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
#	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 13, 2004 10:22:03}
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 truly
44	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	\begin{equation}
57	\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}%
58	\end{equation}
59
60
61	where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three
62	box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the
63	simulation box respectively.
64
65	To find the minimum image, we convert the real vector to its corresponding
66	vector in box space first, \bigskip%
67	\begin{equation}
68	\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}%
69	\end{equation}
70	And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5,
71	\begin{equation}
72	s_{i}^{\prime}=s_{i}-round(s_{i})
73	\end{equation}
74	where
75
76	%
77
78	\begin{equation}
79	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	\end{equation}
87
88
89	For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$.
90
91	Finally, we obtain the minimum image coordinates by transforming back to real space,%
92
93	\begin{equation}
94	\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}%
95	\end{equation}
96
97
98
99	\end{document}