The formulation of the previous paper (paper IV) is extended so that it becomes applicable in an interacting system in the presence of a Bose-Einstein degeneracy. This extension is carried out by the introduction of an $x$ -ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.The formulation of the previous paper (paper IV) is extended so that it becomes applicable in an interacting system in the presence of a Bose-Einstein degeneracy. This extension is carried out by the introduction of an $x$-ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.

Physical ReviewFeb 15, 196059 citations

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

T. D. Lee, C. N. Yang

Birkhäuser Boston eBooksJan 1, 19861 citation

The Gases That Think in Quanta: A Narrative Review of Quantum Statistical Mechanics, from Bose-Einstein Condensation to Degenerate Fermi Gases

Zen Revista, 10 PHYSICS

Quantum statistical mechanics is the arithmetic of indistinguishable particles: bosons that pile into a single state and fermions that refuse to share---statistics whose consequences range from the Bose-Einstein condensate, a macroscopic quantum state, to the Fermi gas's pressure that holds white dwarfs against collapse. This article presents a narrative review of the primary literature of that statistics, from Bose's Planck's law and the light quantum hypothesis and Einstein's quantum theory of the monatomic ideal gas with its 1925 second paper predicting the condensation, through Fermi's quantization of the monatomic perfect gas and Dirac's theory of quantum mechanics, which independently founded the Fermi-Dirac statistics, London's identification of the lambda-phenomenon of liquid helium with the Bose-Einstein degeneracy, the experimental vindication of Anderson, Ensher, Matthews, Wieman, and Cornell's observation of Bose-Einstein condensation in a dilute atomic vapor and Davis and colleagues' sodium condensate, DeMarco and Jin's onset of Fermi degeneracy in a trapped atomic gas, and the syntheses of Leggett's fundamental concepts, Pitaevskii and Stringari's Bose-Einstein condensation, and Bloch, Dalibard, and Zwerger's many-body physics with ultracold gases. The synthesis is organized around three themes: the statistics' invention, in which indistinguishability's consequences were computed before any condensate existed; the condensate's realization, in which the dilute atomic vapors delivered Einstein's sixty-year-old prediction; and the many-body era, in which the ultracold gases became quantum simulators of matter's collective behavior. It is concluded that quantum statistics is the theory whose consequences were verified in reverse---predicted abstractly, found in liquid helium, created in atomic vapors---and that its bosons and fermions remain the universe's two grammars.Quantum statistical mechanics is the arithmetic of indistinguishable particles: bosons that pile into a single state and fermions that refuse to share---statistics whose consequences range from the Bose-Einstein condensate, a macroscopic quantum state, to the Fermi gas's pressure that holds white dwarfs against collapse. This article presents a narrative review of the primary literature of that statistics, from Bose's Planck's law and the light quantum hypothesis and Einstein's quantum theory of the monatomic ideal gas with its 1925 second paper predicting the condensation, through Fermi's quantization of the monatomic perfect gas and Dirac's theory of quantum mechanics, which independently founded the Fermi-Dirac statistics, London's identification of the lambda-phenomenon of liquid helium with the Bose-Einstein degeneracy, the experimental vindication of Anderson, Ensher, Matthews, Wieman, and Cornell's observation of Bose-Einstein condensation in a dilute atomic vapor and Davis and colleagues' sodium condensate, DeMarco and Jin's onset of Fermi degeneracy in a trapped atomic gas, and the syntheses of Leggett's fundamental concepts, Pitaevskii and Stringari's Bose-Einstein condensation, and Bloch, Dalibard, and Zwerger's many-body physics with ultracold gases. The synthesis is organized around three themes: the statistics' invention, in which indistinguishability's consequences were computed before any condensate existed; the condensate's realization, in which the dilute atomic vapors delivered Einstein's sixty-year-old prediction; and the many-body era, in which the ultracold gases became quantum simulators of matter's collective behavior. It is concluded that quantum statistics is the theory whose consequences were verified in reverse---predicted abstractly, found in liquid helium, created in atomic vapors---and that its bosons and fermions remain the universe's two grammars.

Aug 29, 20260 citations

A Simple Model of Bose–Einstein Condensation of Interacting Particles

Yu. M. Poluektov

arXiv:1602.02746Journal of Low Temperature PhysicsDec 8, 201616 citations

Bosonic stimulation and the irreproducibility of condensate fragmentation

P. Kumar

It is pointed out that the quantum statistical phenomenon of Bosonic stimulation, inherent to Bose statistics and the associated Bose-Einstein correlation, can be effectively mapped on to the statistical problem of the Polya urn scheme. Thus, we predict an irreproducibility for the limiting non-degenerate values of the relative populations of two, or more equivalent but separated condensates resulting from the fragmentation of a given source condensate. Experiments are proposed that should verify this prediction $-$ using the dilute gas Bose-Einstein condensates, or a pulsed degenerate multi-mode laser, where one would look for the run-to-run (or the pulse-to-pulse) fluctuations of the relative populations.It is pointed out that the quantum statistical phenomenon of Bosonic stimulation, inherent to Bose statistics and the associated Bose-Einstein correlation, can be effectively mapped on to the statistical problem of the Polya urn scheme. Thus, we predict an irreproducibility for the limiting non-degenerate values of the relative populations of two, or more equivalent but separated condensates resulting from the fragmentation of a given source condensate. Experiments are proposed that should verify this prediction $-$ using the dilute gas Bose-Einstein condensates, or a pulsed degenerate multi-mode laser, where one would look for the run-to-run (or the pulse-to-pulse) fluctuations of the relative populations.

arXiv:cond-mat/0204443Apr 20, 20022 citations

x

-ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.

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using the dilute gas Bose-Einstein condensates, or a pulsed degenerate multi-mode laser, where one would look for the run-to-run (or the pulse-to-pulse) fluctuations of the relative populations.