General thermostatistical formalisms based on parameterized entropic measures

We revisit the concept of generalized thermostatistical formalisms, based on extremizing parameterized entropic functionals subject to appropriate constraints, in order to incorporate an (effective) temperature dependence of the entropic parameters and of parameters characterizing the relevant constraints. Our main aim is to investigate what kinds of temperature dependence of these parameters are consistent with the Legendre-transform structure of thermodynamics. After discussing this problem in a qui te general context, we discuss in detail the important particular example of the q-nonextensive thermostatistical formalism with a temperature-dependent q-parameter. In this special case, our general formalism implies extremizing the concomitant entropy functional S-q, subject to the constraints imposed by normalization and the presumably known exp ectation values of N relevant observables, for arbitrary variations of both the statistical operator (rho) over cap and the parameter q. The ensuing extended variational formalism preserves the usual (Legendre-transform) connection with thermodynamics. For sets of relevant observables that close a Lie semi- algebra with the system's Hamiltonian, we study some features of our approach related to the system's dynamics and show that q is a constant of the motion. Our present developments may be useful for the study of systems whose entropic parameter q is unknown.