# The Legendre Transform in Non-additive Thermodynamics and Complexity

## TL;DR

This paper argues that the standard Legendre transform is unsuitable for non-additive thermodynamics, specifically in systems described by Tsallis entropy, and proposes a modified transform for such cases.

## Contribution

It introduces a modified Legendre transform tailored for non-additive thermodynamics based on Tsallis entropy, challenging the conventional approach.

## Key findings

- Standard Legendre transform is not appropriate for Tsallis entropy-based systems.
- A specific modified Legendre transform is proposed for non-additive thermodynamics.
- Implications extend to the thermodynamics of complex systems.

## Abstract

We present an argument which purports to show that the use of the standard Legendre transform in non-additive Statistical Mechanics is not appropriate. For concreteness, we use as paradigm, the case of systems which are conjecturally described by the (non-additive) Tsallis entropy. We point out the form of the modified Legendre transform that should be used, instead, in the non-additive thermodynamics induced by the Tsallis entropy. We comment on more general implications of this proposal for the thermodynamics of "complex systems".