Intrinsic Attractive and Repulsive Interactions: From Classical to Quantum Gases in the Generalized Maxwell-Boltzmann Distribution

TL;DR

This paper introduces a generalized Maxwell-Boltzmann distribution using Mittag-Leffler functions, linking the parameter to statistical interactions and thermodynamic curvature, bridging classical and quantum gas behaviors.

Contribution

It proposes a new generalized distribution framework that quantifies statistical interactions via a parameter, extending classical and quantum gas models.

Findings

The generalized distribution reduces to classical when parameter equals one.

Parameter values determine the nature of statistical interactions (attractive or repulsive).

Thermodynamic curvature correlates with the distribution's generalized parameter.

Abstract

The thermodynamic parameter space is flat for an ideal classical gas with non-interacting 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, $\alpha$, quantifies the statistical interaction. When $\alpha = 1$, the distribution coincides with the standard classical Maxwell-Boltzmann distribution, where no statistical interaction is present. For $0 < \alpha < 1$ ($\alpha > 1$), the statistical interaction is repulsive (attractive), corresponding to a negative (positive) thermodynamic curvature of the system.