Entropias generalizadas: vínculos termodinâmicos da Terceira Lei

Abstract

Based on the third law of Thermodynamics we ask whether or not generalized entropies satisfy this fundamental property. The third law states that the entropy approaches zero as the temperature (in absolute scale) also approaches zero. However, the entropy can vanish only at absolute zero temperature. In this context, we propose a direct analytical procedure to test if the generalized entropy satisfies the third law, assuming only very general assumptions for the entropy S and energy U of an arbitrary N-level classical system. Mathematically, the method relies on exact calculation of the parameter _ = dS=dU in terms of the microstate probabilities pi. Finally, we determine the relation between the low entropy limit S ! 0 (or more generally Smin) and the low-temperature limit _ ! +1. For comparison, we apply the method to the entropy Boltzmann (standard model), and Kaniadakis Tsallis (generalized models). For the latter two, we illustrate the power of the method by unveiling the ranges of their parameters for which the third law is satisfied. For _-entropy, the values usually assumed in the literature to _ parameter obeys the third law ( - 1 < _ < 1). However, for the q-entropy the same is not true. We show that the q-entropy can vanish at nonzero temperature in certain ranges of q. These results and their consequences are discussed in this thesis. As a concrete example, we consider the paradigmatic one-dimensional Ising model, which is one of the most important models in all of physics. Classically, the Ising model is solved in the canonical ensemble, but it can also solved exactly in nonstandard ensembles using generalized entropies.