Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized $q$-average energy

TL;DR

This paper investigates the relationships between different Tsallis' thermostatistics formalisms, focusing on the connections between standard linear average energy and normalized q-average energy, revealing structural and Legendre transform relations.

Contribution

It uncovers the connections between various Tsallis formalisms, including relations among Lagrange multipliers and the generalized Massieu potential, enhancing understanding of their structural links.

Findings

Normalized q-average energy relates to standard linear average energy.

Legendre transform structures are consistent across formalisms.

Generalized Massieu potential links different entropy and energy definitions.

Abstract

Tsallis' thermostatistics with the standard linear average energy is revisited by employing $S_{2-q}$, which is the Tsallis entropy with $q$ replaced by $2-q$. We explore the connections among the $S_{2-q}$ approach and the other different versions of Tsallis formalisms. It is shown that the normalized $q$-average energy and the standard linear average energy are related to each other. The relations among the Lagrange multipliers of the different versions are revealed. The relevant Legendre transform structures concerning the Lagrange multipliers associated with the normalization of probability are studied. It is shown that the generalized Massieu potential associated with $S_{2-q}$ and the linear average energy is related to one associated with the normalized Tsallis entropy and the normalized $q$-average energy.