Abstract. Modern analysis of the Boltzmann distribution on the basis of probability theory and of Gibbs quantum canonical distribution is proposed. It is shown that the Boltzmann distribution function is neither a reliable result of probability theory nor a consequence of Gibbs quantum canonical distribution. Conclusion is that the Boltzmann distribution function has no statistical and physical meaning, and the concept “Boltzmann distribution” should be abolished.
Abstract. Modern analysis of the Boltzmann distribution on the basis of probability theory and of Gibbs quantum canonical distribution is proposed. It is shown that the Boltzmann distribution function is neither a reliable result of probability theory nor a consequence of Gibbs quantum canonical distribution. Conclusion is that the Boltzmann distribution function has no statistical and physical meaning, and the concept “Boltzmann distribution” should be abolished.
A fundamental postulate of statistical mechanics is that all microstates in an isolated system are equally probable. This postulate, which goes back to Boltzmann, has often been criticized for not having a clear physical foundation. In this note, we provide a derivation of the canonical (Boltzmann) distribution that avoids this postulate. In its place, we impose two axioms with physical interpretations. The first axiom (thermal equilibrium) ensures that, as our system of interest comes into contact with different heat baths, the ranking of states of the system by probability is unchanged. Physically, this axiom is a statement that in thermal equilibrium, population inversions do not arise. The second axiom (energy exchange) requires that, for any heat bath and any probability distribution on states, there is a universe consisting of a system and heat bath that can achieve this distribution. Physically, this axiom is a statement that energy flows between system and heat bath are unrestricted. We show that our two axioms identify the Boltzmann distribution.
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 Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
Using the concept of quantum wave probability, combined with the identity principle, we can derive the Boltzmann distribution, Fermi distribution, and Bose distribution. Different distributions correspond to different conditions. The Boltzmann condition corresponds to the Boltzmann distribution. The Fermi condition corresponds to the Fermi distribution. The Bose condition corresponds to the Bose distribution. This demonstrates that the foundation of these three statistical distributions is quantum wave probability, all originating from quantum mechanics. The Boltzmann distribution is also an independent quantum distribution and is not simply a sparse limit of the Fermi or Bose distributions. The essence of the Boltzmann distribution is a uniform distribution. The Fermi and Bose distributions are deviations from the uniform distribution. Particles that follow the Boltzmann distribution can be called Boltzmannons. Boltzmann entropy based on the quantum wave probability can resolve the Gibbs paradox. We need to rethink the fundamentals of statistical mechanics.
Using the concept of quantum wave probability, combined with the identity principle, we can derive the Boltzmann distribution, Fermi distribution, and Bose distribution. Different distributions correspond to different conditions. The Boltzmann condition corresponds to the Boltzmann distribution. The Fermi condition corresponds to the Fermi distribution. The Bose condition corresponds to the Bose distribution. This demonstrates that the foundation of these three statistical distributions is quantum wave probability, all originating from quantum mechanics. The Boltzmann distribution is also an independent quantum distribution and is not simply a sparse limit of the Fermi or Bose distributions. The essence of the Boltzmann distribution is a uniform distribution. The Fermi and Bose distributions are deviations from the uniform distribution. Particles that follow the Boltzmann distribution can be called Boltzmannons. Boltzmann entropy based on the quantum wave probability can resolve the Gibbs paradox. We need to rethink the fundamentals of statistical mechanics.