In this paper, the effects of boundary and connectivity on ideal gases in two-dimensional confined space and three-dimensional tubes are discussed in detail based on the analytical result. The implication of such effects on the mesoscopic system is also revealed.
The main aim of this paper is twofold: (1) revealing a relation between the counting function N (λ) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K (t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N (λ), for there is no effective method for calculating N (λ) beyond leading order. We suggest a new expression of N (λ) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N (λ) from a nonuniformly convergent expansion of K (t). We calculate N (λ) for D-dimensional boxes, three-dimensional balls, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnet theorem, we generalize the simply-connected heat kernel to the multiply-connected case; this result proves Kac's conjecture on the two-dimensional multiply-connected heat kernel. The approaches for calculating eigenvalue spectra and state densities from N (λ) are introduced.
In this paper we calculate some exact solutions of the grand partition functions for quantum gases in confined space, such as ideal gases in two-and three-dimensional boxes, in tubes, in annular containers, on the lateral surface of cylinders, and photon gases in three-dimensional boxes. Based on these exact solutions, which, of course, contain the complete information about the system, we discuss the geometry effect which is neglected in the calculation with the thermodynamic limit V → ∞, and analyze the validity of the quantum statistical method which can be used to calculate the geometry effect on ideal quantum gases confined in two-dimensional irregular containers. We also calculate the grand partition function for phonon gases in confined space. Finally, we discuss the geometry effects in realistic systems.
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