# Is the Entropy a Function or a Functional? [duplicate]

As in the title, I was wondering whether the entropy of a system (it can be any entropy, from Boltzmann to Renyi etc, it is of no importance) is a function or a functional and why? Since it is mostly defined as: $$S(p)=\sum_{i}g(p_i)$$ for some $g$ that has to be continous etc then it has to be a functional. But then I see that $S_{BG}$ for example, which is defined as $S_{BG}=\sum_i p_i \log p_i$ just needs the value of each $p_i$ in order to be defined, right?

The way I see it, it has to be a functional but it is not clear to me why. Also many authors mention the entropy as a function while others call it a functional.

Thank you!

• What is the difference between function and functional supposed to be? To many, the terms are synonyms. – Did Jul 4 '16 at 8:45

## 1 Answer

A function is a mapping between a set of numbers and another set of numbers. A functional is a mapping between a set of functions and another set of functions. The entropy is defined as the Gibbs functional: $$S(p)=-k\sum_jp_j\log(p_j)$$ where the $p_j$ are functions. So the correct way to define the entropy, following Gibbs, is a functional

• So... $f:\mathbb R^2\to\mathbb R$, $(x,y)\mapsto f(x,y)=x+2y$, is not a function? – Did Jul 4 '16 at 9:16
• – Riccardo.Alestra Jul 4 '16 at 9:42
• Yeah -- but what about the question in my comment? Function or not function? 'Cause if "not function", then I, and a bunch of my colleagues, have to revise our teaching methods on the spot... – Did Jul 4 '16 at 9:47
• @Mitscaype: yes, but the $p_j$ are fuinctions – Riccardo.Alestra Jul 4 '16 at 11:37
• @Mitscaype: correct – Riccardo.Alestra Jul 4 '16 at 11:42