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Journal of High Energy Physics Gravitation and Cosmology

Quantum Wave Probability Derive Thermodynamic Distribution

Xiaolin Li

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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. Boltzmann entropy based on the quantum wave probability can resolve the Gibbs paradox. Using this new approach, we can also derive the results of eigenstate thermalization. The equilibrium state is the state in which all eigenstates have the same temperature. We need to rethink the fundamentals of statistical mechanics.

DOI
10.4236/jhepgc.2026.122036

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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. Boltzmann entropy based on the quantum wave probability can resolve the Gibbs paradox. Using this new approach, we can also derive the results of eigenstate thermalization. The equilibrium state is the state in which all eigenstates have the same temperature. 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. Boltzmann entropy based on the quantum wave probability can resolve the Gibbs paradox. Using this new approach, we can also derive the results of eigenstate thermalization. The equilibrium state is the state in which all eigenstates have the same temperatures. 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.
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.
In the classical mechanical limit in which quantum cells of volume (2πħ)2 are reduced to points in phase space, the Bose and Fermi distributions are shown to approach the Boltzmann distribution. This limit is a continuous limit and therefore does not alter the indistinguishability of bosons (fermions) which is defined in terms of a discontinuous permutation symmetry of the density operator and Hamiltonian for the system. This means that the classical particles which correspond to the identical boson (fermions) remain indistinguishable. In the same limit, the partition function for N identical bosons approaches (2πħ)−3N (N!)−1 f. f d3r1 d3p1…d3rN d3pN exp (–βH)  Zc. The two factors, (2πħ)−3N and (N!)−1 which are often added in an ad hoc manner in many books on statistical mechanics, are thus derived from first principles. In the classical statistical limit in which the number density n and the reciprocal temperature β become small, the quantum statistical distributions also approach the Boltzmann distribution. This is shown by examining the chemical potential μ as a function of n and β. The classical approximation is valid if the thermal de Broglie wavelength is much shorter than the interparticle distance irrespective of any interparticle interaction.
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 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.