Alpha Statistics Framework Bridges Bose, Einstein and Fermi, Dirac Regimes for ≤ 1, Reveals Interaction Signatures

The behaviour of particles collectively dictates the properties of matter, and understanding the rules governing these interactions remains a central challenge in physics. M. H. Naghizadeh Ardabili, Omid Yahyayi Monem, and Morteza Nattagh Najafi, all from the Department of Physics at the University of Mohaghegh Ardabili, alongside Hosein Mohammadzadeh, present a new statistical framework, termed ‘alpha statistics’, that elegantly unifies and extends the well-known Bose-Einstein and Fermi-Dirac behaviours. This innovative approach introduces a parameter which effectively models particle interactions, and crucially, allows exploration of previously inaccessible hyperbosonic regimes. By employing geometrical analysis, the team reveals a direct connection between the strength of these interactions and the curvature of the statistical system, identifying a critical temperature that signals a transition between attractive and repulsive behaviours, and offering new insights into the nature of condensation phenomena.

Alpha Statistics and Generalized Particle Indistinguishability

Researchers propose and investigate alpha statistics, a new quantum statistical framework extending beyond Bose, Einstein, Fermi, and Dirac statistics, even encompassing hyperbosonic behaviours for certain parameter values. Inspired by Haldane’s exclusion principle, this framework introduces a parameter, alpha, which governs how particles interact and are distinguished from one another. The investigation explores the thermodynamic properties of ideal gases within this framework, revealing behaviours not captured by conventional statistical methods. Specifically, the team examines how the gas responds to changes in temperature and the alpha parameter, demonstrating a smooth transition between bosonic and fermionic behaviours. This approach provides a versatile tool for modelling diverse physical systems, including those exhibiting exotic quantum phenomena and strong correlations, and offers new insights into the fundamental principles governing many-particle systems.

The formulation introduces a modified way to calculate particle occupation, encoding effective statistical interactions through the alpha parameter. Using thermodynamic geometry, the team analyses the curvature of the system as a diagnostic of underlying interactions and potential phase structures. A crossover temperature, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose condensation temperature, the ratio of this crossover temperature to the condensation temperature depends universally on the alpha parameter. For hyperbosonic statistics, the team finds curvature singularities.

Riemannian Geometry Reveals Quantum Statistical Framework

This research presents a thorough investigation into a generalized quantum statistical framework, extending beyond traditional Bose-Einstein and Fermi-Dirac statistics. The use of Riemannian geometry, specifically thermodynamic curvature, to characterize the interactions and phase transitions within this framework is a powerful and insightful approach. The novelty of this work lies in the exploration of alpha-statistics and its geometric characterization, providing a unified way to understand different quantum statistics. The mathematical rigor demonstrated, combined with a comprehensive literature review, positions the work within the broader context of existing research. The geometric interpretation of thermodynamic curvature as a measure of interactions and phase transitions is particularly insightful, and the linking of thermodynamic geometry to information geometry adds another layer of depth and potential applications.

Alpha Statistics and Statistical Interactions Revealed

This research introduces alpha statistics, a generalized statistical framework that extends beyond traditional Bose, Einstein, Fermi, and Dirac statistics to encompass a wider range of particle behaviours. The team demonstrates how this framework smoothly interpolates between these established statistics and even extends into the hyperbosonic regime, utilising a parameter to encode effective statistical interactions. By employing geometric analysis, specifically examining the curvature of the system, they identify signatures of underlying interactions and potential phase structures, revealing a crossover temperature that distinguishes between attractive and repulsive statistical regimes. The findings establish a clear link between the fractional exclusion principle and curvature-induced interaction signatures, offering a novel way to characterise statistical systems. Importantly, the research highlights unique condensation phenomena for hyperbosonic statistics, indicated by curvature singularities at specific fugacities, suggesting behaviours distinct from conventional Bose condensation.