trunk/oopsePaper/TrajectoryAnalysis.tex


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

\subsection{\label{subSec:staticProps}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{equation}
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{equation}
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{equation}
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{equation}
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{equation}\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{equation}
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} 
\epsfxsize=6in
\epsfbox{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:	899
Committed:	Tue Jan 6 18:53:58 2004 UTC (20 years, 8 months ago) by mmeineke
Content type:	application/x-tex
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#	User	Rev	Content
1	mmeineke	899
2			\section{\label{sec:analysis}Trajectory Analysis}
3
4			\subsection{\label{subSec:staticProps}Static Property Analysis}
5
6			The static properties of the trajectories are analyzed with the
7			program \texttt{staticProps}. The code is capable of calculating the following
8			pair correlations between species A and B:
9			\begin{itemize}
10			\item $g_{\text{AB}}(r)$: Eq.~\ref{eq:gofr}
11			\item $g_{\text{AB}}(r, \cos \theta)$: Eq.~\ref{eq:gofrCosTheta}
12			\item $g_{\text{AB}}(r, \cos \omega)$: Eq.~\ref{eq:gofrCosOmega}
13			\item $g_{\text{AB}}(x, y, z)$: Eq.~\ref{eq:gofrXYZ}
14			\item $\langle \cos \omega \rangle_{\text{AB}}(r)$:
15			Eq.~\ref{eq:cosOmegaOfR}
16			\end{itemize}
17
18			The first pair correlation, $g_{\text{AB}}(r)$, is defined as follows:
19			\begin{equation}
20			g_{\text{AB}}(r) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle %%
21			\sum_{i \in \text{A}} \sum_{j \in \text{B}} %%
22			\delta( r - \|\mathbf{r}_{ij}\|) \rangle \label{eq:gofr}
23			\end{equation}
24			Where $\mathbf{r}_{ij}$ is the vector
25			\begin{equation*}
26			\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i \notag
27			\end{equation*}
28			and $\frac{V}{N_{\text{A}}N_{\text{B}}}$ normalizes the average over
29			the expected pair density at a given $r$.
30
31			The next two pair correlations, $g_{\text{AB}}(r, \cos \theta)$ and
32			$g_{\text{AB}}(r, \cos \omega)$, are similar in that they are both two
33			dimensional histograms. Both use $r$ for the primary axis then a
34			$\cos$ for the secondary axis ($\cos \theta$ for
35			Eq.~\ref{eq:gofrCosTheta} and $\cos \omega$ for
36			Eq.~\ref{eq:gofrCosOmega}). This allows for the investigator to
37			correlate alignment on directional entities. $g_{\text{AB}}(r, \cos
38			\theta)$ is defined as follows:
39			\begin{equation}
40			g_{\text{AB}}(r, \cos \theta) = \frac{V}{N_{\text{A}}N_{\text{B}}}\langle
41			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
42			\delta( \cos \theta - \cos \theta_{ij})
43			\delta( r - \|\mathbf{r}_{ij}\|) \rangle
44			\label{eq:gofrCosTheta}
45			\end{equation}
46			Where
47			\begin{equation*}
48			\cos \theta_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{r}}_{ij}
49			\end{equation*}
50			Here $\mathbf{\hat{i}}$ is the unit directional vector of species $i$
51			and $\mathbf{\hat{r}}_{ij}$ is the unit vector associated with vector
52			$\mathbf{r}_{ij}$.
53
54			The second two dimensional histogram is of the form:
55			\begin{equation}
56			g_{\text{AB}}(r, \cos \omega) =
57			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
58			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
59			\delta( \cos \omega - \cos \omega_{ij})
60			\delta( r - \|\mathbf{r}_{ij}\|) \rangle \label{eq:gofrCosOmega}
61			\end{equation}
62			Here
63			\begin{equation*}
64			\cos \omega_{ij} = \mathbf{\hat{i}} \cdot \mathbf{\hat{j}}
65			\end{equation*}
66			Again, $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ are the unit
67			directional vectors of species $i$ and $j$.
68
69			The static analysis code is also cable of calculating a three
70			dimensional pair correlation of the form:
71			\begin{equation}\label{eq:gofrXYZ}
72			g_{\text{AB}}(x, y, z) =
73			\frac{V}{N_{\text{A}}N_{\text{B}}}\langle
74			\sum_{i \in \text{A}} \sum_{j \in \text{B}}
75			\delta( x - x_{ij})
76			\delta( y - y_{ij})
77			\delta( z - z_{ij}) \rangle
78			\end{equation}
79			Where $x_{ij}$, $y_{ij}$, and $z_{ij}$ are the $x$, $y$, and $z$
80			components respectively of vector $\mathbf{r}_{ij}$.
81
82			The final pair correlation is similar to
83			Eq.~\ref{eq:gofrCosOmega}. $\langle \cos \omega
84			\rangle_{\text{AB}}(r)$ is calculated in the following way:
85			\begin{equation}\label{eq:cosOmegaOfR}
86			\langle \cos \omega \rangle_{\text{AB}}(r) =
87			\langle \sum_{i \in \text{A}} \sum_{j \in \text{B}}
88			(\cos \omega_{ij}) \delta( r - \|\mathbf{r}_{ij}\|) \rangle
89			\end{equation}
90			Here $\cos \omega_{ij}$ is defined in the same way as in
91			Eq.~\ref{eq:gofrCosOmega}. This equation is a single dimensional pair
92			correlation that gives the average correlation of two directional
93			entities as a function of their distance from each other.
94
95			All static properties are calculated on a frame by frame basis. The
96			trajectory is read a single frame at a time, and the appropriate
97			calculations are done on each frame. Once one frame is finished, the
98			next frame is read in, and a running average of the property being
99			calculated is accumulated in each frame. The program allows for the
100			user to specify more than one property be calculated in single run,
101			preventing the need to read a file multiple times.
102
103			\subsection{\label{dynamicProps}Dynamic Property Analysis}
104
105			The dynamic properties of a trajectory are calculated with the program
106			\texttt{dynamicProps}. The program will calculate the following properties:
107			\begin{gather}
108			\langle \| \mathbf{r}(t) - \mathbf{r}(0) \|^2 \rangle \label{eq:rms}\\
109			\langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle \label{eq:velCorr} \\
110			\langle \mathbf{j}(t) \cdot \mathbf{j}(0) \rangle \label{eq:angularVelCorr}
111			\end{gather}
112
113			Eq.~\ref{eq:rms} is the root mean square displacement
114			function. Eq.~\ref{eq:velCorr} and Eq.~\ref{eq:angularVelCorr} are the
115			velocity and angular velocity correlation functions respectively. The
116			latter is only applicable to directional species in the simulation.
117
118			The \texttt{dynamicProps} program handles he file in a manner different from
119			\texttt{staticProps}. As the properties calculated by this program are time
120			dependent, multiple frames must be read in simultaneously by the
121			program. For small trajectories this is no problem, and the entire
122			trajectory is read into memory. However, for long trajectories of
123			large systems, the files can be quite large. In order to accommodate
124			large files, \texttt{dynamicProps} adopts a scheme whereby two blocks of memory
125			are allocated to read in several frames each.
126
127			In this two block scheme, the correlation functions are first
128			calculated within each memory block, then the cross correlations
129			between the frames contained within the two blocks are
130			calculated. Once completed, the memory blocks are incremented, and the
131			process is repeated. A diagram illustrating the process is shown in
132			Fig.~\ref{fig:dynamicPropsMemory}. As was the case with \texttt{staticProps},
133			multiple properties may be calculated in a single run to avoid
134			multiple reads on the same file.
135
136			\begin{figure}
137			\epsfxsize=6in
138			\epsfbox{dynamicPropsMem.eps}
139			\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.}
140			\label{fig:dynamicPropsMemory}
141			\end{figure}