Group theory, entropy and the third law of thermodynamics

Curado et�al. (2016) 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 three-parameter 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 S_a,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 S_a,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.