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Physical review. E

Statistical model describing Bose-Einstein and Fermi-Dirac statistics

Chi-Tau Yan

Published Dec 14, 2021• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.104.064118

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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.
A new kind of quantum statistics which interpolates between Bose and Fermi statistics is proposed beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles. The quantum 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. The application of this quantum statistics for the ensemble of extremal black holes are discussed.
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
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.
Synopsis: The connection between a particle’s spin and the statistics it obeys is given. Bose–Einstein and Fermi–Dirac statistics are explained and illustrated with products of wave functions. A new type of particle basis state is described, and two possibilities for the commutation properties of the associated raising (“creation”) and lowering (“annihilation”) ladder operators are examined, giving rise to the Bose–Einstein and Fermi–Dirac cases. The relation between particle occupation number and energy is inferred from detailed balance, and cubical enclosures of particles are studied. Two physics mysteries encountered in the earlier chapters (zweideutigkeit and the vanishing electronic component of specific heat) are explained. The Hartree–Fock theoretical framework for the approximate treatment of many-particle atomic systems is constructed.
Synopsis: ƒe connection between a particle’s spin and the statistics it obeys is given. Bose-Einstein and Fermi-Dirac statistics are explained and illustrated with products of wave functions. A new type of particle basis state is described and two possibilities for the commutation properties of the associated raising (“creation”) and lowering (“annihilation”) ladder operators are examined, giving rise to the Bose-Einstein and Fermi-Dirac cases. ƒe relation between particle occupation number and energy is inferred from detailed balance, and cubical enclosures of particles are studied. Two physics mysteries encountered in the earlier chapters (“zweideutigkeit” and the vanishing electronic component of specific heat) are explained. ƒe Hartree-Fock theoretical framework for the approximate treatment of many-particle atomic systems is constructed.
Overview Classical Statistical Mechanics (CSM) ignores the quantum nature of particles. This introduces three errors in the calculation of the partition function. First is the assumption of distinguishability of individual particles not possible in the microscopic world. Second, CSM neglects the discrete nature of system’s energy levels. Third, CSM fails to account for the spin of the particles which divides all particles into two types: boson and fermion, each following its own law of statistical distribution among energy levels. While CSM is impressively successful in describing observed properties of many naturally occurring large scale phenomena (particularly in liquids and biological systems), it fails to describe many phenomena where quantum nature of particles and quantum statistics play important role in determining properties such as electrical and thermal conductivity (particularly in solid materials) and superfluidity in liquid helium. It is not just at low temperatures where quantum nature is important, but also when the particle is intrinsically quantum, such as an electron. Quantum Statistical Mechanics (QSM) is also needed to understand biochemical reactions, like enzyme kinetics and optical properties of DNA. In this chapter, we discuss elementary aspects of quantum Statistical Mechanics, with detailed treatment of ideal Bose gas, leading to Bose-Einstein condensation.