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

$q$-deformed Fermion in Many-Particle Systems and Its Application to BCS Theory