The Validity of the Third Law of Thermodynamics for the Renyi and Homogeneous Entropies

Abstract

Bento et al. (2015) recently proposed to use the third law of thermodynamics as a test for the generalized entropies, since the third law should be respected by any statistical mechanical entropy regardless of the explicit form of the Hamiltonian. We first consider the Renyi entropy which is the sole additive generalized entropy, and show that it violates the third law for q > 1. Interestingly, this is exactly the interval where the Renyi entropy is neither concave nor convex. We then consider the homogeneous entropy and show that it also violates the third law for 0 < q < 1. This implies that the homogeneous entropy cannot be used as a generalized statistical mechanical entropy, since the aforementioned interval is the only interval for which the homogeneous entropy yields normalizable distributions. (C) 2015 Elsevier By. All rights reserved.