  We consider N bosons occupying a discrete set of single-particle quantum
states in an isolated trap. Usually, for a given excitation energy, there are
many combinations of exciting different number of particles from the ground
state, resulting in a fluctuation of the ground state population. As a counter
example, we take the quantum spectrum to be logarithms of the prime number
sequence, and using the fundamental theorem of arithmetic, find that the ground
state fluctuation vanishes exactly for all excitations. The use of the standard
canonical or grand canonical ensembles, on the other hand, gives substantial
number fluctuation for the ground state. This difference between the
microcanonical and canonical results cannot be accounted for within the
framework of equilibrium statistical mechanics.
