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

TL;DR

The paper introduces the Mittag-Leffler Maxwell-Boltzmann (MLMB) distribution, a generalized MB statistic within the superstatistics framework, parameterized by $α$ to quantify statistical interactions. By replacing the exponential with the Mittag-Leffler function, the authors derive thermodynamic quantities via $\mathcal{H}_{n,α}(z)$ and compute the thermodynamic metric and curvature, uncovering fermion-like repulsion for $0<α<1$ and boson-like attraction with a condensation-like divergence at $z_c^{α}$ for $1<α≤1.5$. Key contributions include the explicit expressions for $U$, $N$, and $C_v$, the identification of a critical fugacity and temperature $z_c^{α}$ and $T_c^{α}$, and the interpretation of curvature signs as indicators of intrinsic statistical interactions. The framework offers a tunable bridge between classical MB statistics and quantum-like behaviors, with potential impact on non-equilibrium, long-range interacting, and complex systems modeling where non-exponential statistics prevail.

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, $α$, 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.

Figures (11)

• Figure 1: $n_{\text{ML}}(X)$ of the MLMB distribution as a function of $X$, plotted for the range $0 \leq \alpha \leq 1$. The solid purple line represents the case $\alpha = 1$ (MB distribution), while the other lines correspond to $\alpha = 0.6, 0.7, 0.8, 0.9$.
• Figure 2: $n_{\text{ML}}(X)$ of the MLMB distribution as a function of $X$, plotted for the range $1 \leq \alpha \leq 1.5$. Solid lines indicate positive values of $n_{\text{ML}}(X)$ for $X > X_{\mathrm{th}}^{\alpha}$, while dashed lines denote negative values for $X < X_{\mathrm{th}}^{\alpha}$. Circular markers represent the threshold points $X_{\mathrm{th}}^{\alpha}$ for each $\alpha$, where the sign of $n_{\text{ML}}(X)$ changes. The purple solid line corresponds to $\alpha = 1$, which represents the MB distribution. The inset provides a detailed view near the threshold points, emphasizing the distinct behavior of $n_{\text{ML}}(X)$ for $\alpha = 1.1, 1.2, 1.3, 1.4$.
• Figure 3: Variation of $X$ as a function of $1 < \alpha \leq 1.5$. The dashed purple line represents the values of $X_{\mathrm{th}}^{\alpha}$. The cyan regions above the dashed line indicate where $n_{\mathrm{ML}}(X)$ is positive, while the white regions below the dashed line indicate where $n_{\mathrm{ML}}(X)$ is negative.
• Figure 4: Variation of $z$ with respect to $1 < \alpha \leq 1.5$. The dashed purple line represents $z_{\mathrm{th}}^{\alpha}$ values. The cyan areas (below the dashed line) indicate where $n_{\mathrm{ML}}(X)$ is positive, while the white areas (above the dashed line) indicate where $n_{\mathrm{ML}}(X)$ is negative.
• Figure 5: Thermodynamic curvature as a function of fugacity for an ideal gas, comparing the Bose-Einstein distribution (red solid line) and the Fermi-Dirac distribution (blue solid line), under isothermal conditions ($\beta = 1$).