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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{LastRevised=Tuesday, January 20, 2004 11:37:59} |
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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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\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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\textit{Periodic boundary conditions} are widely used to simulate truly |
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> |
macroscopic systems with a relatively small number of particles. The |
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> |
simulation box is replicated throughout space to form an infinite lattice. |
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> |
During the simulation, when a particle moves in the primary cell, its image in |
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other boxes 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 its images |
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will enter through the opposite face.If the simulation box is large enough to |
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avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the |
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periodic lattice, surface effects can be ignored. Cubic, orthorhombic and |
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parallelepiped are the available periodic cells In OOPSE. We use a matrix to |
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describe the property of the simulation box. Therefore, both the size and |
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shape of the simulation box can be changed during the simulation. The |
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transformation from box space vector $\mathbf{s}$ to its corresponding real |
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space vector $\mathbf{r}$ is defined by |
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\begin{equation} |
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\overrightarrow{r}=H\overrightarrow{s}% |
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\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
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\end{equation} |
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|
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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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|
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where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
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box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the |
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simulation box respectively. |
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|
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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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To find the minimum image of a vector $\mathbf{r}$, we convert the real vector |
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to its corresponding vector in box space first, \bigskip% |
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\begin{equation} |
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\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
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\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{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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And then, each element of $\mathbf{s}$ is wrapped to lie 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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s_{i}^{\prime}=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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% |
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|
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\begin{equation} |
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\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
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round(x)=\left\{ |
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\begin{array} |
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[c]{c}% |
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\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
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\lceil{x-0.5}\rceil & \text{otherwise}% |
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\end{array} |
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\right. |
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\end{equation} |
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For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
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\end{document} |
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Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
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transforming back to real space,% |
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|
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\begin{equation} |
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\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
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\end{equation} |
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|
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\end{document} |
