Locate the ideal Bose condensation threshold

Choose the one-particle ground energy as zero and assume one internal state. At positive temperature, each ideal Bose occupation contains energy minus chemical potential in an exponential. The chemical potential must be negative in a finite grand-canonical system for the ground-state geometric sum to converge. Lowering temperature at fixed density raises fugacity toward one. The zero-energy ground occupation can then grow large even though the continuum density of states vanishes at zero energy.

That last observation explains why a naive continuum integral loses the condensate: multiplying by a square-root density of states assigns zero measure to one special discrete level. Keep its occupation separate. The excited-state integral is controlled by a dimensionless Bose function of order three halves. Its largest value occurs as fugacity approaches one from below and is the Riemann zeta value about 2.612. This finite upper bound is the source of the threshold, not an attractive force between ideal particles.

Set the total density equal to the largest excited density to define the thermodynamic critical temperature. At fixed density, lowering temperature below that value makes the maximum excited population fall as temperature to the three-halves power. The remaining fraction occupies the ground state. In the infinite-volume fixed-density limit the chemical potential approaches the ground energy, while the condensate population remains extensive. Keeping chemical potential exactly at zero in a finite grand sum would instead make that sum diverge.

The low-energy convergence depends on dimension. In a uniform ideal gas with quadratic dispersion, the excited-number integral at zero chemical potential converges only above two spatial dimensions. Thus the same finite-temperature homogeneous ideal-gas condensation mechanism fails in one and two dimensions in the thermodynamic limit. Trapping potentials change the density of states and therefore the conclusion. One cannot transfer the three-dimensional box formula to every laboratory trap merely by inserting its particle number.

Finite systems show a smooth crossover rather than the exact thermodynamic nonanalyticity. When the ground occupation is large, its finite-system fugacity is approximately one minus the reciprocal of that occupation, not exactly one. Moreover, the ideal grand-canonical ground-state number variance can be of order the mean squared. Matching a mean condensate fraction therefore does not prove that every ensemble predicts its fluctuations equally; exact number constraints remain significant for that global observable.

Condensation also does not by itself establish superfluid response. An ideal Bose gas's occupation counting omits the interaction-dependent excitation structure relevant to flow stability and collective modes. The lab can test the quadratic relation between a normalized thermal wavelength and its square, then infer the temperature scaling on paper. It does not calculate a Bose function, detect a condensate, or demonstrate superfluidity, and its clock has no meaning as a cooling protocol.