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Stream: deprecated: thermodynamics

Topic: port-Hamiltonian mechanics

Owen Lynch (Jan 24 2022 at 21:07):

Can you send some material on port hamiltonians?

Owen Lynch (Jan 24 2022 at 22:45):

I'm reading this paper: http://www.math.rug.nl/arjan/DownloadPublicaties/ICMvanderSchaft.pdf

Owen Lynch (Jan 24 2022 at 22:46):

It looks like they get halfway to what I got:

image.png

Owen Lynch (Jan 24 2022 at 22:47):

They consider forces as inputs instead of also flows!

Owen Lynch (Jan 24 2022 at 22:49):

Aha! This is what I was talking about: they remove the necessity of having an even-dimensional symplectic manifold by considering something more general:
image.png

Owen Lynch (Jan 24 2022 at 22:53):

This paper is extremely exciting

Owen Lynch (Jan 24 2022 at 22:54):

If I can make port-Hamiltonians into an operad, I think that I will have done something important

Owen Lynch (Jan 24 2022 at 22:55):

image.png

It's a Lagrangian subspace!

Owen Lynch (Jan 24 2022 at 22:57):

Oh wait, not quite

Owen Lynch (Jan 24 2022 at 22:58):

It's not with respect to symplectic structure, it's with respect to the normal bilinear form

Owen Lynch (Jan 24 2022 at 22:58):

Interesting

Owen Lynch (Jan 24 2022 at 23:05):

OK, this guy actually wrote a more recent book on this: http://www.math.rug.nl/arjan/DownloadVarious/PHbook.pdf

Owen Lynch (Jan 24 2022 at 23:05):

Interestingly enough, he considers switched systems

Owen Lynch (Jan 24 2022 at 23:06):

Which is what I did to derive the Carnot cycle

Owen Lynch (Jan 24 2022 at 23:07):

Ah, and they even do a bit of thermodynamics in this book: http://www.math.rug.nl/arjan/DownloadVarious/PHbook.pdf

Owen Lynch (Jan 24 2022 at 23:07):

Whoops

Owen Lynch (Jan 24 2022 at 23:08):

image.png

Owen Lynch (Jan 24 2022 at 23:18):

He wrote a very recent paper (2020) on thermodynamics: https://arxiv.org/pdf/2010.04213.pdf

John Baez (Jan 24 2022 at 23:44):

It's not "port hamiltonians", it's always "port-hamiltonian". "Port-" is basically a synonym for "open".

The first paper you linked to by van der Schaft is one of the good papers I was going to find for you. Equations (1-5) should remind you of things you told me about... though maybe they're a bit different?

Owen Lynch (Jan 25 2022 at 22:07):

OK, so my "flow systems" are pretty much exactly port-Hamiltonian systems, I just hadn't worked out all the details yet.

Owen Lynch (Jan 25 2022 at 22:07):

I wish you had told me about these earlier... maybe you did but I just didn't listen

Owen Lynch (Jan 25 2022 at 22:45):

OK, so making port-Hamiltonian systems into an operad algebra is almost completely trivial

Owen Lynch (Jan 25 2022 at 22:51):

I'll write it up more formally, but the short answer is that there is a PROP where the morphisms between $[n]$ and $[m]$ are the "Dirac structures" $D \subset (\mathbb{R}^n)^\ast \times \mathbb{R}^n \times (\mathbb{R}^m)^\ast \times \mathbb{R}^m$. These are similar to Lagrangian relations, but the definition is slightly different.

Then there is a lax symmetric monoidal functor from this PROP to Set that sends $[n]$ to the set of port-Hamiltonian systems with interface $[n]$. The laxator takes the product of the state spaces of the various port-Hamiltonian systems, and adds together the Hamiltonians.

Then, we apply the same construction we did in the thermostatics paper to make an operad.

Owen Lynch (Jan 25 2022 at 22:52):

It's almost boring how easy this is

Owen Lynch (Jan 25 2022 at 22:53):

Arjan van der Schaft has done pretty much all of the necessary proofs, he just didn't know about SMCs or operads

John Baez (Jan 25 2022 at 23:49):

Owen Lynch said:

I wish you had told me about these earlier... maybe you did but I just didn't listen

It's a bit hard to know what I should tell you about before it becomes clearly relevant. I believe existing port-Hamiltonian mechanics only handles systems with vector spaces as phase spaces. I'm pretty sure I told you about my paper Open systems in classical mechanics which describes an approach that can handle general symplectic manifolds. The last paragraph in the intro says:

In future work we hope to compare the present approach to the theory of port-Hamiltonian systems, an approach to open systems in classical mechanics widely used in engineering [29].

[29] is a reference to "Port-hamiltonian systems theory: an introductory overview" by van der Schaft and Jeltsema. That's a different paper than the one you're reading, right?

Owen Lynch (Jan 26 2022 at 00:04):

Chapter 3 of the book by van der Schaft handles manifolds: http://www.math.rug.nl/arjan/DownloadVarious/PHbook.pdf#chapter.3

John Baez (Jan 26 2022 at 00:06):

Interesting - my friend Eugene Lerman, who is into symplectic geometry and wrote a paper with Spivak, complained that work on port-Hamiltonian systems assumed a linear phase space! I'd never seen a good nonlinear version. But my coauthor David Weisbart was working on one. (I think he's gotten distracted.)

Owen Lynch (Jan 26 2022 at 01:24):

It is not as easy (I think) to do the operad with the nonlinear version, so there may yet be opportunities for improvement

Owen Lynch (Jan 26 2022 at 01:29):

Also, my impression (which could be off; I've only skimmed the paper) is that for Open systems in classical mechanics, you compose systems by glueing together their phase spaces via pullback. In port-Hamiltonian systems, one takes the product of the phase spaces, and then equalizes forces and fluxes. So perhaps these are complementary approaches.

Owen Lynch (Jan 26 2022 at 17:30):

The fact that this paper treats intensive quantities using projective geometry feels very right to me.

Owen Lynch (Jan 26 2022 at 17:30):

https://arxiv.org/pdf/2102.05493.pdf

Owen Lynch (Jan 28 2022 at 16:33):

I wonder if the projectivization of the symplectic manifold has something to do with the fact that thermodynamics can happen at any speed.

Owen Lynch (Jan 28 2022 at 16:34):

I.e., a quasi-static process is a limit as the path taken becomes infinitely slow.

Markus Lohmayer (Jan 31 2022 at 13:59):

Hi Owen, I don't think so. Maybe I am also trying to tell you something that you know better than I do. The contact structure relates the thermodynamic variables (extensive and intensive) according to some generating function (say internal energy) which is called a thermodynamic potential. There are other thermodynamic potentials (Helmholtz free energy, Gibbs free energy, and enthalpy) which generate the same contact manifold and they are all related via the Legendre transformations. The potential (or the contact manifold that it generates) encodes the material behavior. However, the same material behavior could be encoded by entropy and the corresponding Legendre transformed quantities, sometimes called Massieu functions. The idea of projectivization is to encode the material behavior by (homogeneous) Hamiltonian function and a symplectic manifold. The advantage of that is that energetic and entropic representations of the material behavior are united. This might give you a lot of flexibility in choosing variables and Arjan says that some calculations are "simpler" in the symplectic realm but that might also be his preference, idk. I want to clarify that this has nothing to do whatsoever with port-Hamiltonian systems. However, Arjan and Bernhard have worked on porting the "port-cocept" to these systems, then termed "port thermodynamic systems".

Owen Lynch (Jan 31 2022 at 14:37):

Hi Markus! Thanks for popping in here; I'm not sure if I've met you before but it's always good to hear from someone else interested in thermodynamics.

Mathematically, I know that the projectivization doesn't have anything to do with the quasistatic nature; it's just a way of getting at the geometric structure in a convenient way. But one of my eventual goals is to see if contact geometry can be united with/derived from the statistical mechanics of a symplectic manifold. It is my (unfounded) intuition that time-rescaling may come into play in this derivation, and this seems connected to projectivization in my head.

Markus Lohmayer (Jan 31 2022 at 15:25):

We unfortunately didn't meet yet and I am not that much familiar with statistical mechanics. Although maybe not directly related, maybe you could find something interesting/inspiring in one of the following references: 2020 Montefusco Öttinger Peletier - A Framework of Nonequilibrium Statistical Mechanics ; 2017 Kraaij Azarescu Maes Peletier - Deriving GENERIC from a generalized fluctuation symmetry ; 2021 Patterson Renger Sharma - Variational structures beyond gradient flows A macroscopic fluctuation-theory perspective ; In case unfamiliar, GENERIC is in the first place a different framework for writing thermodynamic systems beyond equilibrium (i.e. with dynamics) that might appear to have less geometric structure. Yet, the equations are coordinate-independent, and GENERIC can be written based on the contact formalism.

Owen Lynch (Jan 31 2022 at 15:36):

These look very interesting! Thanks for the references!