Abstract The classical Maxwell–Boltzmann distribution is an exponential law. Using the Choquet integral framework, one can introduce nonlinear (non-additive) capacities and thereby define new families of distributions belonging to the class of Choquet–exponential distributions. In this paper, we assume that a suitable nonlinear measure can be constructed such that the quantum Bose–Einstein and Fermi–Dirac distributions emerge naturally as Choquet–exponential forms. After calculating this measure, we analyze its mathematical and physical properties. According to our adopted convention, the thermodynamic curvature is positive for intrinsic attractive correlations and negative for repulsive ones. Considering the concavity and convexity of the nonlinear Choquet measure, we propose a direct interpretation of statistical interactions in terms of the geometric behavior of the capacity itself. Finally, the same approach is applied to Tsallis quantum statistics, and the proposed interpretation is verified up to second order in the deviation from the standard cases.
Abstract The classical Maxwell–Boltzmann distribution is an exponential law. Using the Choquet integral framework, one can introduce nonlinear (non-additive) capacities and thereby define new families of distributions belonging to the class of Choquet–exponential distributions. In this paper, we assume that a suitable nonlinear measure can be constructed such that the quantum Bose–Einstein and Fermi–Dirac distributions emerge naturally as Choquet–exponential forms. After calculating this measure, we analyze its mathematical and physical properties. According to our adopted convention, the thermodynamic curvature is positive for intrinsic attractive correlations and negative for repulsive ones. Considering the concavity and convexity of the nonlinear Choquet measure, we propose a direct interpretation of statistical interactions in terms of the geometric behavior of the capacity itself. Finally, the same approach is applied to Tsallis quantum statistics, and the proposed interpretation is verified up to second order in the deviation from the standard cases.
We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for α<0. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter α. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature T∗, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature Tc, the ratio T∗/Tc depends universally on α. For negative α, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.
In this work we present a classical kinetic model of intermediate statistics. In the case of Brownian particles we show that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions can be obtained, just as the Maxwell-Boltzmann (MB) distribution, as steady states of a classical kinetic equation that intrinsically takes into account an exclusion-inclusion principle. In our model the intermediate statistics are obtained as steady states of a system of coupled nonlinear kinetic equations, where the coupling constants are the transmutational potentials ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\kappa}}\mathrm{\ensuremath{\kappa}}\ensuremath{'}}$. We show that, besides the FD-BE intermediate statistics extensively studied from the quantum point of view, we can also study the MB-FD and MB-BE ones. Moreover, our model allows us to treat the three-state mixing FD-MB-BE intermediate statistics. For boson and fermion mixing in a D-dimensional space, we obtain a family of FD-BE intermediate statistics by varying the transmutational potential ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. This family contains, as a particular case, when ${\mathrm{\ensuremath{\eta}}}_{\mathrm{B}\mathrm{F}=0}$, the quantum statistics recently proposed by L. Wu, Z. Wu, and J. Sun [Phys. Lett. A 170, 280 (1992)]. When we consider the two-dimensional FD-BE statistics, we derive an analytic expression of the fraction of fermions. When the temperature T\ensuremath{\rightarrow}\ensuremath{\infty}, the system is composed by an equal number of bosons and fermions, regardless of the value of ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. On the contrary, when T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$ becomes important and, according to its value, the system can be completely bosonic or fermionic, or composed both by bosons and fermions.
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.
The term ‘Verteilungsfunktion’ (distribution function) is used in physics as a semantic container for at least four ontologically distinct mathematical objects: proper probability density functions (PDFs), probability mass functions (PMFs), expectation values of occu-pation numbers, and phase-space measure densities. This conflation arose historically because the statistical mechanics of Boltzmann, Maxwell, Fermi, Dirac and Bose predates Kolmogorov’s axiomatic probability theory (1933) by decades; no systematic back-mapping was ever performed. The central finding concerns the Fermi–Dirac and Bose–Einstein functions specifically: they are not merely mislabeled — they are relative measurements, bounded observable expectation values ⟨nk⟩-normalised against their capacity. Each of the five objects routinely called a ‘distribution function’ is audited against the Kolmogorov taxonomy: the Maxwell–Boltzmann speed distribution is a proper PDF that is merely mislabeled; the Boltzmann canonical distribution is a genuine probability measure; the Boltzmann-transport phase-space density is normalised toNrather than to unity; and the Fermi–Dirac and Bose–Einstein functions are occupation-number expectation values that acquire physical meaning only as convolution kernels against the density of states. A short supporting section shows that this is not an accident of two particular formulas. One elementary theorem establishes that a bounded occupation of capacityMunder Boltz-mann weighting with an additive constraint yields⟨n⟩M= −d ln ZM/dx; atM=1 this is exactly the logistic function, and at M→∞it diverges. Fermi–Dirac and Bose–Einstein are therefore the two branches of a single construction, which is why the first is bounded in [0, 1]and the second admits no bounded normalisation at all. The same unit-capacity form appears outside physics wherever a bounded quantity accumulates additive evidence, which suggests that such objects are best read as relative measurements against a capacity rather than as distributions. The paper reconstructs the historical sequence that produced the terminological divergence and proposes notational conventions to restore conceptual clarity in pedagogy and literature. Keywords: probability density function, distribution function, Fermi–Dirac, Bose–Einstein, Maxwell–Boltzmann, Kolmogorov, Jaynes, ontology, statistical mechanics, probability theory, quantum statistics, foundations of physics,philosophy of science, physics education, information theory, thermodynamics.
Abstract We propose a generalized quantum statistical framework, termed Max–Min statistics, which imposes simultaneous lower ( N min ) and upper ( N max ) bounds on the occupation numbers of single-particle quantum levels. The formalism is developed using two complementary but equivalent approaches: a combinatorial microstate-counting scheme based on modified statistical weights, and an operator-algebraic construction employing a generalized number operator with constrained ladder operations. The resulting distribution function provides a unified description that continuously interpolates between Bose–Einstein, Fermi–Dirac, Gentile, and Ewkon statistics as limiting cases. Thermodynamic consequences of the occupation constraints are examined within the grand canonical ensemble, and the geometric properties of the corresponding thermodynamic state space are analyzed using Ruppeiner geometry. The scalar curvature is shown to vary systematically with the allowable occupation range Δ N = N max − N min , reflecting changes in statistical correlations induced by the imposed bounds. In particular, non-zero lower occupancy constraints lead to distinct thermodynamic behaviors that require regularization via an energy cutoff. These results establish the Max–Min framework as a flexible and mathematically consistent tool for exploring bounded and intermediate quantum statistics from a formal and phenomenological perspective.