Group theory, entropy and the third law of thermodynamics

TL;DR

This paper investigates a three-parameter group-theoretical entropy in relation to the third law of thermodynamics, finding it valid as a two-parameter entropy under specific conditions and exploring its universality classes.

Contribution

It demonstrates that the three-parameter entropy satisfies thermodynamic principles only when reduced to two parameters, clarifying conditions for extensivity and universality classes.

Findings

The three-parameter entropy is valid only with b=0 for thermodynamic consistency.

The two-parameter entropy becomes extensive under the same condition.

Universality class analysis shows similarity to Kaniadakis entropy for certain parameter ranges.

Abstract

Curado \textit{et al.} [Ann. Phys. \textbf{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 $\left\lbrace a,b,r \right\rbrace $ 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 $-1 < r < 0$.