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Physical review. B, Condensed matter

Thermodynamic properties of particles with intermediate statistics

G S Joyce, Sarben Sarkar, Józef Spałek, Krzysztof Byczuk

Published Jan 15, 1996• 23 citations
Publisher

Abstract

Analytic expressions for the distribution function of an ideal gas of particles (exclusons) which have statistics intermediate between Fermi-Dirac and Bose-Einstein are obtained for all values of the Haldane statistics parameter \ensuremath{\alpha}\ensuremath{\in}[0,1]. The analytic structure of the distribution function is investigated and found to have no singularities in the physical region when the parameter \ensuremath{\alpha} lies in the range 0\ensuremath{\le}1. High- and low-temperature series are also derived for the internal energy E and heat capacity ${\mathit{C}}_{\mathit{V}}$ of the D-dimensional excluson gas. The low-temperature series for the thermodynamic properties illustrate the pseudofermion nature of exclusons. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.990

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Analytic expressions for the distribution function of an ideal gas of particles (exclusons) which have statistics intermediate between Fermi-Dirac and Bose-Einstein are obtained for all values of the Haldane statistics parameter \ensuremath{\alpha}\ensuremath{\in}[0,1]. The analytic structure of the distribution function is investigated and found to have no singularities in the physical region when the parameter \ensuremath{\alpha} lies in the range 0\ensuremath{\le}1. High- and low-temperature series are also derived for the internal energy E and heat capacity CV{\mathit{C}}_{\mathit{V}}CV​ of the D-dimensional excluson gas. The low-temperature series for the thermodynamic properties illustrate the pseudofermion nature of exclusons. \textcopyright{} 1996 The American Physical Society.
We overwiev the properties of a quantum gas of particles with the intermediate statistics defined by Haldane. Although this statistics has no direct connection to the symmetry of the multiparticle wave function, the statistical distribution function interpolates continuously between the Fermi-Dirac and the Bose-Einstein limits. We present an explicit solution of the transcendental equation for the didtribution function in a general case, as well as determine the thermodynamic properties in both low- and high-temperature limits.
Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
The calculation of the thermodynamic properties in the Bose-Einstein distribution is presented.The behaviour of the Bose-gas in the Bose-Einstein distribution is analysed based on high and low temperature dependence.At > all particles are in the excited state and the ground state is essentially occupied.But at B T T < particles gradually falls to the ground state and eventually contains all the particles in the system as .0 → TThe analytical expressions for some important parameters such as Bose-Einstein condensation of an ideal Bose gas, the population density of the excited and ground state particles are also presented.Heart capacity of an ideal Bose gas and total internal energy are also calculated and the results presented graphically.The results obtained are consistent with those found in the literatures.
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.
Beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles, a unified quantum statistics, which may interpolate smoothly between Bose-Einstein and Fermi-Dirac statistics, is proposed. The commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. Besides the Bose-Einstein condensation, another kind of particle condensation may appear.
The statistical distribution function of anyon is used to find the eighth viral coefficient in the high-temperature limit and the equation of state in the low-temperature limit. The perturbative results indicate that the thermodynamic quantities, Q(\ensuremath{\alpha}), of the free anyon gas may be factorized in the terms characteristic of the ideal Bose (\ensuremath{\alpha}=0) and fermion (\ensuremath{\alpha}=1) gases, i.e., Q(\ensuremath{\alpha})=\ensuremath{\alpha}Q(1)+(1-\ensuremath{\alpha})Q(0). It is shown that the factorizable property of the thermodynamic quantities, to all orders, can be established from the property of the equivalence between the anyon statistics and statistics in a system with boson-fermion transmutation, which was found by us in a recent paper [W. H. Huang, Phys. Rev. E 51, 3729 (1995)]. \textcopyright{} 1996 The American Physical Society.