Bose-Einstein statistics for a finite number of particles

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