Intrinsic attractive and repulsive interactions: From classical to quantum statistics in the generalized Maxwell-Boltzmann distribution

The thermodynamic parameter space is flat for an ideal classical gas with noninteracting particles. In contrast, for an ideal quantum Bose (Fermi) gas, the thermodynamic curvature is positive (negative), indicating intrinsic attractive (repulsive) interactions. We generalize the classical Maxwell-Boltzmann distribution by employing a generalized form of the exponential function, proposing the Mittag-Leffler Maxwell-Boltzmann distribution within the framework of superstatistics. We demonstrate that the generalization parameter, α, quantifies the statistical interaction. When α=1, the distribution coincides with the standard classical Maxwell-Boltzmann distribution, where no statistical interaction is present. For 0<α<1 (α>1), the statistical interaction is repulsive (attractive), corresponding to a negative (positive) thermodynamic curvature of the system.