Generalized probability distributions for Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics, with unequal source probabilities $q_i$ for each level $i$, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the Pólya urn model, involving neither independent nor identically distributed ("ninid") sampling. The most probable Pólya distribution contains the Acharya-Swamy intermediate statistic.
Generalized probability distributions for Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics, with unequal source probabilities qi for each level i, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the Pólya urn model, involving neither independent nor identically distributed ("ninid") sampling. The most probable Pólya distribution contains the Acharya-Swamy intermediate statistic.
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 : The Fermi-Dirac, Bose-Einstein and, for completeness the Maxwell-Boltzmann, distributions are obtained respectively from considerations of binomial, negative binomial, and Poisson assemblies. The method used has the simplicity of the traditional derivations that are based on combinatorial considerations but involves neither the identification of most probable values with means values nor the invocation of large numbers of particles involved in the use of Stirling's approximation for factorials. The method thereby also relaxes the requirement for large numbers of particles needed in other available derivations that also use mean values, namely, the Darwin-Fowler method of steepest descents and the Khinchin method that employs limit theorems of the theory of probability.
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.
The Maxwell-Boltzmann (MB), Fermi-Dirac FD and Bose-Einstein BE distributions may be obtained by maximizing the ln of energy level degeneracies subject to the constraint of Sum over i ei ni = Eave in the MB case (canonical distribution) and this constraint together with a second one Sum over i ni = Nave (grand canonical case) as shown in (1) page 182 . In particular, in the MB case, a single energy level ei with ni particles is said to have the degeneracy (gi) (power ni) / ni ! (which overcounts). This maximization process leads to the weight gi exp(-ei/T). Setting gi=1, however, leads to the same weight exp(-ei/T) so the MB factor is not a result of a single particle level degeneracy i.e. the direction in which the momentum vector points. To shed physical light on this situation, one may consider two particle scattering in the MB, FD and BE cases. In all three cases, e1+e2 = e3+e4 (with e=pp/2m), but only in the MB case does n(e1)n(e2)=n(e3)n(e4). This suggests that the single energy level ei degeneracy levels ei actually play a physical role in two body scattering for the FD and BE cases, but not in the MB case. Thus, calculating probabilities based on counting still involves the physics related to the interactions in the problem. Degeneracies may exist and may or may not need to be part of the counting of arrangements. As a specific example, we note that in (1) (gi) (power ni) / ni ! is used to count degenerate arrangements for the MB case (on page 182). This same expression is used in (1) on page 194 to describe a method by Fowler and Darwin to obtain the MB single particle probability exp(-ei/T). In that case, gi is taken to be a parameter which is ultimately set to 1 and not the degeneracy of the single particle level. This begs the question: When should single energy ei degeneracy be included in the counting of states?