Geometry Effects In Finite-size Systems

In this paper, we systemically discuss methods on how to calculate the effect of boundary in finite-size systems. We first introduce in detail two works on how to calculate boundary effect in quantum statistical mechanics: one , based on M. Kac's famous mathematical work, obtains the equation of state for ideal gases in two-dimensional confined space with irregular boundaries, and gives the result for three-dimensional long tubes, of which all transverse cross-sections keep the same, and we extend this result to the case of long and narrow irregular three-dimensional containers; the other, for systems whose energy spectra are known, performs the sum over all states in grand potentials directly, and obtains the equation of state exactly. Based on those two works, we calculate and discuss the influence of boundary effects on thermodynamic quantities. In the low-temperature and high-density limit, we obtain the boundary effects on the thermodynamic quantities of two- and three-dimensional ideal quantum gases in finite-size systems. Furthermore, we discuss the essential difference between boundary effects on two-dimensional Bose and Fermi systems: besides the size and shape of the system, the boundary effect on the Bose gas depends almost only on the temperature, while the boundary effect on the Fermi gas depends almost only on the density, and we analyze the reason for such difference. By introducing a modified virial expansion approach, we calculate the boundary effects on the thermodynamic quantities of two- and three-dimensional systems in the high-temperature and low-density limit. Moreover we compare the boundary effects in two- and three-dimensional systems, and find that under the similar conditions, boundary effects in three-dimensional cases are of the same order of those in two-dimensional cases. We then compare boundary effects with statistical fluctuations both for Bose and Fermi gases under various conditions in small-size realistic systems. Finally, we show that when the boundary modification is comparable to the thermodynamic quantities obtained under the thermodynamic limit approximation, the result of the standard statistical mechanics will be invalid.