  In recent decades, there have been increasing interests in quantum statistics
beyond the standard Fermi-Dirac and Bose-Einstein statistics, such as the
fractional statistics, quon statistics, anyon statistics and quantum groups,
since they can provide some new insights into the cosmology, nuclear physics
and condensed matter. In this paper, we study the many-particle system formed
by the $q$-deformed fermions ($q$-fermion), which is realized by deforming the
quantum algebra of the anticommutation relations. We investigate from a
standard perspective of the finite temperature field theory and try to
construct the finite temperature Green's function formalism for the free
many-$q$-fermion system, then generalize it to the well known interacting
fermionic system, the superconductor, and finally obtain a consistent
$q$-deformed BCS ($q$BCS) theory. At low temperature, this theory predicts a
Sarma-like ordered phase, and we call it the $q$-deformed Sarma phase. It also
presents a symmetric phase diagram in the parameter space and new thermodynamic
relations.
