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Journal of Physics A Mathematical and General

Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics

Hiroshi Oshima, Tsunehiro Obata, Hiroaki Hara

Published Aug 26, 1999• 75 citations
Publisher

Abstract

The scalar curvature ( R ) of ideal quantum gases obeying Gentile's statistics is investigated by the method of information geometrical theory. The R value is specified by the fugacity and the maximum number, p , of particles in a state. The lowest case p = 1, corresponds to Fermi-Dirac statistics and the unbounded case, p , to Bose-Einstein statistics. In contrast to R = 0 for ideal classical gases obeying Boltzmann statistics, we find R = (2) 1/2 /32 for p 2 and R = -(2) 1/2 /32 for p = 1, in 0 which is the classical limit. This means that a quantum statistical character is left in R , in the classical limit. Also, a correlation between the sign of R and a quantum mechanical exchange effect is recognized for 0 and >>1. Furthermore, we obtain results that support the instability interpretation of R proposed by Janyszek and Mrugala.

DOI
10.1088/0305-4470/32/36/302

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The scalar curvature ( R ) of ideal quantum gases obeying Gentile's statistics is investigated by the method of information geometrical theory. The R value is specified by the fugacity and the maximum number, p , of particles in a state. The lowest case p = 1, corresponds to Fermi-Dirac statistics and the unbounded case, p , to Bose-Einstein statistics. In contrast to R = 0 for ideal classical gases obeying Boltzmann statistics, we find R = (2) 1/2 /32 for p 2 and R = -(2) 1/2 /32 for p = 1, in 0 which is the classical limit. This means that a quantum statistical character is left in R , in the classical limit. Also, a correlation between the sign of R and a quantum mechanical exchange effect is recognized for 0 and >>1. Furthermore, we obtain results that support the instability interpretation of R proposed by Janyszek and Mrugala.
It is shown that the stability of ideal quantum gases can be measured by means of the Riemann scalar curvature R of the parameter space. The components of the metric tensor were assumed to be the second moments of energy and the number of particle fluctuations. As a result, R is a function of the second and third moments of those quantities. For bosons R is positive and increases monotonically from zero at the classical limit to positive infinity in the condensation region. A system is less stable if R is bigger and vice versa. For fermions R is negative and this means that Fermi gases are more stable than the ideal Bose and ideal classical systems.
We investigate the thermodynamic geometry of an ideal quantum gas obeying inclusion statistics, characterized by a negative statistical parameter g<0g < 0g<0. In this framework the grand-canonical partition function admits a finite maximum fugacity, and the thermodynamic scalar curvature RRR is strictly positive for all g<0g < 0g<0, reflecting effective attractive statistical interactions analogous to those of a bosonic system. As the fugacity approaches its maximum value, RRR diverges, signaling a phase transition of the Bose-Einstein condensation type. A key distinction from the ordinary ideal Bose gas is that the condensation temperature is elevated relative to the bosonic case, and finite-temperature condensation occurs even in the dimensional regime 1/2<D/σ≤11/2 < D/σ\leq 11/2<D/σ≤1 where standard bosons do not condense, while for D/σ≤1/2D/σ\leq 1/2D/σ≤1/2 the transition temperature vanishes. Three independent criteria; divergence of RRR, the maximum fugacity singularity, and the non-analytic cusp in the specific heat, coincide at the same condensation point, confirming the thermodynamic consistency of the transition.
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<0α< 0α<0. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter ααα. 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∗T^{*}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 TcT_{c}Tc​, the ratio T∗/TcT^{*}/T_{c}T∗/Tc​ depends universally on ααα. For negative ααα, 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.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
It is a long held conjecture in the connection between information geometry (IG) and thermodynamics that the curvature endowed by IG diverges at phase transitions. Recent work on the IG of Bose-Einstein (BE) gases challenged this conjecture by saying that in the limit of fugacity approaching unit-where BE condensation is expected-the curvature does not diverge; rather, it converges to zero. However, as the discontinuous behavior that identifies condensation is only observed at the thermodynamic limit, a study of the IG curvature at a finite number of particles, N, is in order from which the thermodynamic behavior can be observed by taking the thermodynamic limit ( N→∞) posteriorly. This article presents such a study. We find that for a trapped gas, as N increases, the values of curvature decrease proportionally to a power of N, while the temperature at which the maximum value of curvature occurs approaches the usually defined critical temperature. This means that, in the thermodynamic limit, the curvature has a limited value where a phase transition is observed, contradicting the forementioned conjecture.
In this paper, we consider the thermodynamic geometry of harmonic trapped Bose and Fermi gases. We construct a thermodynamic geometry for these systems and evaluate the corresponding metric tensor and thermodynamic curvature. By observing that the thermodynamic curvature of the trapped Bose (Fermi) gas is always positive (negative), we conclude that the trapping potential does not affect the intrinsic statistical interactions. By inspecting the singular points of the thermodynamic curvature, we argue that the confinement dimension and the frequency of the trapping potential change the critical value of fugacity for the Bose gas.