title: Third law of thermodynamics as a key test of generalized entropies.
abstract: The laws of thermodynamics constrain the formulation of statistical mechanics at the microscopic level. The third law of thermodynamics states that the entropy must vanish at absolute zero temperature for systems with nondegenerate ground states in equilibrium. Conversely, the entropy can vanish only at absolute zero temperature. Here we ask whether or not generalized entropies satisfy this fundamental property. We propose a direct analytical procedure to test if a 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 β=dS/dU in terms of the microstate probabilities p(i). To illustrate this approach, we present exact results for the two best known generalizations of statistical mechanics. Specifically, we study the Kaniadakis entropy S(κ), which is additive, and the Tsallis entropy S(q), which is nonadditive. We show that the Kaniadakis entropy correctly satisfies the third law only for -1<κ<+1, thereby shedding light on why κ is conventionally restricted to this interval. Surprisingly, however, the Tsallis entropy violates the third law for q<1. Finally, we give a concrete example of the power of our proposed method by applying it to a paradigmatic system: the one-dimensional ferromagnetic Ising model with nearest-neighbor interactions.
authors.affiliation: Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59078-970 Natal RN, Brazil.
authors.forename: E P
authors.initials: EP
authors.lastname: Bento
authors.affiliation: Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59078-970 Natal RN, Brazil.|National Institute of Science and Technology of Complex Systems, Universidade Federal do Rio Grande do Norte, 59078-970 Natal RN, Brazil.
authors.forename: G M
authors.initials: GM
authors.lastname: Viswanathan
authors.affiliation: Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba PR, Brazil.
authors.forename: M G E
authors.initials: MG
authors.lastname: da Luz
authors.affiliation: Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59078-970 Natal RN, Brazil.|Departamento de Física, Universidade do Estado do Rio Grande do Norte, 59610-210 Mossoró RN, Brazil.
authors.forename: R
authors.initials: R
authors.lastname: Silva