Thermodynamic Geometry of Classical and Quantum Statistics in the Relativistic Regime

Abstract: We investigate the thermodynamic geometry of classical and quantum ideal gases in the relativistic regime, with particular emphasis on the effects of particle mass and spatial dimensionality. Relativistic kinematics is incorporated through the full energy-momentum dispersion relation and the corresponding relativistic density of states. Using the Fisher-Rao information metric derived from the partition function, we analyze the thermodynamic curvature for Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Exact analytical expressions are obtained in two spatial dimensions, while the three-dimensional case is studied numerically. We show that the thermodynamic curvature preserves its characteristic sign-positive for bosons and negative for fermions; even in the relativistic regime, reflecting effective attractive and repulsive statistical interactions, respectively. A distinctive relativistic effect is the shift of curvature singularities from the non-relativistic critical point to a mass-dependent threshold at $μ=mc^{2}$. In addition, the relativistic Bose-Einstein condensation temperature is evaluated, revealing explicit mass-dependent corrections to the non-relativistic result. These findings provide a unified geometric perspective on relativistic statistical systems and clarify the interplay between quantum statistics, relativistic kinematics, and critical behavior.