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

Springer eBooks

Distribution Functions in Physics: Fundamentals

Mark S. Hillery, Robert F. O’Connell, Marlan O. Scully, Eugene Paul Wigner

Published Jan 1, 1997• 1,686 citations
Publisher
DOI
10.1007/978-3-642-59033-7_28

Related papers

Distribution functions in physics: Fundamentals

Mark S. Hillery, Robert F. O’Connell, Marlan O. Scully +1 more

Physics Reports•Apr 1, 1984•2,930 citations

Physical Value Primitives are Always Distributions, Never Single-Valued

F. Pat Eblen

Preprint paper abstract: We propose and develop the thesis that every physically real primitive quantity is fundamentally an ontic distribution rather than single-valued in any observable domain. This is not an epistemic claim about measurement uncertainty or observer knowlwdge, but an ontological claim: single-valued physical primitives do not exist in nature. We argue that this view is not merely philosophically tenable but is in fact demanded by three independent pillars of established physics: the Heisenberg Uncertainty Principle (HUP), the impossibility of zero entropy in physical systems, and the Bekenstein bound on information density in finite regions of spacetime [23]. We further develop the implications of this framework for quantum measurement theory, the interpretation of Bell test results [1–3, 31, 32], and the concept of entanglement. In particular, we introduce the concept of a prominence function P(λ,θ) as an ontic measure of the capacity of a distribution’s constituents to physically interact with a given context, and propose that measurement outcomes arise from a distribution-valued product of source distributions and detector context distributions. We argue this framework suggests a reinterpretation of Bell’s theorem in which the assumption of single-valued hidden variables is not merely a simplification but a fundamental category error. This work extends prior analysis of Bell’s theorem in the interval-based framework of [28] and the ontological wave function model of [29].

•Apr 14, 2026•0 citations

Probability Distributions in Statistical Physics

Jan Naudts

Springer eBooks•Jan 1, 2011•0 citations

Generalised partition functions: inferences on phase space distributions

Rudolf A Treumann, W. Baumjohann

Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.

arXiv:1604.05295•Annales Geophysicae•Jun 2, 2016•1 citation

Quantum Mechanical Distribution Functions

Springer eBooks•Dec 19, 2005•1 citation

THEORY OF THE DISTRIBUTION FUNCTIONS FOR MANY-PARTICLE SYSTEMS.

A. Isihara

•Jan 1, 1969•0 citations
Preprint paper abstract: We propose and develop the thesis that every physically real primitive quantity is fundamentally an ontic distribution rather than single-valued in any observable domain. This is not an epistemic claim about measurement uncertainty or observer knowlwdge, but an ontological claim: single-valued physical primitives do not exist in nature. We argue that this view is not merely philosophically tenable but is in fact demanded by three independent pillars of established physics: the Heisenberg Uncertainty Principle (HUP), the impossibility of zero entropy in physical systems, and the Bekenstein bound on information density in finite regions of spacetime [23]. We further develop the implications of this framework for quantum measurement theory, the interpretation of Bell test results [1–3, 31, 32], and the concept of entanglement. In particular, we introduce the concept of a prominence function P(λ,θ) as an ontic measure of the capacity of a distribution’s constituents to physically interact with a given context, and propose that measurement outcomes arise from a distribution-valued product of source distributions and detector context distributions. We argue this framework suggests a reinterpretation of Bell’s theorem in which the assumption of single-valued hidden variables is not merely a simplification but a fundamental category error. This work extends prior analysis of Bell’s theorem in the interval-based framework of [28] and the ontological wave function model of [29].
Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.