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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 Critical exponent ： ウィキペディア英語版 Critical exponent Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on * the dimension of the system, * the range of the interaction, * the spin dimension. These properties of critical exponents are supported by experimental data. The experimental results can be theoretically achieved in mean field theory for higher-dimensional systems (4 or more dimensions). The theoretical treatment of lower-dimensional systems (1 or 2 dimensions) is more difficult and requires the renormalization group. Phase transitions and critical exponents appear also in percolation systems. However, here the critical dimension above which mean field exponents are valid is 6 and higher dimensions. Mean field critical exponents are also valid for random graphs, such as Erdos-Reyni graphd, which can be regarded as infinite dimensional systems. ==Definition== Phase transitions occur at a certain temperature, called the critical temperature $T\_c$. We want to describe the behaviour of a physical quantity $f$ in terms of a power law around the critical temperature. So we introduce the reduced temperature $\backslash tau\; :=\; (T-T\_c)/T\_c$, which is zero at the phase transition, and define the critical exponent $k$. :$k\; \backslash ,\; \backslash stackrel\; \backslash ,\; \backslash lim\_\; \backslash text$ This results in the power law we were looking for. :$f(\backslash tau)\; \backslash propto\; \backslash tau^k,\; \backslash quad\; \backslash tau\backslash approx\; 0\; \backslash text$ It is important to remember that this represents the asymptotic behavior of the function $f(\backslash tau)$ as $\backslash tau\; \backslash to\; 0$. More generally one might expect :$f(\backslash tau)=A\; \backslash tau^k\; (1+b\backslash tau\; ^\; +\; \backslash cdots)\; \backslash text$ 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「Critical exponent」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.