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. For the time
being, Cubic and parallelepiped are the available periodic cells
used in OOPSE.
\bigskip In OOPSE, we use the matrix instead of 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 vector. h_{x},h_{y} and h_{z} represent the
sides of the simulation box respectively. Thus H matrix becomes


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}=s_{i}-round(s_{i})%
\end{equation}
where
\begin{equation}
\round(x)=(x+0.5)(x-0.5)%
\end{equation}

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

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


\end{document}
Revision:	903
Committed:	Wed Jan 7 03:53:53 2004 UTC (21 years, 4 months ago) by tim
Content type:	application/x-tex
File size:	3590 byte(s)
Log Message:	Periodic Boundary Conditions
#	Content
1	\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	%TCIDATA{LastRevised=Tuesday, January 06, 2004 17:35:24}
12	%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	\textit{Periodic boundary conditions} are widely used to simulate
44	truly macroscopic systems with a relatively small number of
45	particles. Simulation box is replicated throughout space to form
46	an infinite lattice. During the simulation, when a particle moves
47	in the primary cell, its periodic image particles in other boxes
48	move in exactly the same direction with exactly the same
49	orientation.Thus, as a particle leaves the primary cell, one of
50	its images will enter through the opposite face.If the simulation
51	box is large enough to avoid "feeling" the symmetric of the
52	periodic lattice,the surface effect could be ignored. For the time
53	being, Cubic and parallelepiped are the available periodic cells
54	used in OOPSE.
55	\bigskip In OOPSE, we use the matrix instead of vector to describe the property
56	of the simulation box. Therefore, not only the size of the
57	simulation box could be changed during the simulation, but also
58	the shape of it.
59	The transformation from box space vector
60	$\overrightarrow{s}$to its corresponding real space vector
61	$\overrightarrow{r}$ is defined by
62	\begin{equation}
63	\overrightarrow{r}=H\overrightarrow{s}%
64	\end{equation}
65
66	where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up
67	of the three box axis vector. h_{x},h_{y} and h_{z} represent the
68	sides of the simulation box respectively. Thus H matrix becomes
69
70
71	To find the minimum image, we need to convert the real vector to
72	its corresponding vector in box space first,
73	\bigskip
74	\begin{equation}
75	\overrightarrow{s}=H^{-1}\overrightarrow{r}%
76	\end{equation}
77	And then, each element of \overrightarrow{s} is casted to lie
78	between -0.5 to 0.5
79	\begin{equation}
80	\s_{i}=s_{i}-round(s_{i})%
81	\end{equation}
82	where
83	\begin{equation}
84	\round(x)=(x+0.5)(x-0.5)%
85	\end{equation}
86
87	For example, round(3.6)=4, round(3.1) = 3, round(-3.6) = -4,
88	round(-3.1)=-3.
89
90	\begin{equation}
91	\overrightarrow{s}=H^{-1}\overrightarrow{r}%
92	\end{equation}
93
94
95
96	\end{document}