The role of constraints within generalized nonextensive statistics

The Gibbs–Jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form S and of physically appropriate constraints. For instance, for the usual canonical ensemble, one adopts (i) S1=−k∑ipi ln pi, (ii) ∑ipi=1, and (iii) ∑ipi i=U1 ({ i}≡ eigenvalues of the Hamiltonian; U1≡ internal energy). Equilibrium consists in optimizing S1 with regard to {pi} in the presence of constraints (ii) and (iii). Within the recently introduced nonextensive statistics, (i) is generalized into Sq=k[1−∑ipiq]/[q−1] (q→1 reproduces S1), (ii) is maintained, and (iii) is generalized in a manner which might involve piq. In the present effort, we analyze the consequences of some special choices for (iii), and their formal and practical implications for the various physical systems that have been studied in the literature. To illustrate some mathematically relevant points, we calculate the specific heat respectively associated with a nondegenerate two-level system as well as with the classical and quantum harmonic oscillators