The parameter space and third law of thermodynamics for the Borges-Roditi, Abe and Sharma-Mittal entropies

In order to investigate the role of the parameter space on the third law of thermodynamics, we have checked the validity intervals of the Borges-Roditi, Abe and Sharma-Mittal entropies in the framework of the third law of thermodynamics. The two-parameter Borges-Roditi entropy conforms to the third law for a < 1, 0 < b < 1 or b < 1, 0 < a < 1 due to its a ? b symmetry, while the Abe entropy satisfies the third law in the interval 1 2 < σ < 1 or 1 < σ < 2. Since the validity interval of the single-parameter Abe entropy can be fully recovered from the two-parameter Borges-Roditi entropy, we note that the third law is immune to the reduction of the parameters in this particular case. The Sharma-Mittal entropy, in this same context, is valid in the interval 0 < q ≤ 1, whereas its second parameter r can take any real values without restriction. The interval 0 < q ≤ 1 is the intersection of the Sharma-Mittal entropy with its reduced expressions, i.e., the Tsallis and Rényi entropies from the third law perspective. This implies that the additional parameter r of the Sharma-Mittal entropy indeed renders it more generalized compared to the one-parameter Rényi and Tsallis entropies when validated by the third law of thermodynamics. Therefore, based on these observations, we deduce that an additional parameter can extend the interval of validity of the third law for a particular generalized entropy only when this additional parameter is not confined by reduction to some other generalized entropy definitions. We also check our theoretical results by using the one-dimensional Ising model with periodic boundary conditions under zero external field.