) \rangle
\end{multline}
Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$
components respectively of vector $\mathbf{r}_{ij}$.

The final pair correlation is similar to
Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega
\rangle_{\text{AB}}(r)$ is calculated in the following way:
\begin{equation}\label{eq:cosOmegaOfR}
\langle \cos \omega \rangle_{\text{AB}}(r)  = 
        \langle \sum_{i \in \text{A}} \sum_{j \in \text{B}}
        (\cos \omega_{ij}) \delta( r - |\mathbf{r}_{ij}|) \rangle
\end{equation}
Here $\cos \omega_{ij}$ is defined in the same way as in
Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair
correlation that gives the average correlation of two directional
entities as a function of their distance from each other.

All static properties are calculated on a frame by frame basis. The
trajectory is read a single frame at a time, and the appropriate
calculations are done on each frame. Once one frame is finished, the
next frame is read in, and a running average of the property being
calculated is accumulated in each frame. The program allows for the
user to specify more than one property be calculated in single run,
preventing the need to read a file multiple times.

\subsection{\label{dynamicProps}Dynamic Property Analysis}

The dynamic properties of a trajectory are calculated with the program
\texttt{dynamicProps}. The program will calculate the following properties:
\begin{gather}
\langle | \mathbf{r}(t) - \mathbf{r}(0) |^2 \rangle \label{eq:rms}\\
\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\
\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr}
\end{gather}

Eq.~\ref{eq:rms} is the root mean square displacement
function. Eq.~\ref{eq:velCorr} and Eq.~\ref{eq:angularVelCorr} are the
velocity and angular velocity correlation functions respectively. The
latter is only applicable to directional species in the simulation.

The \texttt{dynamicProps} program handles he file in a manner different from
\texttt{staticProps}. As the properties calculated by this program are time
dependent, multiple frames must be read in simultaneously by the
program. For small trajectories this is no problem, and the entire
trajectory is read into memory. However, for long trajectories of
large systems, the files can be quite large. In order to accommodate
large files, \texttt{dynamicProps} adopts a scheme whereby two blocks of memory
are allocated to read in several frames each.

In this two block scheme, the correlation functions are first
calculated within each memory block, then the cross correlations
between the frames contained within the two blocks are
calculated. Once completed, the memory blocks are incremented, and the
process is repeated. A diagram illustrating the process is shown in
Fig.~\ref{fig:dynamicPropsMemory}. As was the case with \texttt{staticProps},
multiple properties may be calculated in a single run to avoid
multiple reads on the same file.  

\begin{figure} 
\includegraphics[angle=-90,width=80mm]{dynamicPropsMem.eps}
\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.}
\label{fig:dynamicPropsMemory}
\end{figure}
Revision:	710
Committed:	Fri Aug 22 19:37:54 2003 UTC (22 years, 2 months ago) by mmeineke
Content type:	application/x-tex
File size:	6498 byte(s)
Log Message:	played a little with the headers in DUFF
#	User	Rev	Content
1	mmeineke	710	\section{\label{sec:analysis}Analysis Code}
2	mmeineke	662
3	mmeineke	710	\subsection{\label{subSec:staticProbs}Static Property Analysis}
4	mmeineke	668	The static properties of the trajectories are analyzed with the
5	mmeineke	697	program \texttt{staticProps}. The code is capable of calculating the following
6	mmeineke	695	pair correlations between species A and B:
7	mmeineke	668	\begin{itemize}
8	mmeineke	691	\item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr}
9			\item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta}
10			\item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega}
11			\item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ}
12	mmeineke	668	\item $\langle \cos \omega \rangle_{\text{AB}}(r)$:
13	mmeineke	691	Eq.~\ref{eq:cosOmegaOfR}
14	mmeineke	668	\end{itemize}
15	mmeineke	662
16	mmeineke	695	The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows:
17	mmeineke	691	\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	mmeineke	668	\end{equation}
22	mmeineke	695	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	mmeineke	662
29	mmeineke	695	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	mmeineke	691	\begin{multline}
38	mmeineke	668	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	mmeineke	695	\delta( r - \|\mathbf{r}_{ij}\|) \rangle \label{eq:gofrCosTheta}
43	mmeineke	668	\end{multline}
44	mmeineke	695	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	mmeineke	668
52	mmeineke	695	The second two dimensional histogram is of the form:
53			\begin{multline}
54	mmeineke	668	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	mmeineke	695	\delta( r - \|\mathbf{r}_{ij}\|) \rangle \label{eq:gofrCosOmega}
59	mmeineke	668	\end{multline}
60	mmeineke	695	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	mmeineke	668
67	mmeineke	695	The static analysis code is also cable of calculating a three
68			dimensional pair correlation of the form:
69	mmeineke	668	\begin{multline}\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}
77	mmeineke	695	Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$
78			components respectively of vector $\mathbf{r}_{ij}$.
79	mmeineke	668
80	mmeineke	695	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	mmeineke	668	\begin{equation}\label{eq:cosOmegaOfR}
84	mmeineke	695	\langle \cos \omega \rangle_{\text{AB}}(r) =
85	mmeineke	668	\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	mmeineke	695	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	mmeineke	697	All static properties are calculated on a frame by frame basis. The
94			trajectory is read a single frame at a time, and the appropriate
95			calculations are done on each frame. Once one frame is finished, the
96			next frame is read in, and a running average of the property being
97			calculated is accumulated in each frame. The program allows for the
98			user to specify more than one property be calculated in single run,
99			preventing the need to read a file multiple times.
100
101	mmeineke	710	\subsection{\label{dynamicProps}Dynamic Property Analysis}
102
103	mmeineke	695	The dynamic properties of a trajectory are calculated with the program
104	mmeineke	697	\texttt{dynamicProps}. The program will calculate the following properties:
105			\begin{gather}
106			\langle \| \mathbf{r}(t) - \mathbf{r}(0) \|^2 \rangle \label{eq:rms}\\
107			\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\
108			\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr}
109			\end{gather}
110
111			Eq.~\ref{eq:rms} is the root mean square displacement
112			function. Eq.~\ref{eq:velCorr} and Eq.~\ref{eq:angularVelCorr} are the
113			velocity and angular velocity correlation functions respectively. The
114			latter is only applicable to directional species in the simulation.
115
116			The \texttt{dynamicProps} program handles he file in a manner different from
117			\texttt{staticProps}. As the properties calculated by this program are time
118			dependent, multiple frames must be read in simultaneously by the
119			program. For small trajectories this is no problem, and the entire
120			trajectory is read into memory. However, for long trajectories of
121			large systems, the files can be quite large. In order to accommodate
122			large files, \texttt{dynamicProps} adopts a scheme whereby two blocks of memory
123			are allocated to read in several frames each.
124
125			In this two block scheme, the correlation functions are first
126			calculated within each memory block, then the cross correlations
127			between the frames contained within the two blocks are
128			calculated. Once completed, the memory blocks are incremented, and the
129			process is repeated. A diagram illustrating the process is shown in
130			Fig.~\ref{fig:dynamicPropsMemory}. As was the case with \texttt{staticProps},
131			multiple properties may be calculated in a single run to avoid
132			multiple reads on the same file.
133
134			\begin{figure}
135	mmeineke	709	\includegraphics[angle=-90,width=80mm]{dynamicPropsMem.eps}
136			\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.}
137	mmeineke	697	\label{fig:dynamicPropsMemory}
138			\end{figure}