Relation between grand canonical ensemble, Boltzmann, Fermi-Dirac, and Bose-Einstein distribution: Quantum principle for bosons and bosonic vacuum state, a candidate for dark energy and dark matter

TL;DR

This paper derives the fundamental distributions of quantum statistics from thermal equilibrium principles, revealing that Fermi-Dirac and Bose-Einstein distributions are special cases of the Boltzmann distribution, and proposes bosonic vacuum states as dark energy and dark matter candidates.

Contribution

It introduces a unified derivation of quantum distributions from thermal equilibrium conditions and identifies bosonic vacuum states as potential sources of dark energy and dark matter.

Findings

Fermi-Dirac and Bose-Einstein distributions are special cases of the Boltzmann distribution.

Bosonic vacuum states are proposed as dark energy and dark matter candidates.

The grand canonical ensemble corresponds to the Boltzmann distribution.

Abstract

We obtain the conditions constituting a thermal equilibrium between two energy levels: (i) the total energy is equal in both levels; (ii) the temperature is equal for all particles. Exploiting these conditions, we derive a differential equation of thermal equilibrium that holds for all particles. Integration delivers the Boltzmann distribution, suggesting that it is the general distribution of thermal equilibrium. With excited-state and ground-state population numbers n2 and n1, respectively, Pauli's exclusion principle is formalized as n1=1-n2. Exploiting Einstein's rate-equation approach to Planck's law of blackbody radiation, we derive the equivalent quantum principle for bosons, n1=1+n2. Utilizing either quantum principle, as a boundary condition when integrating the differential equation of thermal equilibrium or by entering it into the Boltzmann distribution, delivers the Fermi-Dirac or Bose-Einstein distribution. The fermionic or bosonic ratios n2/n1 follow the Boltzmann distribution. These results suggest that the Fermi-Dirac and Bose-Einstein distributions are special cases of the Boltzmann distribution, ruled by Pauli's exclusion principle or the quantum principle for bosons. The grand canonical ensemble equals the Boltzmann distribution. The relating principle and the inhibition/enhancement factors for fermions/bosons confirm the quantum principles. The quantum principle for bosons comprises a vacuum state. Consequently, a populated vacuum state should exist for all bosons. Vacuum states are not detectable, therefore, they are dark states. The vacuum states of all bosons contribute to dark energy, while only the matter-boson vacuum states contribute to dark matter, supporting the fact that the amount of dark energy exceeds that of dark matter in the universe.