Thermodynamic geometry of inclusion statistics

Abstract

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 $1/2<D/\sigma \le 1$ where standard bosons do not condense, while for $D/\sigma \le 1/2$ 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.