Bose-Einstein statistics for a finite number of particles

This paper presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation-namely, the fraction of particles in the ground state and the specific heat-are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a three-dimensional box, the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit-here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.

Automated summary

This study presents an exact calculation of thermodynamical quantities related to Bose-Einstein condensation in arbitrary quantum systems, without relying on the thermodynamic limit. Results show that the derivative of specific heat in a three-dimensional box gas reaches smaller values than expected in the thermodynamic limit, confirming the importance of understanding the role of the thermodynamic limit in phase transitions. This approach allows for precise calculations irrespective of temperature, providing a way to further study BE statistics without relying on approximations near critical temperature or the thermodynamic limit.