Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Paragas

Shelomo Izhaq Ben-Abraham

Published Nov 1, 1970• 10 citations
Publisher

Abstract

A hypothetical gas of identical spinless particles obeying r-parastatistics is dealt with. The method allows a unified treatment of both the Fermi and the Bose gases. The partition function, the equation of state, and the thermodynamic functions are simple generalizations of those for the physical quantum gases, which are included as special cases. All paragases have a Fermi energy and a Bose-Einstein condensation. A quantitative criterion of degeneracy for the physical quantum gases is proposed. The fluctuations of occupation numbers and density are also discussed. Expansions for high and low temperatures are given to third order.

DOI
10.1119/1.1976093

Related papers

Quantum ideal gases: Fermi-Dirac and Bose-Einstein statistics

Mark E. Tuckerman

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

•Aug 1, 2023•0 citations

Chemical potentials and thermodynamic characteristics of ideal Bose- and Fermi-gases in the region of quantum degeneracy

Andrii Sotnikov, K. V. Sereda, Yu. V. Slyusarenko

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.

Low Temperature Physics•Jan 1, 2017•13 citations

Quantum principles and free particles

nasa

The quantum principles that establish the energy levels and degeneracies needed to evaluate the partition functions are explored. The uncertainty principle is associated with the dual wave-particle nature of the model used to describe quantized gas particles. The Schroedinger wave equation is presented as a generalization of Maxwell's wave equation; the former applies to all particles while the Maxwell equation applies to the special case of photon particles. The size of the quantum cell in phase space and the representation of momentum as a space derivative operator follow from the uncertainty principle. A consequence of this is that steady-state problems that are space-time dependent for the classical model become only space dependent for the quantum model and are often easier to solve. The partition function is derived for quantized free particles and, at normal conditions, the result is the same as that given by the classical phase integral. The quantum corrections that occur at very low temperatures or high densities are derived. These corrections for the Einstein-Bose gas qualitatively describe the condensation effects that occur in liquid helium, but are unimportant for most practical purposes otherwise. However, the corrections for the Fermi-Dirac gas are important because they quantitatively describe the behavior of high-density conduction electron gases in metals and explain the zero point energy and low specific heat exhibited in this case.

•Feb 15, 2006•0 citations

Degenerate Quantum Gases and Bose-Einstein Condensation

Luca Salasnich

After a brief historical introduction to Bose-Einstein condensation and Fermi degeneracy, we discuss theoretical results we have recentely obtained for trapped degenerate quantum gases by means of a thermal field theory approach. In particular, by using Gross-Pitaevskii and Bogoliubov-Popov equations, we consider thermodynamical properties of two Bosonic systems: a gas of Lithium atoms and a gas of Hydrogen atoms. Finally, we investigate finite-temperature density profiles of a dilute Fermi gas of Potassium atoms confined in a harmonic potential.

arXiv:cond-mat/0012223•Dec 13, 2000•0 citations

Dilute-Degenerate Gases

Franck Laloë

We study the properties of quantum gases which are dilute in terms of the interactions but degenerate; this may be the case if the thermal wavelength of the particles is much larger than the interatomic potential range. We assume that the potential contains a short range repulsive part, which creates strong correlations between particles that cannot be treated by ordinary perturbation theory, even if the region of configuration space where they are dominant is relatively small. Our method combines two basic ingredients: Ursell operators, which generalize the Ursell functions that are usually used to generate quantum cluster expansions, but are not in themselves symmetrized (the operators can be related to an auxiliary system of distinguishable particles); in a distinct step, an exact treatment of particle indistinguishability is given through the inclusion of exchange cycles of arbitrary length. For non-condensed boson systems, as well as for fermion systems, the method provides a compact expression of the partition function that generalizes the well-known Beth–Uhlenbeck formula to arbitrary degeneracy. For the study of the Bose–Einstein condensation, our treatment retains more information on the short range correlations than, for instance, the Hartree–Fock method. It also introduces a natural and explicit distinction between particles and quasiparticles; the latter may condense into a single one-particle state, which obeys a non-linear equation somewhat similar to the Gross–Pitaevskii equation.

Cambridge University Press eBooks•Apr 6, 1995•1 citation

The Bose-Einstein statistics of particles, with special reference to the case of low temperatures

R. Dingle

In this paper, it has been shown that the usual demonstrations of the phenomenon of Bose-Einstein condensation are invalid. Exact formal solutions have been derived for the partition functions and the mean occupation numbers in classical and Bose statistics, and the theory of fluctuations in the case of Bose statistics has been given. In a second paper, convenient approximations will be derived for the mean occupation numbers in Bose statistics, and it will be shown that for temperatures well above the critical temperature of the supposed Bose-Einstein condensation the well-known result for the mean occupation numbers is valid, whilst for temperatures rather lower than this critical value it will be shown that particles will condense into the lowest available state. The critical temperatureT0is given by where the summation is over all energy levels except the lowest.

Mathematical Proceedings of the Cambridge Philosophical Society•Apr 1, 1949•17 citations
A hypothetical gas of identical spinless particles obeying r-parastatistics is dealt with. The method allows a unified treatment of both the Fermi and the Bose gases. The partition function, the equation of state, and the thermodynamic functions are simple generalizations of those for the physical quantum gases, which are included as special cases. All paragases have a Fermi energy and a Bose-Einstein condensation. A quantitative criterion of degeneracy for the physical quantum gases is proposed. The fluctuations of occupation numbers and density are also discussed. Expansions for high and low temperatures are given to third order.
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
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 quantum principles that establish the energy levels and degeneracies needed to evaluate the partition functions are explored. The uncertainty principle is associated with the dual wave-particle nature of the model used to describe quantized gas particles. The Schroedinger wave equation is presented as a generalization of Maxwell's wave equation; the former applies to all particles while the Maxwell equation applies to the special case of photon particles. The size of the quantum cell in phase space and the representation of momentum as a space derivative operator follow from the uncertainty principle. A consequence of this is that steady-state problems that are space-time dependent for the classical model become only space dependent for the quantum model and are often easier to solve. The partition function is derived for quantized free particles and, at normal conditions, the result is the same as that given by the classical phase integral. The quantum corrections that occur at very low temperatures or high densities are derived. These corrections for the Einstein-Bose gas qualitatively describe the condensation effects that occur in liquid helium, but are unimportant for most practical purposes otherwise. However, the corrections for the Fermi-Dirac gas are important because they quantitatively describe the behavior of high-density conduction electron gases in metals and explain the zero point energy and low specific heat exhibited in this case.
After a brief historical introduction to Bose-Einstein condensation and Fermi degeneracy, we discuss theoretical results we have recentely obtained for trapped degenerate quantum gases by means of a thermal field theory approach. In particular, by using Gross-Pitaevskii and Bogoliubov-Popov equations, we consider thermodynamical properties of two Bosonic systems: a gas of Lithium atoms and a gas of Hydrogen atoms. Finally, we investigate finite-temperature density profiles of a dilute Fermi gas of Potassium atoms confined in a harmonic potential.
We study the properties of quantum gases which are dilute in terms of the interactions but degenerate; this may be the case if the thermal wavelength of the particles is much larger than the interatomic potential range. We assume that the potential contains a short range repulsive part, which creates strong correlations between particles that cannot be treated by ordinary perturbation theory, even if the region of configuration space where they are dominant is relatively small. Our method combines two basic ingredients: Ursell operators, which generalize the Ursell functions that are usually used to generate quantum cluster expansions, but are not in themselves symmetrized (the operators can be related to an auxiliary system of distinguishable particles); in a distinct step, an exact treatment of particle indistinguishability is given through the inclusion of exchange cycles of arbitrary length. For non-condensed boson systems, as well as for fermion systems, the method provides a compact expression of the partition function that generalizes the well-known Beth–Uhlenbeck formula to arbitrary degeneracy. For the study of the Bose–Einstein condensation, our treatment retains more information on the short range correlations than, for instance, the Hartree–Fock method. It also introduces a natural and explicit distinction between particles and quasiparticles; the latter may condense into a single one-particle state, which obeys a non-linear equation somewhat similar to the Gross–Pitaevskii equation.
In this paper, it has been shown that the usual demonstrations of the phenomenon of Bose-Einstein condensation are invalid. Exact formal solutions have been derived for the partition functions and the mean occupation numbers in classical and Bose statistics, and the theory of fluctuations in the case of Bose statistics has been given. In a second paper, convenient approximations will be derived for the mean occupation numbers in Bose statistics, and it will be shown that for temperatures well above the critical temperature of the supposed Bose-Einstein condensation the well-known result for the mean occupation numbers is valid, whilst for temperatures rather lower than this critical value it will be shown that particles will condense into the lowest available state. The critical temperatureT0is given by where the summation is over all energy levels except the lowest.