Ideal Quantum Gases with Planck Scale Limitations

TL;DR

This paper explores how imposing a Planck-scale limit on quantum leap energies alters statistical distributions of quantum gases, leading to convergence of fermion and boson behaviors at high energies and implications for cosmology.

Contribution

It introduces a thermodynamic foundation for modified quantum distribution formulas under Planck-scale energy limitations, revealing new behaviors at high energies and temperatures.

Findings

Fermion and boson distributions converge near Planck energy.

Finite upper limit for particles in Bose-Einstein condensate.

Finite energy density limits for photon and neutrino radiation.

Abstract

A thermodynamic system of non-interacting quantum particles changes its statistical distribution formulas if there is a universal limitation for the size of energetic quantum leaps (magnitude of quantum leaps smaller than Planck energy). By means of a restriction of the a priori equiprobability postulate one can reach a thermodynamic foundation of these corrected distribution formulas. The number of microstates is determined by means of a suitable counting method and combined with thermodynamics via the Boltzmann principle. The result is that, for particle energies that come close to the Planck energy, the thermodynamic difference between fermion and boson distribution vanishes. Both distributions then approximate a Boltzmann distribution. The wave and particle character of the quantum particles, too, can be influenced by choosing the size of the temperature and particle energy parameters relative to the Planck energy, as you can see from the associated fluctuation formulas. In the case of non-relativistic degeneration, the critical parameters Fermi momentum (fermions) and Einstein temperature (bosons) vanish as soon as the rest energy of the quantum particles reaches the Planck energy. For the Bose-Einstein condensation there exists, in the condensation range, a finite upper limit for the number of particles in the ground state, which is determined by the ratio of Planck mass to the rest mass of the quantum particles. In the relativistic high-temperature range, the energy densities of photon and neutrino radiation have finite limit values, which is of interest with regard to the start of cosmic expansion.