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.
In a previous note (1) we argued that for a wavefunction W(x,t)= Sum over n an exp(iEnt) Wn(x), the entropy for a particle in a box with infinite potential walls from (2) is: S/(-k)= Sum over n |anan| ln |anan| ((1)) and should be modified to: S/(-k)= Sum over n |anan| ln|anan| + |anan| Sp(n,L) + |anan| Sx(L) ((2)) where Sp and Sx are the pure state momentum and spatial entropies i.e Sp/-k = Integral dp an(p)an(p) ln|an(p)an(p)| and Sx/(-k) = Integral dx W*n(x)Wn(x) ln[Wn*Wn|. In (2) the Clausius relationship: dQ/T = dS is used where dQ = Sum over n En d|anan| and T is temperature to find an expression for |anan|. In general this would be: |anan| = exp(-En/T) , but with ((2)) we find: |anan| = exp(-En/kT) exp(-1-B(n)) ((3)) where B(n)= Sp(n)+Sx. L portions form Sp and Sx cancel so there is no L dependence, L being the size of the box. Given the different form ((3)) we suggest that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions may change if one uses the grand partition function approach. For example, exp(-ni ei/kT) is replaced by: exp(-ni ei/kT) exp(-ni(1+B(n)). This leads to changes in the FD and BD distributions: zexp(-Ei/kT)exp(-1-B(i) / [1 +/- exp(-u/kT) exp(-ei/kT) exp(-1-B(i))] where z=exp(u/kT), u being the chemical potential. In the high n limit B(n)--> constant (3) so the Maxwell-Boltzmann (MB) distribution remains unchanged. The MB distribution has been verified to high precision and so modifications related to 1+B(n) might have been noticed had B(n) remained a function of n.
Equilibrium distribution functions are obtained for boson and fermion ensembles with a limited number of particles. It is shown that the number-of-particle distribution functions in different quantum states are statistically dependent; this dependence disappears only for a large number of particles in the ensemble. The distributions are transformed into the Boltzmann distribution at a high temperature and into the Bose-Einstein and Fermi-Dirac distributions for a large number of particles in the ensemble.
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.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.