Features of constrained entropic functional variational problems

TL;DR

This paper explores the general features of constrained entropic variational problems, demonstrating that the first thermal law holds universally in systems subject to Legendre extremization, with broad applications across various functionals.

Contribution

It provides a general framework for constrained entropic variational problems, showing the universal validity of the first thermal law in such systems.

Findings

The first thermal law is obeyed in all constrained entropic variational problems.

The framework encompasses various functionals like entropy, Fisher information, and relative entropy.

The approach applies broadly to generalized statistical systems.

Abstract

We describe in great generality features concerning constrained entropic, functional variational problems that allow for a broad range of applications. Our discussion encompasses not only entropies but, potentially, any functional of the probability distribution, like Fisher-information or relative entropies, etc. In particular, in dealing with generalized statistics in straightforward fashion one may sometimes find that the first thermal law $\frac{dS}{d\beta}=\beta\frac{d<U>}{d\beta}$ seems to be not respected. We show here that, on the contrary, it is indeed obeyed by any system subject to a Legendre extremization process, i.e., in all constrained entropic variational problems.