Miguel wrote:

A second, more technical question: the construction of the “phase space” (the cotangent bundle of the space of equilibrium states) is rather natural. However, the construction of the analogous contact manifold does not seem that natural to me: we have to add “by hand” the extra dimension Could the first jet bundle of the space of equilibrium states be the “natural” -dimensional space?

Thanks—yes, that’s a somewhat nicer way to think about what’s going on. As you can see from my article, the extra dimension keeps track of the value of entropy, while a point in keeps track of a point in and the differential of the entropy at the point. Putting these all together we get a 1-jet.

I wouldn’t say the 1-jet approach is “more natural”, not in a technical sense anyway, because the first jet bundle of a manifold is *naturally* isomorphic to the direct sum of its cotangent bundle and a trivial line bundle. And here by “trivial” I don’t mean just “trivializable”: I mean *trivialized*, equipped with a natural trivialization.

This is because the first jet of a function at some point splits naturally into the differential and the value . So, we get a natural isomorphism between the first jet bundle and as bundles over

Jet bundles become a lot more exciting with the second jet bundle: this splits as a direct sum of 3 vector bundles, but not naturally.

But still, the jet bundle viewpoint is nice!

]]>Thank you for your reply! I am looking forward to reading about these new ideas.

]]>Allen’s question could be related to my question about ‘quantizing’ thermodynamics, in the sense of replacing Poisson brackets on the symplectic manifold of extensive and intensive variables by commutators. I wrote about this here:

• Classical mechanics versus thermodynamics (part 2), 23 January 2012.

I didn’t make much progress because I couldn’t figure out the physical significance of this sort of ‘quantization’. Now that I’m spending more time on thermodynamics, I’m inclined to plunge ahead and see what this quantization gives without worrying too much about what it means. That might be the way to figure out what it means.

All of that was phrased in terms of symplectic manifolds, but one should also be able to quantize contact manifolds, and I think the math Allen points out should help. I bet people have already quantized contact manifolds—I should check.

]]>One reason it’s hard to use contact geometry to do new things in equilibrium thermodynamics is that Gibbs already *knew* contact geometry, implicitly, when formulating his approach to thermodynamics. This at least is V. I. Arnol’d’s claim.

However, there’s been a lot of mathematical work on contact geometry since Gibbs, so there could be ways to use theorems about contact geometry to do something new. Unfortunately a lot of the deeper theorems are about contact manifolds not of the form But maybe even these more complicated manifolds have some relevance to thermodynamics, just as more complicated symplectic manifolds (not even cotangent bundles) turn out to be relevant to classical mechanics.

I have *other* ideas for applying contact and symplectic geometry to thermodynamics in new ways, but I’ll talk about those in future blog posts! I’m reviewing this old stuff to set the stage.

Tardy … I see what you did there.

So the first multiplication is one in which (∂/∂𝑥)² means the second partial derivative (so (∂/∂𝑥)²𝑓 = ∂²𝑓/∂𝑥²), while the second is one in which (∂/∂𝑥)² means the square of the first derivative (so (∂/∂𝑥)²𝑓 = (∂𝑓/∂𝑥)², which we can also think of as acting on d𝑓 rather than on 𝑓 itself). And so (𝑥 ∂/∂𝑥)² expands to 𝑥 ∂/∂𝑥 + 𝑥² (∂/∂𝑥)² in the first case but just to 𝑥² (∂/∂𝑥)² in the second case. We get the algebraic structure of the second multiplication by purging all terms with the wrong grade from the structure of the first multiplication. (And we get the Lie bracket, another important way to multiply vector fields, by antisymmetrizing the first multiplication; this is the only operation that keeps the result within vector fields.)

Anyway, the contact manifold has, in addition to coordinates like 𝑥 and ∂/∂𝑥, the coordinate 𝑆. Although we don't have any particular equation in mind yet, we anticipate expressing 𝑆 as a function of the coordinates like 𝑥. So if we want to multiply ∂/∂𝑥 and 𝑆 in a way analogous to the first kind of multiplication, then I think that the answer should involve a symbol ∂𝑆/∂𝑥, specifically ∂/∂𝑥 𝑆 = ∂𝑆/∂𝑥 + 𝑆 ∂/∂𝑥. More explicitly, if 𝑓 is a function on 𝑄 × ℝ, then 𝑆 𝑓, ∂/∂𝑥 𝑓, and 𝑆 ∂/∂𝑥 𝑓 are also such functions, while ∂/∂𝑥 𝑆 𝑓 = ∂𝑆/∂𝑥 𝑓 + 𝑆 ∂/∂𝑥 𝑓 cannot be interpreted this way. But it lies in wait; as soon as you specify a Legendre submanifold, then all of these can be interpreted as functions on 𝑄. In this way, we know how the multiplication operation works. And it's still graded, and it's still true that dropping all terms of the wrong grade recovers the algebra of functions on the contract manifold.

]]>Thanks, I’ll fix it!

]]>A second, more technical question: the construction of the “phase space” (the cotangent bundle of the space of equilibrium states) is rather natural. However, the construction of the analogous contact manifold does not seem that natural to me: we have to add “by hand” the extra dimension Could the first jet bundle of the space of equilibrium states be the “natural” -dimensional space?

]]>Do you know a corresponding noncommutative picture of the contact manifold ?

BTW one of your LaTeXs (after “We get a 1-form”) is tardy.

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