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Stream: deprecated: thermodynamics

Topic: Legendre-Fenchel transform

John Baez (Nov 09 2021 at 19:13):

You might check out Soichiro Fujii's thesis, mentioned in Willerton's post on the Legendre-Fenchel transform and enriched profunctors, since this thesis defines addition and subtraction for $[-\infty,\infty]$, and it seems their rules are different from ours: they say

$\infty + (\infty) = \infty$

while we say

$\infty + (-\infty) = -\infty$

But they are studying convex functions, while we've been studying concave ones, so I'm hoping that our approach is just like theirs upside-down!

John Baez (Nov 09 2021 at 19:14):

They make $[-\infty,\infty]$ into a closed symmetric monoidal category where the monoidal structure is addition and the closed structure is subtraction! (We haven't been talking about subtraction.)

Owen Lynch (Nov 09 2021 at 19:50):

John Baez said:

But they are studying convex functions, while we've been studying concave ones, so I'm hoping that our approach is just like theirs upside-down!

I think this is exactly right!!

John Baez (Nov 09 2021 at 19:54):

Great!! I had completely forgotten this business when we reinvented the commutative monoid for $[-\infty,\infty]$ based on our application to thermodynamics.

John Baez (Nov 09 2021 at 19:55):

It's interesting that they're making it into closed symmetric monoidal category (with numbers as objects, and a single morphism from $x$ to $y$ when $x \ge y$, which is probably the opposite of what we'd use.)

John Baez (Nov 09 2021 at 19:56):

What's interesting to me is that we can get a closed symmetric monoidal category from an abelian group, taking the group elements as objects and having only identity morphisms. Subtraction in the group gives the closed structure.

John Baez (Nov 09 2021 at 19:57):

$[-\infty,\infty]$ is not an abelian group, but they're still getting a closed structure! They must need the non-identity morphisms - coming from the ordering on $[-\infty,\infty]$ - to get this to work.

John Baez (Nov 09 2021 at 19:58):

But we also need the ordering on $[-\infty,\infty]$ for physics, in order to talk about entropy maximization.

Owen Lynch (Nov 10 2021 at 22:05):

In the Willerton n-cafe post, he starts with the pairing of the dual $\langle -,- \rangle \colon V \times V^\ast \to \overline{\mathbb{R}}$.

We can generalize this from vector spaces to convex spaces by defining $X^\ast$ for a convex space to simply be the convex space of convex-linear functions $X \to \overline{\mathbb{R}}$. The rest of the construction of the Legendre transform works for this.

Owen Lynch (Nov 10 2021 at 22:16):

One interesting thing here is that the dual is not cancelative.

John Baez (Nov 11 2021 at 00:02):

One weird thing about this generalization is that when you take your convex space to be a vector space, this new dual doesn't reduce to the usual dual.

Owen Lynch (Nov 11 2021 at 00:03):

Yes, but it almost does

Owen Lynch (Nov 11 2021 at 00:03):

The convex dual of a vector space has as elements linear functionals + constants

Owen Lynch (Nov 11 2021 at 00:04):

And adding a constant to a linear functional doesn't change the maximization behavior

John Baez (Nov 11 2021 at 00:05):

Oh, when you said "convex functions" I thought you meant "convex functions". But now I think you meant "convex-linear functions".

Owen Lynch (Nov 11 2021 at 00:06):

Oh, yeah, that's totally my bad

Owen Lynch (Nov 11 2021 at 00:07):

That is what I meant

John Baez (Nov 11 2021 at 00:38):

Okay, now I'm starting to get it.

John Baez (Nov 11 2021 at 02:23):

I'd still have to check to see if this makes sense.

By the way, for getting some more category theory into the game we should read this paper:

• Simon Willerton, The Legendre-Fenchel transform from a category theoretic perspective.

Simon Willerton (Nov 13 2021 at 08:03):

Sorry to come in in the middle of the conversation! If you are using $\mathbb{R}$ with $\le$ as a closed monoidal category to think about entropy, then you should check out Lawvere's State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions.

Owen Lynch (Nov 13 2021 at 21:33):

Thanks Simon! Lawvere is always relevant!

John Baez (Nov 14 2021 at 00:27):

Never be sorry about joining the conversation, Simon. I was sort of trying to summon you by saying the phrase "Legendre-Fenchel".

Owen Lynch (Nov 14 2021 at 14:24):

@Simon Willerton, what do you think of taking the dual of a convex space to be the space of all convex-linear functions to $\bar{\mathbb{R}}$?