trunk/oopsePaper/pbc.tex

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\begin{document}
\section{\label{Sec:pbc}Periodic Boundary Conditions}

\textit{Periodic boundary conditions} are widely used to simulate truly
macroscopic systems with a relatively small number of particles. Simulation
box is replicated throughout space to form an infinite lattice. During the
simulation, when a particle moves in the primary cell, its periodic image
particles in other boxes move in exactly the same direction with exactly the
same orientation.Thus, as a particle leaves the primary cell, one of its
images will enter through the opposite face.If the simulation box is large
enough to avoid "feeling" the symmetric of the periodic lattice,the surface
effect could be ignored. Cubic and parallelepiped are the available periodic
cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe
the property of the simulation box. Therefore, not only the size of the
simulation box could be changed during the simulation, but also the shape of
it. The transformation from box space vector $\overrightarrow{s}$ to its
corresponding real space vector $\overrightarrow{r}$ is defined by
\begin{equation}
\overrightarrow{r}=H\overrightarrow{s}%
\end{equation}


where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the
simulation box respectively.

To find the minimum image, we need to convert the real vector to its
corresponding vector in box space first, \bigskip%
\begin{equation}
\overrightarrow{s}=H^{-1}\overrightarrow{r}%
\end{equation}
And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5
to 0.5,
\begin{equation}
s_{i}^{\prime}=s_{i}-round(s_{i})
\end{equation}
where%

\begin{equation}
round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{
}x\geqslant0
\end{equation}
%

\begin{equation}
round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0
\end{equation}


For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$.

Finally, we could get the minimum image by transforming back to real space,%

\begin{equation}
\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}%
\end{equation}


\end{document}
Revision:	904
Committed:	Wed Jan 7 15:21:00 2004 UTC (21 years, 4 months ago) by tim
Content type:	application/x-tex
File size:	3836 byte(s)
Log Message:	* empty log message *
#	User	Rev	Content
1	tim	903	\documentclass{article}%
2			\usepackage{amsfonts}
3			\usepackage{amsmath}
4			\usepackage{amssymb}
5			\usepackage{graphicx}%
6			\setcounter{MaxMatrixCols}{30}
7			%TCIDATA{OutputFilter=latex2.dll}
8			%TCIDATA{Version=5.00.0.2552}
9			%TCIDATA{CSTFile=40 LaTeX article.cst}
10			%TCIDATA{Created=Friday, September 19, 2003 08:29:53}
11	tim	904	%TCIDATA{LastRevised=Wednesday, January 07, 2004 10:20:42}
12	tim	903	%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
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15			%TCIDATA{ComputeDefs=
16			%$H$
17			%}
18			\newtheorem{theorem}{Theorem}
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20			\newtheorem{algorithm}[theorem]{Algorithm}
21			\newtheorem{axiom}[theorem]{Axiom}
22			\newtheorem{case}[theorem]{Case}
23			\newtheorem{claim}[theorem]{Claim}
24			\newtheorem{conclusion}[theorem]{Conclusion}
25			\newtheorem{condition}[theorem]{Condition}
26			\newtheorem{conjecture}[theorem]{Conjecture}
27			\newtheorem{corollary}[theorem]{Corollary}
28			\newtheorem{criterion}[theorem]{Criterion}
29			\newtheorem{definition}[theorem]{Definition}
30			\newtheorem{example}[theorem]{Example}
31			\newtheorem{exercise}[theorem]{Exercise}
32			\newtheorem{lemma}[theorem]{Lemma}
33			\newtheorem{notation}[theorem]{Notation}
34			\newtheorem{problem}[theorem]{Problem}
35			\newtheorem{proposition}[theorem]{Proposition}
36			\newtheorem{remark}[theorem]{Remark}
37			\newtheorem{solution}[theorem]{Solution}
38			\newtheorem{summary}[theorem]{Summary}
39			\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
40			\begin{document}
41			\section{\label{Sec:pbc}Periodic Boundary Conditions}
42
43	tim	904	\textit{Periodic boundary conditions} are widely used to simulate truly
44			macroscopic systems with a relatively small number of particles. Simulation
45			box is replicated throughout space to form an infinite lattice. During the
46			simulation, when a particle moves in the primary cell, its periodic image
47			particles in other boxes move in exactly the same direction with exactly the
48			same orientation.Thus, as a particle leaves the primary cell, one of its
49			images will enter through the opposite face.If the simulation box is large
50			enough to avoid "feeling" the symmetric of the periodic lattice,the surface
51			effect could be ignored. Cubic and parallelepiped are the available periodic
52			cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe
53			the property of the simulation box. Therefore, not only the size of the
54			simulation box could be changed during the simulation, but also the shape of
55			it. The transformation from box space vector $\overrightarrow{s}$ to its
56			corresponding real space vector $\overrightarrow{r}$ is defined by
57	tim	903	\begin{equation}
58			\overrightarrow{r}=H\overrightarrow{s}%
59			\end{equation}
60
61
62	tim	904	where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three
63			box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the
64			simulation box respectively.
65	tim	903
66	tim	904	To find the minimum image, we need to convert the real vector to its
67			corresponding vector in box space first, \bigskip%
68	tim	903	\begin{equation}
69			\overrightarrow{s}=H^{-1}\overrightarrow{r}%
70			\end{equation}
71	tim	904	And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5
72			to 0.5,
73	tim	903	\begin{equation}
74	tim	904	s_{i}^{\prime}=s_{i}-round(s_{i})
75	tim	903	\end{equation}
76	tim	904	where%
77
78	tim	903	\begin{equation}
79	tim	904	round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{
80			}x\geqslant0
81	tim	903	\end{equation}
82	tim	904	%
83	tim	903
84	tim	904	\begin{equation}
85			round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0
86			\end{equation}
87	tim	903
88	tim	904
89			For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$.
90
91			Finally, we could get the minimum image by transforming back to real space,%
92
93	tim	903	\begin{equation}
94	tim	904	\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}%
95	tim	903	\end{equation}
96
97
98
99	tim	904	\end{document}