  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.
