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. The
simulation box is replicated throughout space to form an infinite lattice.
During the simulation, when a particle moves in the primary cell, its image 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 \textquotedblleft feeling\textquotedblright\ the symmetries of the
periodic lattice, surface effects can be ignored. Cubic, orthorhombic and
parallelepiped are the available periodic cells In OOPSE. We use a matrix to
describe the property of the simulation box. Therefore, both the size and
shape of the simulation box can be changed during the simulation. The
transformation from box space vector $\mathbf{s}$ to its corresponding real
space vector $\mathbf{r}$ is defined by
\begin{equation}
\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{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 three sides of the
simulation box respectively.

To find the minimum image of a vector $\mathbf{r}$, we convert the real vector
to its corresponding vector in box space first, \bigskip%
\begin{equation}
\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}%
\end{equation}
And then, each element of $\mathbf{s}$ is wrapped 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)=\left\{
\begin{array}
[c]{c}%
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\
\lceil{x-0.5}\rceil & \text{otherwise}%
\end{array}
\right.
\end{equation}


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

Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by
transforming back to real space,%

\begin{equation}
\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}%
\end{equation}

\end{document}
Revision:	968
Committed:	Tue Jan 20 16:49:22 2004 UTC (21 years, 3 months ago) by tim
Content type:	application/x-tex
File size:	3816 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	968	%TCIDATA{LastRevised=Tuesday, January 20, 2004 11:37:59}
12	tim	903	%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
13			%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
14			%TCIDATA{<META NAME="DocumentShell" CONTENT="Standard LaTeX\Standard LaTeX Article">}
15			%TCIDATA{ComputeDefs=
16			%$H$
17			%}
18			\newtheorem{theorem}{Theorem}
19			\newtheorem{acknowledgement}[theorem]{Acknowledgement}
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	tim	928	macroscopic systems with a relatively small number of particles. The
45			simulation box is replicated throughout space to form an infinite lattice.
46			During the simulation, when a particle moves in the primary cell, its image in
47			other boxes move in exactly the same direction with exactly the same
48			orientation.Thus, as a particle leaves the primary cell, one of its images
49			will enter through the opposite face.If the simulation box is large enough to
50	tim	968	avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the
51			periodic lattice, surface effects can be ignored. Cubic, orthorhombic and
52			parallelepiped are the available periodic cells In OOPSE. We use a matrix to
53			describe the property of the simulation box. Therefore, both the size and
54			shape of the simulation box can be changed during the simulation. The
55			transformation from box space vector $\mathbf{s}$ to its corresponding real
56			space vector $\mathbf{r}$ is defined by
57	tim	903	\begin{equation}
58	tim	968	\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}%
59	tim	903	\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	tim	928	box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the
64	tim	904	simulation box respectively.
65	tim	903
66	tim	968	To find the minimum image of a vector $\mathbf{r}$, we convert the real vector
67			to its corresponding vector in box space first, \bigskip%
68	tim	903	\begin{equation}
69	tim	968	\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}%
70	tim	903	\end{equation}
71	tim	928	And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5,
72	tim	903	\begin{equation}
73	tim	904	s_{i}^{\prime}=s_{i}-round(s_{i})
74	tim	903	\end{equation}
75	tim	928	where
76	tim	904
77			%
78	tim	903
79	tim	904	\begin{equation}
80	tim	928	round(x)=\left\{
81			\begin{array}
82			[c]{c}%
83			\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\
84			\lceil{x-0.5}\rceil & \text{otherwise}%
85			\end{array}
86			\right.
87	tim	904	\end{equation}
88	tim	903
89	tim	904
90			For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$.
91
92	tim	968	Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by
93			transforming back to real space,%
94	tim	904
95	tim	903	\begin{equation}
96	tim	968	\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}%
97	tim	903	\end{equation}
98
99	tim	904	\end{document}