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Stream: deprecated: thermodynamics

Topic: extending concave functions

John Baez (Nov 10 2021 at 18:09):

Here's an idea I don't want to forget, @Owen Lynch! (I want us to work on different ideas as different "topics" in this stream. This will help me not forget them.)

Suppose we have a convex space $X$ obeying the cancellation axiom. Then we can choose an inclusion $X \subseteq \mathbb{R}^n$. I claim any concave function

$f: X \to [-\infty,\infty]$

extends uniquely to a concave function

$f: \mathbb{R}^n \to [-\infty,\infty]$

taking the value $-\infty$ on the complement of $X$. Does this sound correct to you?

John Baez (Nov 10 2021 at 18:10):

I think I've seen this idea (in upside-down form, for convex functions) in Rockafellar's book Convex Analysis.

John Baez (Nov 10 2021 at 18:10):

The idea, as usual, is that entropy $-\infty$ means "impossible".

John Baez (Nov 10 2021 at 18:12):

If this is true, as long we only consider thermostatic systems whose state spaces are convex spaces obeying the cancellation axiom, we don't lose a huge amount by considering only the spaces $\mathbb{R}^n$. And that's nice, because then they form a PROP!

Owen Lynch (Nov 10 2021 at 21:56):

Yes, this is true, and in fact we can derive it from our operad algebra! I.e., this is just applying the functor Ent to the graph of the inclusion $X \subseteq \mathbb{R}^n$.

Owen Lynch (Nov 10 2021 at 22:11):

The problem with only considering thermostatic systems whose state spaces are convex spaces is that we miss out on what I was just talking about in the other stream with duals!! I.e., the dual of $\Delta^n$ is some quotient of the convex dual of $\mathbb{R}^{n+1}$

Owen Lynch (Nov 10 2021 at 22:12):

And in general, I think that thinking about $\mathbb{R}^n$ is problematic because it's very tempting to fall into thinking about the standard basis, when we really want to do constructions that are basis-invariant!

Owen Lynch (Nov 10 2021 at 22:13):

And inclusion-invariant!

Owen Lynch (Nov 10 2021 at 22:14):

I.e., perhaps this is good for knowing that we aren't missing out on a lot w.r.t. examples; it suffices to consider $\mathbb{R}^n$, but I would worry about only using $\mathbb{R}^n$ in the formalism.

John Baez (Nov 10 2021 at 23:53):

Nice!!! That's a slick approach.

So I think we should investigate an approach where we only use the convex spaces $\mathbb{R}^n$, with arbitrary concave entropy functions taking values in $[-\infty,\infty]$, and then bring in Legendre transforms. (Sorry, this is a bit vague, but it's saying "try seeing what we can do without worrying about other convex spaces").

Owen Lynch (Nov 10 2021 at 23:54):

But we can do Legendre transforms on arbitrary convex spaces!

John Baez (Nov 10 2021 at 23:56):

If you take the Legendre transform of a function $f: K \to [-\infty,\infty]$ where $K$ is an arbitrary convex space, you get a function defined on... what space?

John Baez (Nov 10 2021 at 23:58):

In the case where $K = V^\ast$ is a vector space, the Legendre transform of $f$ is a function defined on the dual $V^\ast$.

John Baez (Nov 10 2021 at 23:58):

I don't know what's the dual of a more general convex space!

Owen Lynch (Nov 10 2021 at 23:58):

Did you see my other thread?

Owen Lynch (Nov 10 2021 at 23:58):

https://categorytheory.zulipchat.com/#narrow/stream/306433-practice.3A-thermodynamics/topic/Generalizing.20Legendre.20Transform

John Baez (Nov 11 2021 at 00:01):

No, I hadn't. I'm merged that thread with the thread "Legendre-Fenchel transform".