Features of constrained entropic functional variational problems

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.

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