### Rényi Entropy and Free Energy

#### Posted by John Baez

David Corfield used to study statistical inference and machine learning. This led him to become interested in some variants of the ordinary notion of entropy — for example, Tsallis entropy and Rényi entropy. They’re very odd-looking concepts when you first see them, especially if you’ve spent a lot of time studying the ordinary Boltzmann–Gibbs–Shannon–von Neumann entropy. So, I’ve never been happy with them. Whenever I saw them, I thought: *what do they mean?*

I just got a lot happier with Renyi entropy. Thanks to conversations at the Entropy Club here at the Centre for Quantum Technologies, especially with Oscar Dahlsten, I noticed that Renyi entropy is just a slightly disguised version of a more familiar concept in thermodynamics: free energy.

Unsurprisingly, somebody already knew this.

However, further discussions revealed that Rényi entropy is just the natural *q-deformation* of the ordinary notion of entropy!

For more, check out my blog post and brief paper. I’m only reblogging here to get David’s attention — it makes more sense to put all the comments in one place.

And for even more, see David’s blog posts on Tsallis entropy and Rényi entropy. Amusingly, I didn’t remember these blog posts, or the comments I made on them, until just now!