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We study the thermodynamics of massive Gross-Neveu models with explicitly broken discrete or continuous chiral symmetries for finite temperature and fermion densities. The large N limit is discussed bearing attention to the no-go theorems for symmetry breaking in two dimensions which apply to the massless cases. The main purpose of the study is to serve as analytical orientation for the more complex problem of chiral transition in 4−dimensional QCD with quarks. For any non-vanishing fermion mass we find, at finite densities, lines of first order phase transitions. For small mass values traces of would-be second order transitions and a tricritical point are recognizable. We study the thermodynamics of these models, and in the model with broken continuous chiral symmetry we examine the properties of the pion like state.

We study the phases of a two-flavor Nambu-Jona-Lasinio model at finite temperature T, and baryon and isospin chemical potentials B ϭ( u ϩ d )/2, I ϭ( u Ϫ d )/2. This study completes a previous analysis where only small isospin chemical potentials I were considered.

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