An ideal quantum gas in a finite-sized container

The role of finite-size effects in determining the thermodynamic behavior of an ideal gas is critically examined. While classical statistics do not produce any perceptible effects, quantum statistics do yield results that play a crucial role in determining the low-temperature behavior of the given system. For illustration, we carry out an exact analysis of the ideal Bose gas in one dimension and show that at least in this case (i) the bulk term (customarily obtained by replacing the summation-over-states by an integration) yields results that are, at best, misleading, whereas (ii) the correct behavior of the system is determined almost entirely by terms representing finite-size effects. The subtle, yet distinctive, role played by the boundary conditions imposed on the system is also explored.