  This article 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 3-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.
