In the case of high densities and/or low temperatures,
the wave functions of the various particles (partly) overlap.
This gives rise to specific quantum effects which influence
the distributions of the elementary particles.
Such situations are quite frequently encountered when cosmic
phenomena are being investigated, like for instance neutron
stars, the cosmic microwave background radiation or the sites
where cosmic rays are produced.
To enable a proper description and understanding of these situations,
the Fermi-Dirac and Bose-Einstein statistical distributions are
derived from first principles.
Using the Fermi-Dirac and Bose-Einstein statistics, the concept of
Quantum Mechanical Phase-Space is introduced and the distribution
functions for a Free Electron Gas and Photon Gas are derived explicitly.
The latter is used in an application to determine the mean free path
of high-energy protons that are traveling through the cosmic microwave
background radiation. This gives rise to the so called GZK cutoff
for the most energetic cosmic rays.
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