Commit f925edbb authored by COULAUD Olivier's avatar COULAUD Olivier
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Add the relationship between Ewald method and Macro structure approach

parent 152f5db4
@article{Schmidt1991,
abstract = {The Rokhlin-Greengard fast multipole algorithm for evaluating Coulomb and multipole potentials has been implemented and analyzed in three dimensions. The implementation is presented for bounded charged systems and systems with periodic boundary conditions. The results include timings and error characterizations},
author = {Schmidt, K. E. and Lee, Michael a.},
doi = {10.1007/BF01030008},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/Schmidt, Lee - 1991 - Implementing the fast multipole method in three dimensions.pdf:pdf},
issn = {0022-4715},
journal = {Journal of Statistical Physics},
keywords = {Ewald summation,Periodic boundary conditions,fast multipole method,macro summation,many-body problem,n-body,spherical harmonique},
mendeley-groups = {FMM},
month = jun,
number = {5-6},
pages = {1223--1235},
title = {{Implementing the fast multipole method in three dimensions}},
url = {http://link.springer.com/10.1007/BF01030008},
volume = {63},
year = {1991}
}
@article{Heyes1981,
abstract = {The Coulomb potentials and fields in infinite point charge lattices are represented by series expansions in real and reciprocal space using a method similar to that devised by Bertaut. A systematic investigation is made into the relative convergence characteristics of series derived by varying the charge spreading function which is required in the theory. A number of formulas which are already in use are compared with new expressions derived by this method.},
author = {Heyes, D. M.},
doi = {10.1063/1.441285},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/Heyes - 1981 - Electrostatic potentials and fields in infinite point charge lattices.pdf:pdf},
issn = {00219606},
journal = {The Journal of Chemical Physics},
keywords = {Electrostatics,Lattice theory},
mendeley-groups = {Chimie},
mendeley-tags = {Electrostatics,Lattice theory},
number = {3},
pages = {1924},
title = {{Electrostatic potentials and fields in infinite point charge lattices}},
url = {http://link.aip.org/link/?JCP/74/1924/1\&Agg=doi},
volume = {74},
year = {1981}
}
@article{DeLeeuw1980,
author = {{De Leeuw}, S W and Perram, J W and Smith, E R},
file = {:Users/coulaud/Recherche/Bibliographie/Mendeley/De Leeuw, Perram, Smith - 1980 - Simulation of electrostatic systems in periodic boundary conditions. I. Lattice sums and dielectric con.pdf:pdf},
journal = {Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences},
keywords = {Periodic Boundary Conditions,dieclectric constant,lattice sums},
mendeley-groups = {Chimie},
mendeley-tags = {Periodic Boundary Conditions,dieclectric constant,lattice sums},
number = {1752},
pages = {27--56},
title = {{Simulation of electrostatic systems in periodic boundary conditions. I. Lattice sums and dielectric constants}},
url = {http://www.jstor.org/stable/2397238 http://rspa.royalsocietypublishing.org/content/373/1752/27.short},
volume = {373},
year = {1980}
}
......@@ -77,19 +77,22 @@
\begin{minipage}{0.4\textwidth}
\begin{flushleft} \large
\emph{Author:}\\
Berenger \textsc{Bramas} % Your name
Berenger \textsc{Bramas} \\% Your name
Olivier \textsc{Coulaud} % Your name
\end{flushleft}
\end{minipage}
~
\begin{minipage}{0.4\textwidth}
\begin{flushright} \large
\emph{Supervisor:} \\
\textsc{} % Supervisor's Name
%\emph{Supervisor:} \\
%\textsc{} % Supervisor's Name
\end{flushright}
\end{minipage}\\[4cm]
{\large \today}\\[3cm] % Date, change the \today to a set date if you want to be precise
{\large \today}\\[1cm] % Date, change the \today to a set date if you want to be precise
{\Large \textbf{Working note}}
\vfill % Fill the rest of the page with whitespace
\end{titlepage}
......@@ -387,6 +390,16 @@ and the force on atom $x_i$
$$
f(x_i) = \frac{q_i }{4 \pi\epsilon_0}\sum_{j=0,i\neq j}^{N}{q_j\frac{x_i-x_j}{\|x_i-x_j\|^3}}
$$
The Ewald method is equivalent to consider a surrounding conductor ($\epsilon=\infty$). The method described in this document and implemented in ScalFMM consider a finite set of the original box surrounded by a vacuum with dielectric constant $\epsilon=1$. The results by these two approaches are different \cite{DeLeeuw1980, Heyes1981} and for a very large sphere surrounding all boxes we have the following equation
\begin{equation}
E_{Ewald} = E_{FMM} -\frac{2\pi}{3V}|\sum_{i=1}^{N}{q_i x_i}|^2.
\end{equation}
By tacking the gradient of this equation, the relationship between FMM forces and Ewald forces is
\begin{equation}
F_{Ewald}(x_k) = -\nabla_k E_{Ewald} = F_{FMM}(x_k) + \frac{4\pi}{3V} q_k \sum_{i=1}^{N}{q_i x_i}.
\end{equation}
\subsubsection{DL\_Poly comparisons}
DL\_POLY\_2 uses the following internal molecular units \\
......@@ -427,7 +440,8 @@ Force &$C_{force}$& -138935.4835 & $10\,J mol^{-1}\, \AA^{-1}$\\
\end{center}
The force unit is the internal DL\_Poly unit.
The first test consists in a small crystal $4\times 4\times 4$ of NaCl. It is composed of 128 atoms The second tests is a larger crystal $10\times 10\times 10$ of NaCl and have 2000 atoms. The positions and the forces are stored in the \texttt{REVCON} file and the energy in the \texttt{STATIS} file. The EWald's parameter are chosen such that the Ewald sum precision is $10^{-08}$.
The first test consists in a small crystal $4\times 4\times 4$ of NaCl. It is composed of 128 atoms The second tests is a larger crystal $10\times 10\times 10$ of NaCl and have 2000 atoms. The positions and the forces are stored in the \texttt{REVCON} file and the energy in the \texttt{STATIS} file. The Ewald's parameter are chosen such that the Ewald sum precision is $10^{-08}$.
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
......@@ -446,6 +460,8 @@ A new approach has been proposed to compute a periodic FMM.
This approach can be used with any kernel and without the need of developing new operators.
Moreover, it is possible to compute an important repetition size for a small cost.
\bibliographystyle{plain}
\bibliography{fmm}
\newpage
\section*{Appendix 1 }
......@@ -457,24 +473,17 @@ Moreover, it is possible to compute an important repetition size for a small cos
\begin{verbatim}
timestep 1.e-15
steps 1
temperature 0.0
ensemble nve
delr 0.5
cutoff 8.0
ewald precision 1.e-8
no vdw
traj 1,1,2
print 1
stats 1
job time 3600
close time 10
finish
\end{verbatim}
\end{document}
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