752 |
|
\begin{equation} |
753 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
754 |
|
\label{introEquation:timeReversible} |
755 |
< |
\end{equation},appendixFig:architecture |
755 |
> |
\end{equation} |
756 |
|
|
757 |
|
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
758 |
|
The classical equation for a system consisting of interacting |
1110 |
|
particle $j$ in the system |
1111 |
|
\[ |
1112 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1113 |
< |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \fract{\rho |
1113 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
1114 |
|
(r)}{\rho}. |
1115 |
|
\] |
1116 |
|
Note that the delta function can be replaced by a histogram in |
1117 |
< |
computer simulation. Figure |
1118 |
< |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1119 |
< |
distribution function for the liquid argon system. The occurrence of |
1120 |
< |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1121 |
< |
to find particles at certain radial values than at others. This is a |
1122 |
< |
result of the attractive interaction at such distances. Because of |
1123 |
< |
the strong repulsive forces at short distance, the probability of |
1124 |
< |
locating particles at distances less than about 3.7{\AA} from each |
1125 |
< |
other is essentially zero. |
1117 |
> |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
1118 |
> |
the height of these peaks gradually decreases to 1 as the liquid of |
1119 |
> |
large distance approaches the bulk density. |
1120 |
|
|
1127 |
– |
%\begin{figure} |
1128 |
– |
%\centering |
1129 |
– |
%\includegraphics[width=\linewidth]{pdf.eps} |
1130 |
– |
%\caption[Pair distribution function for the liquid argon |
1131 |
– |
%]{Pair distribution function for the liquid argon} |
1132 |
– |
%\label{introFigure:pairDistributionFunction} |
1133 |
– |
%\end{figure} |
1121 |
|
|
1122 |
|
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
1123 |
|
Properties}} |
1555 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1556 |
|
\] |
1557 |
|
where $g_\alpha$ are the coupling constants between the bath |
1558 |
< |
coordinates ($x_ \apha$) and the system coordinate ($x$). |
1558 |
> |
coordinates ($x_ \alpha$) and the system coordinate ($x$). |
1559 |
|
Introducing |
1560 |
|
\[ |
1561 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |