Basic properties of the q-entropy Sq[ρ]=(q−1)−1(1−tr(ρq)) (0<q≠1) for states of a quantum system are established: concavity, quasi-convexity, continuity, and failure of ‘‘additivity’’ and ‘‘subadditivity’’ for composite systems.

For a large class of quantum models of mean-field type the thermodynamic limit of the free energy density is proved to be given by the Gibbs variational principle. The latter is shown to be equivalent to a noncommutative version of Varadhan's asymptotic formula.

We consider the quantum thermal statistics à la Gibbs-Shannon-Szilard-Jaynes based on q-entropies S q ͓͔ϭ(qϪ1) Ϫ1 "1Ϫtr(q)… (0Ͻq 1) and the nonstandard ''internal energy'' functionals U q ͓͔ϭtr(q H) proposed by C. Tsallis ͓J.

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The thermodynamic limit is discussed for a class of inhomogeneous (“site-dependent coupling”) mean-field models: we give a variational expression for the limiting free-energy density, and obtain general properties of the equilibrium states. The class of models includes the full BCS-model in its quasi-spin formulation; this model is chosen here to illustrate the results.