Erratum: Third law of thermodynamics as a key test of generalized entropies [Phys. Rev. E 91,022105 (2015)]

In this letter we discuss two aspects of non-Gaussian statistics. In the first, we show that Maxwell's first derivation of the stationary distribution function for a dilute gas can be extended in the context of Kaniadakis statistics. In the second, by investigating the stellar system, we study the Kaniadakis analytical relation between the entropic parameter κ and the stellar polytrope index n. We compare also the Kaniadakis relation n = n(κ) with n = n(q) proposed in the Tsallis framework.

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.

RESUMOEste artigo apresenta uma metodologia para resolução de problemas de configuração de redes de distribuição, com aplicação à minimização de perdas elétricas. O método utiliza um Algoritmo Genético (AG) básico e algumas de suas variantes para seus operadores genéticos: Seleção, Cruzamento e Mutação. O emprego do AG possibilita a análise de redes reais, sem necessidade de simplificações ou aproximações, o que permite a obtenção de soluções otimizadas em tempos de execução compatíveis para aplicações em atividades de planejamento da expansão e da operação. As variantes do AG permitem melhorar a robustez dos resultados e reduzir os tempos de execução, compatíveis com a aplicação em redes de distribuição. A metodologia proposta é aplicada, inicialmente, a uma rede de distribuição hipotética, onde é conhecida a configuração de perdas mínimas. Diversas alternativas de parametrizações do AG são efetuadas, o que permite a comparação do desempenho de cada alternativa. As melhores alternativas são aplicadas na resolução de uma rede real, mostrando o grande potencial do modelo. PALAVRAS-CHAVE:Minimização de perdas elétricas em redes de distribuição, Otimização, Algoritmos Genéticos. ABSTRACTThis paper presents a methodology to evaluate the optimal configuration of radial electric power distribution systems. Network loss minimization is set as the main objective to demonstrate the potential in solving real problems. The proposed framework is based on genetic algorithms (GA), which makes possible the analysis of real sized networks. This allows for attaining optimized solutions in an affordable computation time, especially for expansion and operation planning applications. The method requires neither simplifications nor approximations to the original problem formulation, thus improving the quality of the obtained results. Basic (or canonical) GA and some variants are presented, highlighting main functional characteristics of each alternative and respective parameterization. The proposed methodology is first applied to a hypothetical network, in which the minimum losses configuration is known in advance. Several simulations make possible comparing the performance of each GA parameter. Finally, the best parameter settings are applied to a real sized network, illustrating the promising potential of this methodology.

In Bento et al. [Phys. Rev. E 91, 039901 (2015)] we develop a method to verify if an arbitrary generalized statistics does or does not obey the third law of thermodynamics. As examples, we address two important formulations, Kaniadakis and Tsallis. In their Comment on the paper, Bagci and Oikonomou suggest that our examination of the Tsallis statistics is valid only for q≥1, using arguments like there is no distribution maximizing the Tsallis entropy for the interval q<0 (in which the third law is not verified) compatible with the problem energy expression. In this Reply, we first (and most importantly) show that the Comment misses the point. In our original work we have considered the now already standard construction of the Tsallis statistics. So, if indeed such statistics lacks a maximization principle (a fact irrelevant in our protocol), this is an inherent feature of the statistics itself and not a problem with our analysis. Second, some arguments used by Bagci and Oikonomou (for 0