Group theory, entropy and the third law of thermodynamics

Abstract

Curado et al. [Ann. Phys. 366 (2016) 22] have recently studied the axiomatic structure and the universality of a three-parameter trace-form entropy inspired by the group-theoretical structure. In this work, we study the group-theoretical entropy S a,b,r in the context of the third law of thermodynamics where the parameters {a, b, r} are all independent. We show that this threeparameter entropy expression can simultaneously satisfy the third law of thermodynamics and the three Khinchin axioms, namely continuity, concavity and expansibility only when the parameter b is set to zero. In other words, it is thermodynamically valid only as a two-parameter generalization Sa,r. Moreover, the restriction set by the third law i.e., the condition b = 0, is important in the sense that the so obtained two-parameter group-theoretical entropy becomes extensive only when this condition is met. We also illustrate the interval of validity of the third law using the one-dimensional Ising model with no external field. Finally, we show that the Sa,r is in the same universality class as that of the Kaniadakis entropy for 0 < r < 1 while it has a distinct universality class in the interval −1 < r < 0.

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