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

Distribution Functions of Finite-Temperature Ideal Gases Described by Parastatistics and Analysis Using the Two-Point Summation Method

Jun Iizuka, Yoshiki Ueoka

Published Feb 1, 2026• 0 citations
Publisher

Abstract

This is a preprint of a theoretical study in statistical mechanics. We propose a generalized statistical framework based on para-statistics,introducing a modified grand partition function and generalized distributionfunctions. The formulation systematically extends conventional Bose–Einsteinand Fermi–Dirac statistics. This work is independent of the authors’ previously published studies andpresents a new theoretical framework. This preprint is intended for submission to arXiv in the category ofstatistical mechanics.

DOI
10.5281/zenodo.18445970

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

DISTRIBUTION OF PARASTATISTICS FUNCTIONS: AN OVERVIEW OF THERMODYNAMICS PROPERTIES

R. Yosi Aprian Sari, Wipsar Sunu Brams Dwandaru

This study aims to determine the thermodynamic properties of the parastatistics system of order two. The thermodynamic properties to be searched include the Grand Canonical Partition Function (GCPF) Z, and the average number of particles N. These parastatistics systems is in a more general form compared to quantum statistical distribution that has been known previously, i.e.: the Fermi-Dirac (FD) and Bose-Einstein (BE). Starting from the recursion relation of grand canonical partition function for parastatistics system of order two that has been known, recuresion linkages for some simple thermodynamic functions for parastatistics system of order two are derived. The recursion linkages are then used to calculate the thermodynamic functions of the model system of identical particles with limited energy levels which is similar to the harmonic oscillator. From these results we concluded that from the Grand Canonical Partition Function (GCPF), Z, the thermodynamics properties of parastatistics system of order two (paraboson and parafermion) can be derived and have similar shape with parastatistics system of order one (Boson and Fermion). The similarity of the graph shows similar thermodynamic properties. Keywords: parastatistics, thermodynamic properties

Jurnal Sains Dasar•May 20, 2016•0 citations

General forms of statistical mechanics with special reference to the requirements of the new quantum mechanics

Ralph Howard Fowler

Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Dec 1, 1926•40 citations

Many-Body Problem in Quantum Statistical Mechanics. I. General Formulation

T. D. Lee, Chongming Yang

A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain $U$ functions defined for the same system with Boltzmann statistics. It is then shown that these $U$ functions can be evaluated in successive approximations in terms of a binary kernel $B$ which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.

Physical Review•Mar 1, 1959•210 citations

Many-Body Problem in Quantum Statistical Mechanics. IV. Formulation in Terms of Average Occupation Number in Momentum Space

T.D. Lee, Chongming Yang

Starting from Rules $A$ and $B$ of a previous paper (I), it is shown that the grand partition function can be evaluated in terms of the statistical averages of the occupation number in momentum space. The final formulation is in terms of a simple variational principle. The procedure represents a concise and complete separation of the effect of the Bose-Einstein or Fermi-Dirac statistical character of the particles from the dynamical problem. In the case of Bose statistics, this formulation makes possible a systematic computation of all thermodynamic functions near the Bose-Einstein transition point in the gaseous phase. Applications to a system of hard spheres are discussed.

Physical Review•Jan 1, 1960•124 citations

Boltzmann, Gibbs, and Darwin-Fowler Approaches in Parastatistics

Roberto Leon Ponczek, C. C. Yan

There are essentially three methods used in equilibrium statistical mechanics, namely, the Boltzmann method of identifying the equilibrium sta te with the most probable one; the Gibbs ensemble method of postulating a canonical distribution; and the Darwin-Fowler method' of identifying the equilibrium state with the average state. Most textbooks on statis tical mechanics include discussions of the first two methods. The in clusion into the text of the last method, which schrudinger2 seems to advocate, is less usual . The equivalence of those methods is usual 1 y demonstrated for the i deal Boltzmann, Bose - Einstein, and Fermi-Dirac sta tistics. However, the equivalence is not guaranteed when the methodsare applied to different statistics such as parastatistics3, where eachener gy leve1 can afford up to some number k of particles, and can be regar ded as a generalization of the above mentioned statistics. It seems desirable to confirm t he equivalence of those methods for parastatistics. In most textbooks however discussions on parastatistics are r ather rare.

Revista brasileira de fisica•Jan 1, 1976•0 citations
This is a preprint of a theoretical study in statistical mechanics. We propose a generalized statistical framework based on para-statistics,introducing a modified grand partition function and generalized distributionfunctions. The formulation systematically extends conventional Bose–Einsteinand Fermi–Dirac statistics. This work is independent of the authors’ previously published studies andpresents a new theoretical framework. This preprint is intended for submission to arXiv in the category ofstatistical mechanics.
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
This study aims to determine the thermodynamic properties of the parastatistics system of order two. The thermodynamic properties to be searched include the Grand Canonical Partition Function (GCPF) Z, and the average number of particles N. These parastatistics systems is in a more general form compared to quantum statistical distribution that has been known previously, i.e.: the Fermi-Dirac (FD) and Bose-Einstein (BE). Starting from the recursion relation of grand canonical partition function for parastatistics system of order two that has been known, recuresion linkages for some simple thermodynamic functions for parastatistics system of order two are derived. The recursion linkages are then used to calculate the thermodynamic functions of the model system of identical particles with limited energy levels which is similar to the harmonic oscillator. From these results we concluded that from the Grand Canonical Partition Function (GCPF), Z, the thermodynamics properties of parastatistics system of order two (paraboson and parafermion) can be derived and have similar shape with parastatistics system of order one (Boson and Fermion). The similarity of the graph shows similar thermodynamic properties. Keywords: parastatistics, thermodynamic properties
Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.
A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain UUU functions defined for the same system with Boltzmann statistics. It is then shown that these UUU functions can be evaluated in successive approximations in terms of a binary kernel BBB which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.
Starting from Rules AAA and BBB of a previous paper (I), it is shown that the grand partition function can be evaluated in terms of the statistical averages of the occupation number in momentum space. The final formulation is in terms of a simple variational principle. The procedure represents a concise and complete separation of the effect of the Bose-Einstein or Fermi-Dirac statistical character of the particles from the dynamical problem. In the case of Bose statistics, this formulation makes possible a systematic computation of all thermodynamic functions near the Bose-Einstein transition point in the gaseous phase. Applications to a system of hard spheres are discussed.
There are essentially three methods used in equilibrium statistical mechanics, namely, the Boltzmann method of identifying the equilibrium sta te with the most probable one; the Gibbs ensemble method of postulating a canonical distribution; and the Darwin-Fowler method' of identifying the equilibrium state with the average state. Most textbooks on statis tical mechanics include discussions of the first two methods. The in clusion into the text of the last method, which schrudinger2 seems to advocate, is less usual . The equivalence of those methods is usual 1 y demonstrated for the i deal Boltzmann, Bose - Einstein, and Fermi-Dirac sta tistics. However, the equivalence is not guaranteed when the methodsare applied to different statistics such as parastatistics3, where eachener gy leve1 can afford up to some number k of particles, and can be regar ded as a generalization of the above mentioned statistics. It seems desirable to confirm t he equivalence of those methods for parastatistics. In most textbooks however discussions on parastatistics are r ather rare.