Interacting quantum gases in confined space: Two- and three-dimensional equations of state

Abstract

In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for three-dimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.