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Stream: deprecated: thermodynamics

Topic: port-thermodynamic systems

Owen Lynch (Feb 20 2022 at 19:16):

To stay grounded in where I want to go, I returned to the van der Schaft papers on thermodynamics, and I found that any input-state-output port-Hamiltonian system can be seen as a port-Thermodynamical system.

And by port-Hamiltonian system, he includes a resistive structure! So he is actually describing the resistive structure here as conversion into heat and dumping into a heat bath, which he calls the "internal energy".

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Owen Lynch (Feb 20 2022 at 19:17):

This was my dream! To represent resistors as dumping heat into a heat bath!

Owen Lynch (Feb 20 2022 at 19:18):

The thing is, it doesn't look like he does this for general port-Hamiltonian systems; only for those factored into input-state-output form

Owen Lynch (Feb 20 2022 at 19:18):

It seems like his formulation of port-Thermodynamic system is restricted to this explicit factoring, as far as I can tell

John Baez (Feb 20 2022 at 19:28):

That's good in a way because it leaves you something to do. In real-world systems the so-called "output" can often affect the so-called "input". Jan Willems has some important rants about this.

Owen Lynch (Feb 20 2022 at 19:30):

I feel like at some point it might be a good idea to reach out to Arjan and talk to him about this stuff, though after I understand enough of it that I won't make a fool out of myself

Owen Lynch (Feb 20 2022 at 19:32):

I.e., I bet he's thought about a more general definition of port-Thermodynamical systems

John Baez (Feb 20 2022 at 20:37):

Yes, it would be good to talk to him.

Robin Piedeleu (Feb 21 2022 at 09:36):

John Baez said:

That's good in a way because it leaves you something to do. In real-world systems the so-called "output" can often affect the so-called "input". Jan Willems has some important rants about this.

Yes! I feel that this is an underestimated point. A lot of constraints in physics are fundamentally relational, and do not seem to impose any intrinsic distinction between inputs and outputs. The distinction only seems to be relevant from the external perspective of agents who interact with a (necessarily open) system. Nevertheless, I still think that input and output are useful notions - and so does Jan Willems despite his famous rants - and I'd be interested to read work that defines them carefully, if you know of any.

This is now taking us far from the purpose of this stream (sorry for hijacking it), but I do think that conceptual clarity about these ideas would also benefit computer science, where systems are built precisely for the purpose of exhibiting a specific input/output behaviour. Perhaps that's a controversial claim, but I get the sense that the field is generally confused about these notions (in particular in theoretical areas stemming from concurrency, distributed systems, coalgebra, automata etc.), though this is likely more a reflection of my own ignorance than of the field itself.

Robin Piedeleu (Feb 21 2022 at 09:51):

Owen Lynch said:

This was my dream! To represent resistors as dumping heat into a heat bath!

This might not be exactly what you're looking for, but the bond graph literature also has so-called RS-elements, to represent systems which convert energy in some mechanical/electrical/other domain to the entropy domain. I believe resistors (when one also wants to keep track of energy dissipated in the form of heat) should fit in this category. The idea can be found in J. Thoma's Introduction to Bond Graphs and their Applications for example.

Owen Lynch (Feb 21 2022 at 13:36):

Oh, excellent, thank you

John Baez (Feb 21 2022 at 16:32):

Owen has been sneaking up on bond graphs through his study of Dirac structures, which are a kind of elite mathematical approach to what engineers do with bond graphs. I seem to recall that Thoma's book is one of the best on bond graphs: at one point I read all the books I could get my hands on.

John Baez (Feb 21 2022 at 16:35):

A lot of constraints in physics are fundamentally relational, and do not seem to impose any intrinsic distinction between inputs and outputs. The distinction only seems to be relevant from the external perspective of agents who interact with a (necessarily open) system. Nevertheless, I still think that input and output are useful notions - and so does Jan Willems despite his famous rants - and I'd be interested to read work that defines them carefully, if you know of any.

I think it's an interesting physics question how despite the reciprocity principle which holds quite generally in physics - roughly, the symmetry between inputs and outputs - we are able to build physical systems that act as if the input affects the output and not vice versa.

Owen Lynch (Feb 21 2022 at 16:37):

It seems like the Willems approach is to treat systems as fundamentally relational, but then consider input-output "factorizations", which are non-unique for a particular system

Owen Lynch (Feb 21 2022 at 16:39):

Something interesting that I've seen in the bond-graph literature is to build directionality into relational systems, so that although the systems are not strictly input-output, the computations are still directed

Owen Lynch (Feb 21 2022 at 16:39):

Something feels very homotopy-theoretic about the non-canonicity of input-output factorizations...

John Baez (Feb 21 2022 at 16:40):

Oh?

Owen Lynch (Feb 21 2022 at 16:42):

I don't really have deep thoughts on this, it's just that my homotopy spider senses always trigger when there are lots of equivalent choices to be made

Owen Lynch (Feb 21 2022 at 16:42):

Perhaps a more precise word would be "weakness"

Owen Lynch (Feb 21 2022 at 16:43):

Kind of like how we end up with a double category for structured cospans because pullbacks are only up to isomorphism

Owen Lynch (Feb 21 2022 at 16:45):

I have a similar intuition that something like this is going to show up if we define a Dirac structure as being defined by its so-called kernel representation, i.e. given as the kernel of some matrix.

Owen Lynch (Feb 21 2022 at 16:45):

Which is the form that we need Dirac structures in if you want to compute with them

Owen Lynch (Feb 21 2022 at 16:46):

And I expect there are special ways of writing that matrix that make computing with Dirac relations easier

Owen Lynch (Feb 21 2022 at 16:47):

But, undoubtedly, whatever categorical construct we come up with that has a matrix as part of the definition of an object will need to be treated "weakly"

Robin Piedeleu (Feb 21 2022 at 17:54):

John Baez said:

I think it's an interesting physics question how despite the reciprocity principle which holds quite generally in physics - roughly, the symmetry between inputs and outputs - we are able to build physical systems that act as if the input affects the output and not vice versa.

I would love to see a definition of input/output in the most general setting one can imagine in physics. Is it possible to formally define the two notions? Is it just work performed on/by a given system?

John Baez (Feb 21 2022 at 19:03):

That's really tough. In control theory they have thoughts about how one physical system can control another through 'information': for example if you flip a switch to turn on a light it can take very little work, but in some approximation you are affecting it while it is not affecting you. But these thoughts were never tied to physics very carefully in anything I read.

Robin Piedeleu (Feb 22 2022 at 09:11):

It would be great to tie them to physics then! Seems like there might even be links with thermodynamics: in general, setting an input to some value (flipping the switch) should be a not-necessarily reversible operation, involving exchange of entropy between the system and the agent/environment who sets the input. This is highly speculative, but perhaps the directionality of inputs and outputs can be clarified in this setting.

John Baez (Feb 22 2022 at 17:28):

I agree that thermodynamics has to come into it. A switch like a light switch is supposed to be reversible, but it takes a little energy to flip the switch: you're pushing something over a potential barrier, and when it comes back down that energy gets lots to heat. The energy should be big enough that the switch doesn't flip by accident. It would be nice to work this out clearly.

Actually in control theory they are more interested in something like a thermostat that measures something about the system and uses the measurement to change something else; this too deserves a good analysis. In particular: how is it that the temperature in the room affects the thermostat but the thermostat doesn't affect the temperature in the room - or at least, not enough to matter?

John Baez (Feb 22 2022 at 17:28):

(The furnace controlled by the thermostat affects the temperature in the room, but not the thermostat itself.)

Owen Lynch (Feb 22 2022 at 17:34):

So, in systems+control theory, the input/output distinction is far less philosophical than all of this. An input/state/output factorization is a factorization where you can choose any path for your inputs, and an initial state, and then there exists a unique path for the outputs.

Owen Lynch (Feb 22 2022 at 17:35):

In a system with no state, where the behavior is to set the output equal to a bijection of the input, then you could just as easily say that the "output" is the input and the "input" is the output

Owen Lynch (Feb 22 2022 at 17:36):

This is why output/input is non-canonical

John Baez (Feb 22 2022 at 17:36):

I don't think I was being "philosophical" (which sounds like an insult here).

Owen Lynch (Feb 22 2022 at 17:36):

Perhaps "physically determined" would be a better adjective

John Baez (Feb 22 2022 at 17:37):

But anyway, that's a good way to try to define an input/state/output factorization. I think it only exists in some limiting case, in reality - that is, a limit where you ignore some effects that are supposed to be 'negligible'.

Owen Lynch (Feb 22 2022 at 17:38):

What do you mean?

Owen Lynch (Feb 22 2022 at 17:39):

Here's an example, by the way. If you have a capacitor, then you can choose any assignment of currents through time, i.e. function $I(t)$, and given an initial charge $Q_0$ on the capacitor, you get an output $V(t)$.

John Baez (Feb 22 2022 at 17:40):

Well, like I said, in a good thermostat we're allowed to pretend that the temperature of the room affects the dial on the thermostat and not vice versa. This is a good approximation, but it's not perfectly true. So part of the task of system design is trying to make systems where 'back-reaction' is negligible.

Owen Lynch (Feb 22 2022 at 17:41):

Ah, this is the subtle bit. Just because there is a back-reaction, doesn't mean that there doesn't exist a unique output for each input.

Owen Lynch (Feb 22 2022 at 17:42):

I.e., even if in the physics, the causality runs in both directions, we can still mathematically make the factorization

John Baez (Feb 22 2022 at 17:42):

Yeah. I'd like to think about what happens when you try to use the 'output' voltage to affect some other system. The other system will affect the output voltage and thus the 'input' current.

John Baez (Feb 22 2022 at 17:43):

So I think there are a couple of concepts here....

Owen Lynch (Feb 22 2022 at 17:44):

Hmm, I don't know how another system could affect the output voltage...

Owen Lynch (Feb 22 2022 at 17:44):

The voltage is purely determined by charge on the capacitor

Owen Lynch (Feb 22 2022 at 17:44):

You'd have to change the charge on the capacitor, and the only way to do that is to apply a current

Owen Lynch (Feb 22 2022 at 17:45):

You are right though in that in most electronic systems, the "output voltage" is not actually a pure output

Owen Lynch (Feb 22 2022 at 17:46):

I just don't think that's the case for a pure capacitor

Robin Piedeleu (Feb 22 2022 at 17:46):

What do you call a "pure output"?

Owen Lynch (Feb 22 2022 at 17:47):

I.e., if you put different resistors between the "output" of a circuit, you will get different voltages

Owen Lynch (Feb 22 2022 at 17:47):

It cannot supply infinite current in the limit

Owen Lynch (Feb 22 2022 at 17:48):

For an ideal capacitor, however, it does supply infinite current in the limit

Owen Lynch (Feb 22 2022 at 17:48):

At least, for an instant

Robin Piedeleu (Feb 22 2022 at 17:56):

Owen Lynch said:

In a system with no state, where the behavior is to set the output equal to a bijection of the input, then you could just as easily say that the "output" is the input and the "input" is the output

This assumes reversible dynamics, right? If you introduce some coarsening of the state-space, perhaps by ignoring some "negligible" effects as John suggested, you might have some genuine outputs that cannot be turned into inputs?

Owen Lynch (Feb 22 2022 at 17:58):

Definitely

Owen Lynch (Feb 22 2022 at 17:59):

The underlying model will constrain which input/output factorizations are possible

Owen Lynch (Feb 22 2022 at 17:59):

In general, there could be a unique one, many, or none

Owen Lynch (Feb 22 2022 at 18:00):

I think it's a very interesting question to also look at approximate input/output factorizations

Owen Lynch (Feb 22 2022 at 18:00):

Which might hold, for instance, as long as you don't put too much "load" on the output

Joe Moeller (Feb 22 2022 at 18:34):

I'm reminded of the rocket equation. The scenario is that you're trying to determine the fuel needed for a rocket, so you calculate that based on the weight. That gives some amount of fuel that has its own weight, so now you have to recalculate.

John Baez (Feb 22 2022 at 19:33):

"It's not rocket science." :upside_down: