Mittag-Leffler Quantum Statistics and Thermodynamic Anomalies

TL;DR

This work generalizes quantum statistics by replacing the exponential weight in BE/FD distributions with a Mittag–Leffler function, introducing a deformation parameter $\alpha$ that interpolates between fermionic and bosonic behavior and encodes nonequilibrium effects. The authors formulate the Mittag–Leffler Bose–Einstein (MLBE) and Mittag–Leffler Fermi–Dirac (MLFD) distributions, derive thermodynamic quantities via generalized integrals $\mathcal{F}^{a}_{n,\alpha}(z)$, and analyze the thermodynamic geometry through the curvature $R$, revealing fermionic-to-bosonic crossovers, condensation-like divergences at $z=1$, and $\alpha$-dependent anomalies in heat capacity. They show that MLBE can exhibit BE-like condensation and that MLFD can display negative $C_V$ at low $T$ for $\alpha>1$, highlighting a statistical origin of such anomalies. The paper also compares ML statistics with Tsallis and Kaniadakis formalisms, applies MLBE to a generalized Debye solid, and discusses experimental relevance and future directions in relativistic fields, multi-parameter ML extensions, and ultracold/nanoscale systems.

Abstract

Building upon the framework established in our recent work [M. Seifi et al., Phys. Rev. E 111, 054114 (2025)], wherein a generalized Maxwell Boltzmann distribution was formulated using the Mittag Leffler function within the superstatistical formalism, we extend this approach to the quantum domain. Specifically, we introduce two statistical distributions,termed the Mittag Leffler Bose Einstein (MLBE) and Mittag Leffler Fermi Dirac (MLFD) distributions, constructed by generalizing the conventional Bose-Einstein and Fermi-Dirac distributions through the Mittag-Leffler function. This generalization incorporates a deformation parameter (α), which facilitates a continuous interpolation between bosonic and fermionic statistics, while inherently capturing nonequilibrium effects and generalized thermodynamic behavior. We analyze the thermodynamic geometry associated with these distributions and identify significant departures from standard statistical models. Notably, the MLBE distribution manifests a Bose-Einstein-like condensation even in the absence of interactions, whereas the MLFD distribution exhibits unconventional features, such as negative heat capacity in the low-temperature regime. These findings highlight the pivotal role of statistical deformation in determining emergent macroscopic thermodynamic phenomena.

Figures (10)

• Figure 1: The occupation number $n_{\text{ML}}(x)$ associated with the MLFD distribution is presented as a function of the variable $x$ for several values of the parameter $\alpha$. The solid orange curve corresponds to the standard FD distribution, obtained when $\alpha = 1$.
• Figure 2: The occupation number $n_{\text{ML}}(x)$ associated with the MLBE distribution is presented as a function of the variable $x$ for several values of the parameter $\alpha$. The solid orange curve corresponds to the standard BE distribution, obtained when $\alpha = 1$.
• Figure 3: The thermodynamic curvature associated with the MLBE distribution is presented as a function of fugacity for the interval $0 < \alpha \leq 1$, under isothermal conditions ($\beta = 1$). Dashed lines correspond to specific cases where $\alpha = 0.3, 0.5, 0.8$, and the respective values of $z^{*}$ are indicated by solid circles for each value of $\alpha$.
• Figure 4: Thermodynamic curvature of an MLBE distribution as a function of fugacity, plotted for the range $(1 < \alpha)$ under isothermal conditions ($\beta = 1$). Dashed lines represent the values $\alpha = 1.1, 1.2, 1.3$.
• Figure 5: Thermodynamic curvature of an MLFD distribution as a function of fugacity, plotted for the range $1 < \alpha$ under isothermal conditions ($\beta = 1$). Dashed lines represent the values $\alpha = 0.3, 0.5, 0.8$.