Mittag-leffler Quantum Statistics Enable Bosonic and Fermionic Interpolation with Generalized Distributions

The behaviour of matter at extreme conditions often deviates from predictions based on standard statistical models, prompting scientists to explore new theoretical frameworks, and a team led by Maryam Seifi and Zahra Ebadi from University of Mohaghegh Ardabili, alongside Hamzeh Agahi and Hossein Mehri-Dehnavi from Babol Noshirvani University of Technology, now investigates a generalized approach to quantum statistics. Building on previous work, the researchers introduce novel statistical distributions, termed Mittag-Leffler Bose-Einstein and Mittag-Leffler Fermi-Dirac, which incorporate a parameter allowing for a smooth transition between behaviours expected of bosons and fermions, and importantly, capture effects arising from systems far from equilibrium. Their analysis reveals that these distributions predict unexpected phenomena, including Bose-Einstein-like condensation without interactions and negative heat capacity at low temperatures, demonstrating the crucial role of statistical deformation in shaping macroscopic properties and offering new insights into the behaviour of complex systems. These findings represent a significant advance in statistical mechanics, potentially impacting our understanding of diverse physical phenomena from condensed matter physics to astrophysics.

Mittag-Leffler Functions Model Non-Extensive Systems

Scientists have developed a powerful new framework for understanding complex systems by applying the Mittag-Leffler function to statistical mechanics and thermodynamics. This research explores how these functions, which extend the properties of exponential functions, accurately describe systems exhibiting non-extensive behaviour, memory effects, and long-range interactions, moving beyond the limitations of traditional approaches. The work provides a more realistic model for a wide range of physical phenomena. The core of this research lies in the Mittag-Leffler function, a mathematical tool that generalizes the exponential function and allows for the description of systems that deviate from standard Boltzmann-Gibbs statistics.

Non-extensive statistical mechanics modifies the traditional assumption that a system’s entropy is directly proportional to the number of accessible microstates, accounting for long-range interactions and correlations. Researchers explored how these functions define thermodynamic potentials like energy, entropy, and free energy in non-extensive systems, and utilized geometric approaches to analyze stability and phase transitions. Potential applications of this framework span diverse fields, including astrophysical plasmas, disordered systems like glasses and amorphous materials, nuclear matter at high densities and temperatures, complex fluids, and ultracold quantum gases. This work connects to a broad range of research areas, including non-extensive statistical mechanics, fractal geometry, long-range interactions, memory effects, complex systems, and geometric thermodynamics. The mathematical rigor and novel insights presented in this paper open up new avenues for research in statistical mechanics and thermodynamics.

Deformed Fermi-Dirac and Bose-Einstein Statistics Explained

Scientists have developed a novel statistical framework by generalizing conventional Bose-Einstein and Fermi-Dirac distributions using the Mittag-Leffler function, creating the Mittag-Leffler Bose-Einstein (MLBE) and Mittag-Leffler Fermi-Dirac (MLFD) distributions. This generalization incorporates a deformation parameter, alpha, which allows for a continuous transition between bosonic and fermionic statistics and inherently captures non-equilibrium effects. The study pioneers an approach to analyze the geometry associated with these new distributions, revealing significant departures from standard statistical models and highlighting the role of statistical deformation in determining macroscopic phenomena. Researchers investigated the microcanonical formalism to extract heat capacity from fluctuations in the kinetic energy of nuclear fragments.

The team related observed kinetic energy fluctuations to the curvature of the entropy surface, by leveraging model systems. These combined experimental and theoretical results provide robust evidence for the existence of negative heat capacity in finite, isolated systems, establishing it as a diagnostic of non-extensive or first-order phase transition behaviour. To contextualize these findings, scientists compared the thermodynamic geometry and heat capacity of the ML-based statistics with those of Tsallis and Kaniadakis statistics. The team explained that the Tsallis function modifies the intensity of statistical interactions, while the Kaniadakis function reveals more nuanced geometric behaviour. The ML statistics further generalize this picture, allowing for both modification of interaction strengths and qualitative crossovers between bosonic- and fermionic-like behaviour.

Deformed Statistics Reveal Novel Condensation and Heat Capacity

This work extends established statistical frameworks by introducing the Mittag Leffler Bose Einstein (MLBE) and Mittag Leffler Fermi Dirac (MLFD) distributions, generalizations of the conventional Bose-Einstein and Fermi-Dirac statistics. These new distributions incorporate a deformation parameter, alpha, which allows for a continuous transition between bosonic and fermionic behaviour while also capturing effects beyond standard equilibrium conditions. Analysis reveals that the MLBE distribution exhibits Bose-Einstein-like condensation even without interactions, a significant departure from traditional models, while the MLFD distribution displays unconventional features, including negative heat capacity at low temperatures. Researchers calculated internal energy and total particle number using integrals involving the Mittag-Leffler function, defining a generalized integral function to simplify these calculations.

This formulation provides a unified approach for analyzing quantum gases under these generalized statistical conditions. Further investigation employed thermodynamic geometry to explore the statistical properties of these distributions. Calculations of the metric tensor components and their derivatives were then used to compute the Ricci scalar curvature, providing insights into the geometric structure and thermodynamic implications of the MLBE and MLFD distributions.

Mittag-Leffler Statistics Reveal Novel Thermodynamic Behaviour

This research introduces two new statistical distributions, the Mittag Leffler Bose Einstein (MLBE) and Mittag Leffler Fermi Dirac (MLFD), which extend conventional quantum statistics by incorporating a deformation parameter. By utilizing the Mittag-Leffler function, the team successfully generalized the Bose-Einstein and Fermi-Dirac distributions, creating a framework capable of modelling systems exhibiting non-equilibrium effects and behaviours beyond those predicted by standard statistical models. Analysis reveals that the MLBE distribution exhibits Bose-Einstein-like condensation even without interactions, a significant departure from traditional models.