Haldane-Inspired Generalized Statistics

TL;DR

alpha statistics introduces a continuous interpolation between Bose--Einstein and Fermi--Dirac statistics through a single parameter $\alpha$, extending to hyperbosonic behavior for $\alpha<0$. The authors employ thermodynamic geometry to compute the two-dimensional curvature $R$ in $(\beta,\gamma)$ space, revealing a crossover temperature $T^*$ at which the curvature changes sign and showing that $T^*/T_c$ depends uniquely on $\alpha$. In the $\alpha<0$ regime, curvature singularities occur at fugacities $z_c<1$, indicating modified condensation phenomena beyond standard Bose condensation. The work links generalized exclusion principles to curvature-driven interactions, provides a practical framework to compute thermodynamic quantities via a $w$-variable transformation, and suggests rich phase behavior in generalized quantum gases.

Abstract

We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for $α< 0$. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter $α$. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature $T^{*}$, 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--Einstein condensation temperature $T_{c}$, the ratio $T^{*}/T_{c}$ depends universally on $α$. For negative $α$, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.

Figures (7)

• Figure 1: The number of possible configurations for distributing $N_i = 55$ particles among $G_i = 94$ states as a function of the fractional parameter $\alpha$. The solid line represents the Haldane generalization, and the dashed line represents the proposed generalization.
• Figure 2: Distribution function of alpha statistics as a function of $\beta(\epsilon - \mu)$ for different values of $\alpha$.
• Figure 3: Thermodynamic curvature as a function of fugacity for isothermal processes ($\beta = 1$) in a 3-dimensional ideal $\alpha$-statistics gas with $\sigma = 2$ (non-relativistic regime).
• Figure 4: Thermodynamic curvature as a function of fugacity for isothermal processes ($\beta = 1$) in a 2-dimensional ideal $\alpha$-statistics gas with $\sigma = 1$ (ultra-relativistic regime).
• Figure 5: The dependence of $z^*$ on $\alpha$ delineates two regions: a fermion-like region and a boson-like region. These regions are separated by the $z^*$ curve, corresponding to the points where the thermodynamic curvature changes sign.