In this work we show that the introduction of a U(1) symmetry breaking field in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC) for strictly negative values of the chemical potential
μ. Finally, the proof of equivalence of limit pressures is extended to a class of full-diagonal models.
We investigate the thermodynamic geometry of an ideal quantum gas obeying inclusion statistics, characterized by a negative statistical parameter
g<0. In this framework the grand-canonical partition function admits a finite maximum fugacity, and the thermodynamic scalar curvature
R is strictly positive for all
g<0, reflecting effective attractive statistical interactions analogous to those of a bosonic system. As the fugacity approaches its maximum value,
R diverges, signaling a phase transition of the Bose-Einstein condensation type. A key distinction from the ordinary ideal Bose gas is that the condensation temperature is elevated relative to the bosonic case, and finite-temperature condensation occurs even in the dimensional regime
1/2<D/σ≤1 where standard bosons do not condense, while for
D/σ≤1/2 the transition temperature vanishes. Three independent criteria; divergence of
R, the maximum fugacity singularity, and the non-analytic cusp in the specific heat, coincide at the same condensation point, confirming the thermodynamic consistency of the transition.
It is well known that Bose–Einstein condensation cannot occur in a free two-dimensional (2D) system. Recently, several studies have showed that BEC can occur on the surface of a sphere. We investigate BEC on the surface of cylinder on both sides of which atoms are confined in a one-dimensional (1D) harmonic potential. In this work, only the non-interacting Bose gas is considered. We determine the critical temperature and the condensate fraction in the geometry using the semi-classical approximation. Moreover, the thermodynamic properties of ideal bosons are also studied using the grand canonical partition function.