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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 Bose gas ： ウィキペディア英語版 Bose gas An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for photons, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate. == Thomas–Fermi approximation == The thermodynamics of an ideal Bose gas is best calculated using the grand partition function. The grand partition function for a Bose gas is given by: :$\backslash mathcal(z,\backslash beta,V)\; =\; \backslash prod\_i\; \backslash left(1-ze^\backslash right)^$ where each term in the product corresponds to a particular energy εi ; gi  is the number of states with energy εi ; ''z '' is the absolute activity (or "fugacity"), which may also be expressed in terms of the chemical potential μ by defining: :$z(\backslash beta,\backslash mu)=\; e^$ and β defined as: :$\backslash beta\; =\; \backslash frac$ where ''k '' is Boltzmann's constant and ''T '' is the temperature. All thermodynamic quantities may be derived from the grand partition function and we will consider all thermodynamic quantities to be functions of only the three variables ''z '', β (or ''T ''), and ''V ''. All partial derivatives are taken with respect to one of these three variables while the other two are held constant. It is more convenient to deal with the dimensionless grand potential defined as: :$\backslash Omega=-\backslash ln(\backslash mathcal)\; =\; \backslash sum\_i\; g\_i\; \backslash ln\backslash left(1-ze^\backslash right).$ Following the procedure described in the gas in a box article, we can apply the Thomas–Fermi approximation which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral: :$\backslash Omega\backslash approx\; \backslash int\_0^\backslash infty\; \backslash ln\backslash left(1-ze^\backslash right)\backslash ,dg.$ The degeneracy ''dg '' may be expressed for many different situations by the general formula: :$dg\; =\; \backslash frac\backslash ,\backslash frac\}\; ~dE$ where α is a constant, $E\_c$ is a "critical energy", and Γ is the Gamma function. For example, for a massive Bose gas in a box, α=3/2 and the critical energy is given by: :$\backslash frac=\backslash frac$ where Λ is the thermal wavelength. For a massive Bose gas in a harmonic trap we will have α=3 and the critical energy is given by: :$\backslash frac=\backslash frac$ where ''V(r)=mω2r2/2 '' is the harmonic potential. It is seen that ''Ec '' is a function of volume only. We can solve the equation for the grand potential by integrating the Taylor series of the integrand term by term, or by realizing that it is proportional to the Mellin transform of the Li1(z exp(-β E)) where Lis(x) is the polylogarithm function. The solution is: :$\backslash Omega\backslash approx-\backslash frac(z)\}.$ The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the Bose–Einstein condensate and will be dealt with in the next section. 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「Bose gas」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.