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A thermodynamic system of non-interacting quantum particles changes itsstatistical distribution formulas if there is a universal limitation for thesize of energetic quantum leaps (magnitude of quantum leaps smaller than Planckenergy). By means of a restriction of the a priori equiprobability postulateone can reach a thermodynamic foundation of these corrected distributionformulas. The number of microstates is determined by means of a suitablecounting 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 andparticle character of the quantum particles, too, can be influenced by choosingthe size of the temperature and particle energy parameters relative to thePlanck energy, as you can see from the associated fluctuation formulas. In thecase of non-relativistic degeneration, the critical parameters Fermi momentum(fermions) and Einstein temperature (bosons) vanish as soon as the rest energyof the quantum particles reaches the Planck energy. For the Bose-Einsteincondensation there exists, in the condensation range, a finite upper limit forthe number of particles in the ground state, which is determined by the ratioof Planck mass to the rest mass of the quantum particles. In the relativistichigh-temperature range, the energy densities of photon and neutrino radiationhave finite limit values, which is of interest with regard to the start ofcosmic expansion.

Ideal Quantum Gases with Planck Scale Limitations