Quantum statistical mechanics
Statistical mechanics is the description of a gas of
(usually indistinguishable) particles. In cases where the
temperature is low enough or the concentration is high
enough then the quantum
mechanical symmetry properties
of the particles become important.
Particles with integral spin (bosons) obey
Bose-Einstein statistics and particles with
half-integral spin (fermions) obey
Said another way, whether the particles' wave functions
are symmetric or anti-symmetric under particle interchange
determines which quantum statistics are applicable.