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:	972
Committed:	Wed Jan 21 18:40:38 2004 UTC (21 years, 9 months ago) by mmeineke
Content type:	application/x-tex
File size:	3719 byte(s)
Log Message:	added changes into the Empirical Energy section
#	User	Rev	Content
1	mmeineke	972	\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 20, 2004 11:37:59}
12			%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			\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 \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			\begin{equation}
58			\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}%
59			\end{equation}
60
61
62			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 three sides of the
64			simulation box respectively.
65
66			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			\begin{equation}
69			\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}%
70			\end{equation}
71			And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5,
72			\begin{equation}
73			s_{i}^{\prime}=s_{i}-round(s_{i})
74			\end{equation}
75			where
76
77			%
78
79			\begin{equation}
80			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			\end{equation}
88
89
90			For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$.
91
92			Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by
93			transforming back to real space,%
94
95			\begin{equation}
96			\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}%
97			\end{equation}
98
99			\end{document}