Thermodynamic Geometry of Alpha Statistics: A Unified Quantum Statistical Framework

We propose and study a generalized quantum statistical framework–referred to as alpha statistics–that continuously interpolates between Bose–Einstein and Fermi–Dirac statistics and naturally extends into the hyperbosonic regime for [Formula: see text]. Inspired by Haldane’s exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter [Formula: see text]. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature [Formula: see text], at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose–Einstein condensation temperature [Formula: see text], the ratio [Formula: see text] depends universally on [Formula: see text]. For negative [Formula: see text], corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.