The n-Category Café

Indeed, the whole ‘ordinary analysis / idempotent analysis’ analogy is very relevant to the business I’m talking about now.

I’ve been avoiding discussing it, because it multiplies the whole discussion by two in complexity — roughly — and I’m already struggling to fit together all the puzzle pieces I’m laying out. Having too many puzzle pieces can be confusing! (Though having too few can be even worse.)

For the most part, you can say that “Legendre transforms show up in classical mechanics, while Fourier transforms show up in quantum mechanics and Laplace transforms show up in statistical mechanics”, with classical mechanics being the $\hbar \to 0$ limit of quantum mechanics and the $T \to 0$ limit of statistical mechanics, and with quantum mechanics and statistical mechanics linked by $i \hbar \cong T$.

So, in particular, if you take the Laplace transform

$$\widehat{f}(k) = \int_0^\infty exp(- k x) f(x) d x$$

and replace ordinary analysis by idempotent analysis, you get the Legendre transform:

$$\widehat{f}(k) = inf_{k \in [0,\infty)} f(x) - k x$$

Or something like that. If you push me harder by asking more specific questions, I can try to answer them, and then fall on my face, and then figure things out.

There are plenty of anomalies making the puzzle hard to solve. Here’s one:

The engineers like this big chart:

                displacement    flow          momentum      effort
                     q           q'              p            p'
Mechanics       position       velocity       momentum      force
(translation)
Mechanics       angle          angular        angular       torque
(rotation)                     velocity       momentum
Electronics     charge         current        flux          voltage
                                              linkage
Hydraulics      volume         flow           pressure      pressure
                                              momentum
Thermodynamics  entropy        entropy        temperature   temperature
                               flow           momentum
Chemistry       moles          molar          chemical      chemical
                               flow           momentum      potential

Typically in engineering each row in the chart is treated ‘classically’ — but we can also treat them quantum-mechanically, or statistical-mechanically. In the first two rows this is a famous thing to do. There’s also some interesting work on quantization of electrical circuits, which I’ll discuss someday if I live long enough. But the row called thermodynamics is a bit disturbing, since thermodynamics is the macroscopic story that’s ‘explained’ by statistical mechanics! In week289 I asked about the idea of quantizing thermodynamics. It’s an odd thing to do from a physics perspective. For example, I’ve never heard people talk about a ‘temperature operator’ and ‘entropy operator’ that may fail to commute. But mathematically it’s very easy, thanks to this analogy chart!

But just as easily as you can quantize thermodynamics, you can also replace $i \hbar$ by $T$ and ‘statistical-mechanize thermodynamics’. And that’s even more odd! You get two different things called temperature, for example.

But of course these curious problems are part of what makes analogies so much fun.

Interdepartmental communicating of my interdisciplinary PhD research 1973-1977 was difficult, because the Biologists on my Dissertation Committee didn’t know about Laplace Transforms (as I was composing subsystems into systems with feedback), the Computer Scientists didn’t know any Metabolomics (as basic as the Michaelis-Menten equations, ODEs for chemical flux in systems of enzymes), the Mathematicians didn’t see the point in the interactive computer graphics front end that I built for my simulations, and so forth. This ignorance by each discipline about each other discipline is why each developed different vocabulary and notation, and why this half century of unifying meta-theory was so urgently needed.

Daniele wrote:

I really missed in your recent weeks the so called “memristor”. It’s been the focus of a lot of press as well as new venues for innovation. What is its place in this whole scheme?

It’s coming, along with some other funky 1-ports I haven’t gotten to yet.

From one viewpoint, I already have enough 1-ports, 2-ports and 3-ports to illustrate the overall scheme, and extra examples just need to be put in their proper slots. Are memristors active or passive? If passive, are they conservative or dissipative? Are they linear or nonlinear? Once we know that, we know where they belong in the grand scheme.

From another viewpoint, each individual $n$-port deserves its own detailed study! I would need to learn more about memristors to do them justice.

Thanks — fixed! The dangers of naive cut-and-paste.

Not to get into Econophysics (I know Economists and Physicists who love the term, and others who hate it). But a simple back-of envelope question: How cost-efficient is power from burning one-dollar bills? Say, several tons at a time, to boil water for a steam plant? What emissions are there? Do they have to be beneficiated by removing a metal wire, or is that only in higher denominations bills? This could be a useful baseline/normalization for public exposition, or business plans, on power generation systems.

Doug wrote:

I like seeing the unification of particle physics with engineering techniques as in your “big analogy chart”.

Thanks! But I didn’t really say much about particle physics, except for the humble case of classical point particles. There is indeed an analogy between circuit diagrams and Feynman diagrams, but I haven’t said much about that yet.

I would add that it is relatively easy to also incorporate economics techniques, since basically physics and engineering deal with energy economics.

As the ‘smart grid’ grows and matures, this may become a big deal.

In week289 I mentioned a version of bond graphs where instead of

$$position, momentum,$$ $$velocity and force$$

we have

$$inventory, economic momentum,$$ $$product flow and price$$

But I haven’t studied this version and I don’t know how much sense it makes.

James Dolan also pointed out an analogy where time and interest are canonically conjugate variables, with the present value of a future income stream $f(t)$ at interest rate $k$ being the Laplace transform

$$\widehat{f}(k) = \int_0^\infty exp(-k t) f(t) d t$$

Jonathan wrote:

But a simple back-of envelope question: How cost-efficient is power from burning one-dollar bills?

Here’s what I learned online. Maybe it’s true, maybe not.

A dollar bill weighs about a gram.

Colored office paper has a gross calorific value of about 6300 BTU per pound.

A British Thermal Unit is about 1055 joules, and a pound is about 454 grams.

That gives me about 14,600 joules per dollar bill.

That seems like a shockingly large number at first, so let me compare: a cup of whole milk has about 150 ‘calories’, but these are really kilocalories, and a kilocalorie is 4184 joules, so that’s 628,000 joules.

So, okay, that sounds about right: you’d need to eat about 40 dollar bills to get the caloric equivalent of a cup of milk, assuming you could metabolize cellulose and extract all the energy you could from burning it! For example, if you were a termite…

Now let’s compare gasoline. For example, we can imagine a car that runs on dollar bills, and see how it would stack up against a normal petrol-powered car.

Conveniently, it says here that “If it were possible for human beings to digest gasoline, a gallon would contain about 31,000 food calories – the energy in a gallon of gasoline is equivalent to the energy in about 110 McDonalds hamburgers!”

So, a gallon of gasoline, which costs about $3 in California right now, contains about 130,000,000 joules. That’s about 40,000,000 joules per dollar. Whereas the dollar bill itself provides only about 628,000 joules if you burn it.

That’s about a factor of 64.

any chance you could comment on Verlinde’s new paper

There was some discussion in the comments to Week 191.

I might get around to talking about Verlinde’s paper sometime. If I were trying to ‘boost my ratings’ and get as many people to read This Week’s Finds as possible, I would have done it already!

For now, I again urge everyone to Jacobson’s paper incredibly interesting, back in 1995 — so I again urge everyone to read that. It’s short and you can get a lot out of it even if you skip the 6 numbered equations.

For now, you can think of a linear circuit as one made of linear resistances, capacitances and inductances, along with all the 2-ports and 3-ports I’ve introduced (which are are all linear). I defined resistances, capacitances and inductances in week290, and I said what it meant for them to be ‘linear’.

More generally, an $n$-port is defined by $n$ equations relating $n$ efforts, $n$ flows, and perhaps the time variable $t$. If these equations are all linear in the efforts and flows (for each fixed time $t$), we say the $n$-port is linear. Then, we say a circuit is linear if it’s built from linear $n$-ports.

Of course this definition only works for circuits built from $n$-ports.

I suppose I should fix week292 a bit so that it includes an answer to your question.

Patience! I tried to answer your second question this morning, but I didn’t succeed in posting my reply. I posted it just now.

Jim wrote:

Am I the only one to be puzzled by your treatment of bond graphs compared to Paytner’s? His diagrams you show us manifestly have e.g. 2-inputs and one output??

An $n$-port has a total of $n$ inputs and outputs. I drew a resistor, which is a 1-port:

           e       \
    ----------------- R
           f

This has 1 input at left, and no outputs: the resistor R is not an output: ‘the buck stops here’.

Paynter’s picture also includes 2-ports and 3-ports:

The reservoir is a 1-port, the fluid power transmission is a 2-port, and the speed governor is a 3-port.

Is that what’s bothering you? That I only drew a 1-port? I wanted to draw some 2-ports and 3-ports myself, but I discovered it would take some work to draw them nicely. So, I decided to use Paynter’s picture for those.

If you’re having trouble understanding this stuff, I’m sure lots of other people are, too. So please let me know what’s the problem.