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<ノート>多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(11)量子統計力学(3)

邦夫 橋爪

Published Dec 1, 2009• 0 citations
OpenAlex

Abstract

ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~

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ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~
distribution in an external potential directly probes the equation of state under the local density approximation. The equation of state is obtained in two ways: from the local density of the gas, reconstructed using the inverse Abel transformation; and from the local pressure, directly deduced from the 2D column density via an exact relation including the Gaussian potential. Regions of low density allow us to extract the chemical potential and the temperature using the virial expansion of the equation of state. We validate our method using the non-interacting Fermi gas. The experimental results are compared to recent Monte-Carlo calculations. 1 AFOSR-MURI, DARPA-OLE, Alfred P. Sloan Foundation, NSF
Abstract In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padé approximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.
The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality ddd. In particular, the convergence radius ρc(d)ρ_c(d)ρc​(d) of the expansion is particulary large at {\em exactly\/} d=3d=3d=3 dimensions, ρc(3)=7.1068…×lim⁡d→3ρc(d)ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)ρc​(3)=7.1068…×limd→3​ρc​(d). The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.
The internal energy and chemical potential of a noninteracting Fermi gas has been evaluated in three ways: numerically (applicable to arbitrary temperatures), through an asymptotic power series expansion (suitable for low temperatures), and by means of standard Padé approximants (appropriate for low and moderate temperatures). The main conclusion of this work is that the Padé approximant, which is constructed from the asymptotic series employing only minimal labor, is far superior to the truncated series expansion.