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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク radial distribution function ： ウィキペディア英語版 radial distribution function In statistical mechanics, the radial distribution function, (or pair correlation function) $g(r)$ in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If a given particle is taken to be at the origin O, and if $\backslash rho\; =N/V$ is the average number density of particles, then the local time-averaged density at a distance $r$ from O is $\backslash rho\; g(r)$. This simplified definition holds for a homogeneous and isotropic system. A more general case will be considered below. In simplest terms it is a measure of the probability of finding a particle at a distance of $r$ away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of r and r+dr away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and blue particles are those which are within the circular shell, dotted in orange. The RDF is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The histogram is then normalized with respect to an ideal gas, where particle histograms are completely uncorrelated. For three dimensions, this normalization is the number density of the system multiplied by the volume of the spherical shell, which mathematically can be expressed as $g(r)\_I\; =\; 4\backslash pi\; r^2\backslash rho\; dr$, where $\backslash rho$ is the number density. Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein-Zernike equation, using approximative closure relations like the Percus-Yevick approximation or the Hypernetted Chain Theory. It can also be determined experimentally, by radiation scattering techniques or by direct visualization for large enough (micrometer-sized) particles via traditional or confocal microscopy. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. Moreover, by the reversion of the Kirkwood-Buff theory, it is possible to attain the microscopic details of the radial distribution function from the macroscopic properties. ==Definition== Consider a system of $N$ particles in a volume $V$ (for an average number density $\backslash rho\; =N/V$) and at a temperature $T$ (let us also define $\backslash textstyle\; \backslash beta\; =\; \backslash frac$). The particle coordinates are $\backslash mathbf\_$, with $\backslash textstyle\; i\; =\; 1,\; \backslash ,\; \backslash ldots,\; \backslash ,\; N$. The potential energy due to the interaction between particles is $\backslash textstyle\; U\_\; (\backslash mathbf\_\backslash ,\; \backslash ldots,\; \backslash ,\; \backslash mathbf\_)$ and we do not consider the case of an externally applied field. The appropriate averages are taken in the canonical ensemble $(N,V,T)$, with $\backslash textstyle\; Z\_\; =\; \backslash int\; \backslash cdots\; \backslash int\; \backslash mathrm^\; \backslash mathbf\_1\; \backslash cdots\; \backslash mathrm\; \backslash mathbf\_N$ the configurational integral, taken over all possible combinations of particle positions. The probability of an elementary configuration, namely finding particle 1 in $\backslash textstyle\; \backslash mathrm\; \backslash mathbf\_1$, particle 2 in $\backslash textstyle\; \backslash mathrm\; \backslash mathbf\_2$, etc. is given by } \, \mathrm \mathbf_1 \cdots \mathrm \mathbf_N\, .|}} The total number of particles is huge, so that $P^$ in itself is not very useful. However, one can also obtain the probability of a reduced configuration, where the positions of only particles are fixed, in $\backslash textstyle\; \backslash mathbf\_\backslash ,\; \backslash ldots,\; \backslash ,\; \backslash mathbf\_$, with no constraints on the remaining $N-n$ particles. To this end, one has to integrate () over the remaining coordinates $\backslash mathbf\_\backslash ,\; \backslash ldots,\; \backslash ,\; \backslash mathbf\_$: : $P^(\backslash mathbf\_1,\backslash ldots,\backslash mathbf\_n)\; =\backslash frac\; \backslash int\; \backslash cdots\; \backslash int\; \backslash mathrm^\; \backslash ,\; \backslash mathrm\; \backslash mathbf\_\; \backslash cdots\; \backslash mathrm\; \backslash mathbf\_N\; \backslash ,$. The particles being identical, it is more relevant to consider the probability that ''any'' $n$ of them occupy positions $\backslash textstyle\; \backslash mathbf\_\backslash ,\; \backslash ldots,\; \backslash ,\; \backslash mathbf\_$ in ''any'' permutation, thus defining the $n$-''particle density'' For $n=1$, () gives the one-particle density which, for a crystal, is a periodic function with sharp maxima at the lattice sites. For a (homogeneous) liquid, it is independent of the position $\backslash textstyle\; \backslash mathbf\_1$ and equal to the overall density of the system: : $\backslash frac\; \backslash int\; \backslash rho^(\backslash mathbf\_1)\; \backslash ,\; \backslash mathrm\; \backslash mathbf\_1\; =\; \backslash rho^\; =\; \backslash frac\; =\; \backslash rho\; \backslash ,$ It is now time to introduce a correlation function $g^$ by $g^$ is called a correlation function, since if the atoms are independent from each other $\backslash rho^$ would simply equal $\backslash rho^$ and therefore $g^$ corrects for the correlation between atoms. From () and () it follows that 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「radial distribution function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.