  We investigate the thermodynamics at finite density of a generalized $(2 +
1)$-dimensional Gross-Neveu model of $N$ fermion species with various types of
four-fermion interactions. The motivation for considering such a generalized
schematic model arises from taking the Fierz-transformation of an effective
Coulomb current-current interaction and certain symmetry breaking interaction
terms, as considered for graphene-type models in Ref. [16]. We then apply
path-integral bosonization techniques, based on the large $N$ limit, to derive
the thermodynamic potential. This includes the leading order mean-field (saddle
point) contribution as well as the next-order contribution of Gaussian
fluctuations of exciton fields. The main focus of the paper is then the
investigation of the thermodynamic properties of the resulting fermion-exciton
plasma. In particular, we derive an extended Beth-Uhlenbeck form of the
thermodynamic potential, discuss the Levinson theorem and the decomposition of
the phase of the exciton correlation into a resonant and scattering part.
