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Stream: deprecated: thermodynamics

Topic: compositional thermostatics

John Baez (Nov 19 2021 at 02:28):

This thread is for discussing this paper:

• John Baez, Joe Moeller and Owen Lynch, Compositional thermostatics.

Abstract. We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical mechanics, quantum statistical mechanics, and also generalized probabilistic theories of the sort studied in quantum foundations. It also allows us to treat a heat bath as a thermostatic system on an equal footing with any other. We construct an operad whose operations are convex relations from a product of convex spaces to a single convex space, and prove that thermostatic systems are algebras of this operad. This gives a general, rigorous formalism for combining thermostatic systems, which captures the fact that such systems maximize entropy subject to whatever constraints are imposed upon them.

John Baez (Nov 22 2021 at 16:05):

I blogged about this paper:

• John Baez, Compositional thermostatics, Azimuth.

John Baez (Nov 22 2021 at 16:07):

So did @Owen Lynch:

• Owen Lynch, Compositional thermostatics, Topos Institute Blog.

Mine gets more into the definition of "convex space" and "thermostatic system", while his goes further in developing the motivations from physics, and how we use our setup to actually compose thermostatic systems.

John Baez (Nov 23 2021 at 21:20):

On Twitter someone reading about our paper recommended this book:

• Jan Brinkhuis, Convex Analysis for Optimization.

He said there's projective geometry lurking around here. I don't know how. Guess I'll have to look at the book!

Owen Lynch (Nov 23 2021 at 21:22):

I have thought before that extensive coordinates are kind of projective, because scaling all of them uniformly does not change the behavior of the system.

John Baez (Nov 23 2021 at 22:14):

Maybe that's what is hinted at. There's also a connection between contact geometry and projective geometry. But until I look at the book, these are just guesses!

Owen Lynch (Nov 23 2021 at 22:40):

I've started to read the book, and so far it's living up to its subtitle of "A Unified Approach"

John Baez (Nov 24 2021 at 01:15):

Good?

Owen Lynch (Nov 24 2021 at 01:51):

Yeah, it's tying together some things that have been confusing me

Owen Lynch (Nov 24 2021 at 15:25):

Their "unified method" is to translate to convex cones. Which reminds me a lot of using unnormalized probability distributions....

Owen Lynch (Nov 24 2021 at 15:52):

They give a different definition of a dual of a convex set, which requires the convex set to be embedded in $\mathbb{R}^n$. Namely, the dual of $C \subseteq \mathbb{R}^n$ consists of all linear functionals $\phi \colon \mathbb{R}^n \to \mathbb{R}$ such that $\phi(C) \subseteq \mathbb{R}_{\geq 0}$.

Owen Lynch (Nov 24 2021 at 15:52):

With this definition, the double-dual is in fact the original space.

Owen Lynch (Nov 24 2021 at 15:53):

Even though the dual is not generally the "same dimension" as the original space.

Owen Lynch (Nov 24 2021 at 16:06):

A convex function is a function whose epigraph is convex, where the epigraph is all of the points lying above the graph of the function.

The necessity of functions taking values in $[-\infty,\infty]$ is also because we want to allow empty and unbounded epigraphs. I.e. the constant function at $\infty$ has an empty epigraph, and a function that takes values of $-\infty$ has an epigraph that contains arbitrarily low points.

Owen Lynch (Nov 24 2021 at 16:09):

They figure out how to define all of the common binary operations on convex sets (minkowski sum, intersection, convex hull of the union, and inverse sum) using translation to convex cones and the operations of $+ \colon X \times X \to X$ and $\Delta \colon X \to X \times X$. I feel like this ought to be able to be categorified using frobenius monoids.

Robin Piedeleu (Nov 24 2021 at 16:17):

Owen Lynch said:

They figure out how to define all of the common binary operations on convex sets (minkowski sum, intersection, convex hull of the union, and inverse sum) using translation to convex cones and the operations of $+ \colon X \times X \to X$ and $\Delta \colon X \to X \times X$. I feel like this ought to be able to be categorified using frobenius monoids.

It has! https://arxiv.org/abs/2105.10946

Owen Lynch (Nov 24 2021 at 16:20):

Amazing!

John Baez (Nov 24 2021 at 17:46):

Owen Lynch said:

They give a different definition of a dual of a convex set, which requires the convex set to be embedded in $\mathbb{R}^n$. Namely, the dual of $C \subseteq \mathbb{R}^n$ consists of all linear functionals $\phi \colon \mathbb{R}^n \to \mathbb{R}$ such that $\phi(C) \subseteq \mathbb{R}_{\geq 0}$.

With this definition, the double-dual is in fact the original space.

I don't believe that unless your original space $C$ was a convex cone.

John Baez (Nov 24 2021 at 17:47):

With this definition of "dual", the dual of any subset $C \subseteq \mathbb{R}^n$ is a convex cone. That's pretty easy to show straight from the definition.

John Baez (Nov 24 2021 at 17:48):

So, the double dual of $C$ cannot be isomorphic to $C$ unless $C$ is a convex cone.

John Baez (Nov 24 2021 at 17:48):

On the other hand, I am willing to bet that if $C \subseteq \mathbb{R}^n$ is a convex cone, its double dual is isomorphic to $C$.

John Baez (Nov 24 2021 at 17:49):

So, I am willing to bet that with this definition of "dual", $C \subseteq \mathbb{R}^n$ is isomorphic to $C$ iff $C$ is a convex cone. Is that right?

Owen Lynch (Nov 24 2021 at 21:33):

Ah, not linear functionals, I'm wrong, affine functionals!

Owen Lynch (Nov 24 2021 at 21:33):

But for linear functionals, yes, a cone