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 "feeling" 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{\underline{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, we convert the real vector to its corresponding
vector in box space first, \bigskip%
\begin{equation}
\mathbf{s}=\underline{\underline{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 by transforming back to real space,%

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


\end{document}
Revision:	928
Committed:	Tue Jan 13 15:24:22 2004 UTC (21 years, 3 months ago) by tim
Content type:	application/x-tex
File size:	3745 byte(s)
Log Message:	revision
#	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	928	%TCIDATA{LastRevised=Tuesday, January 13, 2004 10:22:03}
12	tim	903	%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
13			%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
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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			avoid "feeling" the symmetries of the periodic lattice, surface effects can be
51			ignored. Cubic, orthorhombic and parallelepiped are the available periodic
52			cells In OOPSE. We use a matrix to describe the property of the simulation
53			box. Therefore, both the size and shape of the simulation box can be changed
54			during the simulation. The transformation from box space vector $\mathbf{s}$
55			to its corresponding real space vector $\mathbf{r}$ is defined by
56	tim	903	\begin{equation}
57	tim	928	\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}%
58	tim	903	\end{equation}
59
60
61	tim	904	where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three
62	tim	928	box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the
63	tim	904	simulation box respectively.
64	tim	903
65	tim	928	To find the minimum image, we convert the real vector to its corresponding
66			vector in box space first, \bigskip%
67	tim	903	\begin{equation}
68	tim	928	\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}%
69	tim	903	\end{equation}
70	tim	928	And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5,
71	tim	903	\begin{equation}
72	tim	904	s_{i}^{\prime}=s_{i}-round(s_{i})
73	tim	903	\end{equation}
74	tim	928	where
75	tim	904
76			%
77	tim	903
78	tim	904	\begin{equation}
79	tim	928	round(x)=\left\{
80			\begin{array}
81			[c]{c}%
82			\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\
83			\lceil{x-0.5}\rceil & \text{otherwise}%
84			\end{array}
85			\right.
86	tim	904	\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	tim	928	Finally, we obtain the minimum image coordinates by transforming back to real space,%
92	tim	904
93	tim	903	\begin{equation}
94	tim	928	\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}%
95	tim	903	\end{equation}
96
97
98
99	tim	904	\end{document}