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\documentclass{article}%
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{graphicx}%
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\setcounter{MaxMatrixCols}{30}
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%TCIDATA{OutputFilter=latex2.dll}
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%TCIDATA{Version=5.00.0.2552}
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%TCIDATA{CSTFile=40 LaTeX article.cst}
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%TCIDATA{Created=Friday, September 19, 2003 08:29:53}
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%TCIDATA{LastRevised=Tuesday, January 06, 2004 17:35:24}
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%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
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%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
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%TCIDATA{<META NAME="DocumentShell" CONTENT="Standard LaTeX\Standard LaTeX Article">}
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%TCIDATA{ComputeDefs=
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%$H$
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%}
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\newtheorem{theorem}{Theorem}
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\newtheorem{acknowledgement}[theorem]{Acknowledgement}
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\newtheorem{algorithm}[theorem]{Algorithm}
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\newtheorem{axiom}[theorem]{Axiom}
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\newtheorem{case}[theorem]{Case}
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\newtheorem{claim}[theorem]{Claim}
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\newtheorem{conclusion}[theorem]{Conclusion}
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\newtheorem{condition}[theorem]{Condition}
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\newtheorem{conjecture}[theorem]{Conjecture}
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{criterion}[theorem]{Criterion}
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{example}[theorem]{Example}
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\newtheorem{exercise}[theorem]{Exercise}
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{notation}[theorem]{Notation}
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\newtheorem{problem}[theorem]{Problem}
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{remark}[theorem]{Remark}
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\newtheorem{solution}[theorem]{Solution}
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\newtheorem{summary}[theorem]{Summary}
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\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
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\begin{document}
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\section{\label{Sec:pbc}Periodic Boundary Conditions}
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\textit{Periodic boundary conditions} are widely used to simulate
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truly macroscopic systems with a relatively small number of
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particles. Simulation box is replicated throughout space to form
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an infinite lattice. During the simulation, when a particle moves
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in the primary cell, its periodic image particles in other boxes
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move in exactly the same direction with exactly the same
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orientation.Thus, as a particle leaves the primary cell, one of
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its images will enter through the opposite face.If the simulation
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box is large enough to avoid "feeling" the symmetric of the
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periodic lattice,the surface effect could be ignored. For the time
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being, Cubic and parallelepiped are the available periodic cells
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used in OOPSE.
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\bigskip In OOPSE, we use the matrix instead of vector to describe the property
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of the simulation box. Therefore, not only the size of the
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simulation box could be changed during the simulation, but also
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the shape of it.
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The transformation from box space vector
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$\overrightarrow{s}$to its corresponding real space vector
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$\overrightarrow{r}$ is defined by
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\begin{equation}
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\overrightarrow{r}=H\overrightarrow{s}%
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\end{equation}
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where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up
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of the three box axis vector. h_{x},h_{y} and h_{z} represent the
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sides of the simulation box respectively. Thus H matrix becomes
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To find the minimum image, we need to convert the real vector to
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its corresponding vector in box space first,
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\bigskip
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\begin{equation}
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\overrightarrow{s}=H^{-1}\overrightarrow{r}%
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\end{equation}
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And then, each element of \overrightarrow{s} is casted to lie
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between -0.5 to 0.5
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\begin{equation}
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\s_{i}=s_{i}-round(s_{i})%
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\end{equation}
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where
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\begin{equation}
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\round(x)=(x+0.5)(x-0.5)%
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\end{equation}
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For example, round(3.6)=4, round(3.1) = 3, round(-3.6) = -4,
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round(-3.1)=-3.
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\begin{equation}
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\overrightarrow{s}=H^{-1}\overrightarrow{r}%
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\end{equation}
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\end{document}
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