In week296 of This Week’s Finds, you can get a free book on how Felix Klein used the icosahedron to solve the quintic equation. And then we’ll try to construct a compact dagger-category where the morphisms are electrical circuits made of resistors!
A great photo of Eyjafjallajökull, taken by Marco Fulle of Stromboli Online:
TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2208
The trouble is in the existence of the bent short-circuits. At first, we only get short-circuits at all by throwing them in ad hoc as identities, to make the semicategory into a full-fledged category. That process doesn’t give you bent short-circuits, and it’s at this point that John makes the claim that composites of non-identities are non-identities. It’s only later that John says that tossing in the bent short-circuits doesn’t cause any contradictions (meaning, I assume, that the inclusion functor is faithful?), and at this point your argument shows that the underlying category no longer arises from a semicategory.
Long time no see, John!
John Armstrong wrote:
Neither of these two diagrams is an identity – indeed, they don’t even seem go from one object back to itself – and yet their composite is the short-circuit. What have I misunderstood?
Owen explained, but let me take a crack at it. I start by describing a ‘category’ whose morphisms are circuits made of wires with positive resistance. Alas, this ‘category’ has no identity morphisms — it’s just a semicategory.
I point out that this can be cured simply by giving every object an identity morphism. This method for turning a semicategory into a category clearly produces precisely those categories that have this special property: the composite of non-identity morphisms is always a non-identity morphism.
Alas, the category we get this way lacks the cup:
| | | | \_/
the cap:
_
/ \
| |
| |
and the “symmetry”:
\ /
\ /
/
/ \
/ \
And as you suggest, this is inevitable — since these are non-identity morphisms which when composed with each other (in the correct way) give identity morphisms!
So, back to the drawing board…
We could throw these morphisms in by hand, but I take another route, and get identities, caps, cups and the symmetry in a less artificial way by treating electrical circuits as ‘linear canonical relations’.
‘Canonical relations’ (otherwise known as Lagrangian correspondences) are a slight generalization of the ‘canonical transformations’ (otherwise known as symplectomorphisms) beloved in classical mechanics.
Oh, I’ve been around, but there hasn’t been much for the apostate to add to the discussion.
I followed the semicategory -> category thing, but once you add the desired structure (which can clearly be built in terms of actual electronic circuits) it seems not to have come from a semicategory anymore. On the other hand, maybe this is a place where we really do gain something by considering planar algebras instead of monoidal categories? That is, there’s a “semi-(planar algebra)” structure, which can be similarly completed to a planar algebra.
Incidentally, I don’t recall seeing anyone mention this recent XKCD, which I hope can be brought into the categorical rubric.
do we think the “holding pen” in there contains velociraptors?
I followed the semicategory -> category thing, but once you add the desired structure (which can clearly be built in terms of actual electronic circuits) it seems not to have come from a semicategory anymore.
Right. He abandons that approach and treats the circuits as linear canonical relations instead.
Are you hinting with the discussion of the lumped/distributed distinction that we might want to consider 2-categories rather than identify morphisms corresponding to the same function?
David wrote:
Are you hinting with the discussion of the lumped/distributed distinction that we might want to consider 2-categories rather than identify morphisms corresponding to the same function?
I wasn’t trying to hint that — I was trying to hint that next week I might give a mathematical description of the category whose morphisms are distributed electrical circuits made of resistors.
But, it’s a neat idea. There are lots of options, depending on which moves between circuits you want to treat as 2-morphisms, and which you want to keep as equations.
In fact Larry Harper here at the math department of UCR surprised me a few weeks back by saying he was trying to categorify the theory of electrical circuits. He hadn’t been reading This Week’s Finds and didn’t know what I’ve been up to….
He’s a combinatorist who works on optimization problems. One of his favorite tricks is taking a bunch of problems of a given type and making them into the objects of a category, where the morphisms are ways of reducing one problem to another.
He’s used this idea to study the Ford-Fulkerson algorithm. This algorithm tackles a very nice problem. Imagine a network of pipes. Each pipe has an upper limit on how much water can flow through it. You’re trying to pump as much water from one point to another. What’s the maximum you can achieve?
Mathematically, you’ve got a ‘flow network’: a graph where each edge is labelled with a nonnegative number describing the capacity of that edge. Larry Harper came up with a concept of ‘flow morphism’ between flow networks, which makes these problems into a category. And now he’s generalized the whole idea, including these flow morphisms, to a class of electrical circuits.
But I would like to think of flow networks as themselves morphisms in a category. So then you’d have a 2-category!
Here’s a reference… the first one I was able to quickly turn up:
J. D. Chavez, A natural notion of morphism for linear programming problems, Journal of Combinatorial Theory, Series A, 52 (November 1989), 206–227.
Abstract. In this paper we present a morphism for linear programming problems, called block reduction, and construct the category LP* which has block reductions as morphisms and linear programming problems as objects. Block reduction is a natural notion of morphism for linear programming problems in that it is an extension of the flow morphism for network flow problems (presented by Harper in Adv. in Appl. Math. 1 (1980), 158–181), and at the same time it is a restriction of the reduction on linear programs (presented by Harper in J. Combin. Theory Ser. A 32 (1982), 281–298). After establishing this, we prove the existence of pushouts for block reductions.
That is the subject of the thesis Chavez wrote under the supervision of Harper.
One of his [Harper’s] favorite tricks is taking a bunch of problems of a given type and making them into the objects of a category, where the morphisms are ways of reducing one problem to another.
Interesting. I had conversations with machine learning colleagues about the idea of reducing one task to another. Sometimes this could be done for seemingly very different looking problems. So I guess the tricky thing is to carve out a useful type of problem, and then to come up with a sensible notion of reduction.
David wrote:
I had conversations with machine learning colleagues about the idea of reducing one task to another. Sometimes this could be done for seemingly very different looking problems. So I guess the tricky thing is to carve out a useful type of problem, and then to come up with a sensible notion of reduction.
Yes — I guess this gets hard if we consider a large class of problems that look very different… but Larry has focused on cases where we have a class of problems that all look fairly similar. Like: all network flow problems, or all linear programming problems. And he gets some mathematically tractable and interesting categories this way.
Could he be encouraged to make this work more easily available? Guest post?
I don’t think he’s a “guest post” kind of guy, but I can ask. I can also see if he’s willing to make some of his papers freely accessible!
If I succeed, you’ll hear about it here.
You conjecture that the functions Q giving power as a function of the potentials at the exposed wires from some collection of wires and resistors are all non-negative quadratic forms that do not change if you add the same number to each potential. I believe that this conjecture is false.
Consider Q(x, y, z) = (x + y - 2z)^2. If you put in voltages of -1, 1 and 0 you’ll get 0 power, indicating no net current and infinite resistance within the circuit. But voltages of 1, 1 and 0 gives you a power of 4, when I’d expect a power of 0 again from the infinite resistance we saw earlier. This doesn’t seem physically plausible.
In fact let me go farther.
Here is an approach that may lead to a proof and a correct characterization of all possible circuits. Take a circuit with 3 exposed wires. Apply voltages of 1, 0, 0. Let c_12 be the current flowing out of wire 2, and c_13 be the current flowing out of wire 3. Next apply voltages of 0, 1, 0. Let c_21 be the current flowing out of wire 1, and c_23 be the current flowing out of wire 3.
Now I strongly suspect that c_12 and c_21 must be the same. Can this be proven? Does this generalize? In an n exposed wire circuit can it be shown that for any distinct i and j, that in a configuration where wire i is at 1 volt and everything else is at 0, the current c_ij flowing from i to j is the same as the current c_ji flowing from j to i if j is at 1 volt and everything else is 0?
If it can be, then for any circuit at all we can collect all of the measurements of the currents flowing in and out when each of the first n-1 wires are at 1 volt and everything else is at ground. Note that these measured currents are all non-negative. By linearity if we apply any voltages at all we can figure out all of the currents in and out of all of the wires. We can therefore figure out how much work must be applied to keep the voltages there, and therefore how much work is being spent in the circuit. When you calculate it and expand the terms out the work is the sum, over all unordered pairs {i, j} with i not equal to j, of c_ij(x_i - x_j)^2.
An immediate consequence is that all cross terms x_i*x_j must be non-positive. Which tells us that (x+y-2z)^2 can’t work because it has a positive cross term of 2xy.
Note that all polynomials written out that way can easily be realized by real circuits by just connecting wires with appropriate resistances directly between the exposed wires. Each term tells us the current flowing at 1 volt between two exposed wires, which therefore tells us the resistance we need to have. (A current of 0 means no wire.)
I leave finding a nicer characterization of all polynomials of that form to the reader.
Thanks for taking a stab at this, Ben!
Say we have an electrical circuit with inputs. (There’s no big difference between inputs and inputs for this sort of category, so let’s call them all inputs.) And suppose it’s built by hooking each input to each other input by a resistor. What kind of quadratic form describes its power consumption?
Say we connect the th input to the th input with a resistor of resistance . Then the power consumed by the circuit is
where is the electrostatic potential of the th input.
As you note, is a quadratic form where the coefficients of all the cross-terms are negative… or at least non-positive, if we allow .
So if every circuit built from resistors is equivalent to one of this form, your conjecture is true.
Here is another idea for how to prove that.
We know that all circuits with no interior free nodes have power consumption that fits that form. Perhaps we can prove it by induction on the number of interior free nodes?
So suppose that we have a circuit whose power consumption fits that description. Take one node, and disconnect it, turning it into an interior free node. The voltage at that node will become some weighted average of the voltages of all of the remaining inputs. Do you wind up with a circuit whose power consumption is of the same form? This should be a straightforward, though messy, calculation.
If the answer is yes, then by induction that kind of power consumption is true of any finite circuit. And the result is proven.
In the meantime we wrote about circuits here:
http://ncategory.wordpress.com/2010/03/18/loose-ends/#comment-158
I’ll repost those comments here now…
I’m interested in the discussion of circuits in This Week’s Finds 296, so I hope someone is reading here whilst comments on that blog are down.
The reason that (or in fact more simply is not the power form of a circuit is that the power must not increase when the potentials are brought closer together.
Rigourously, define . Then a Dirichlet form must satisfy , i.e. apply to each coordinate separately.
All that come from a circuit must be Dirichlet, but I don’t know if the converse is true. I suspect so.
Dirichlet forms are important in probability theory. The link between stochastic processes and circuits is, as usual, the Laplacian.
I’m interested in this categorization of circuits, because circuits are related to interacting particle models, such as the Ising model, and percolation.
This is an interesting book on the topic of circuits:
In it, the author shows that Dirichlet forms that I defined above correspond exactly with “Laplacians” which are the linear maps from potentials to currents that you wrote about in week296, John.
The condition on Laplacians which corresponds to the Markov condition () of Dirichlet forms is exactly Ben Tilly’s intuition that the off-diagonal elements must be non-positive (he has the inverse sense so he writes non-negative).
He also demonstrates that every Dirichlet form, i.e. every Laplacian, comes from an electrical network.
Thanks very much for offering a solution to my problem, Tom! This sounds wonderful!
Alan Weinstein pointed me to this article, which you may enjoy too:
It proposes a class of quadratic forms which it uses as a way to formalize electrical circuits made of resistors. It does a lot of interesting things with quadratic forms of this type. However, I didn’t see a proof that every circuit made of resistors gives a quadratic form of this type, or conversely. I should reread it and compare it to the reference you provide.
John, that paper contains a reference to this one:
Existence and uniqueness of diffusions on finitely ramified self-similar fractals also by C. Sabot.
http://linkinghub.elsevier.com/retrieve/pii/S001295939789934X
In particular the latter contains (p609, i.e. p5 in the PDF):
“With each bond conductivity matrix J [circuit] we associate a positive bilinear form A on E x E by [the energy form]
The following result is well-known:
PROPOSITION 1.7. – The quadratic form A is a Dirichlet form on F … The map J |-> A is bijective from the set of bond conductivity matrices [circuits] to the set of Dirichlet forms.”
I rather dislike these “This result is well known” appeals to folk theorems, but oh well, at least you’ve got another reference for your result!
Hello Brother Baez,
You wrote:
”We can study quantum-mechanical systems, or classical ones. ”
In relation to circuits is kirchoffs law in some sense a classical approximation to a quantum system? In which case in what sense does it approximate it?
Kim wrote:
In relation to circuits is Kirchoff’s law in some sense a classical approximation to a quantum system?
Kirchoff’s laws are laws, not systems, so they aren’t ‘classical approximations to quantum systems’.
But: an electrical circuit, treated classically, is a classical system. So, we can ask if it’s an approximation to some quantum system.
If our circuit is built out of linear capacitors and inductors, I know the answer is yes. The reason is that such circuits are mathematically isomorphic to systems built from masses and springs, and I know how to quantize these using standard techniques.
If however our circuit also contains resistors, things get more tricky. Why? Because such such circuits are mathematically isomorphic to systems built from masses and springs with friction. And, I don’t know much about how to quantize systems with friction.
But, in “week296” I gave a reference on the quantization of electrical circuits containing resistors. So, people have thought about this. I need to learn more about this.
And, I don’t know much about how to quantize systems with friction.
With a time dependent Hamiltonian:
A time-dependent Hamiltonian is one option… I’d like to see what results that gives! It seems most people prefer the Caldeira–Leggett model, where you couple a harmonic oscillator to an infinite collection of harmonic oscillators that serve as a ‘heat bath’, and then trace out over these extra degrees of freedom.
That addresses a completely different question, namely: how do you model apparently irreversible physics (dissipation) with an intrinsically time-reversal-invariant microscopic Hamiltonian.
If all you want to do is study the damped harmonic oscillator (and don’t care where the dissipation came from), then the above time-dependent (and explicitly not time-reversal-invariant) Hamiltonian suffices.
I wouldn’t go so far as to call this a completely different question. When we see a damped quantum oscillator in nature, we typically suspect that its irreversible behavior is due to ignoring some degrees of freedom. In other words, we suspect that its damping is fundamentally due to interaction with its environment. If this is true, as time passes, pure states of the oscillator will evolve into mixed states (when we trace over the environment’s degrees of freedom).
The model you present seeks to ignore the environment and describe a time evolution that maps pure states to pure states. The Caldeira–Leggett model takes the environment into account, but assumes an environment of a particularly bland and tractable sort: an infinite collection of harmonic oscillators, coupled to the oscillator of interest in the simplest possible way. Neither option is likely to be the full story for realistic situations, so it’s an interesting question—at least to me—which one is a better approximation in which situations.
In short, while it’s a completely different model, I don’t think it addressses a completely different question.
JB:
If our circuit is built out of linear capacitors and inductors, I know the answer is yes. […] If however our circuit also contains resistors, things get more tricky. I don’t know much about how to quantize systems with friction.
The point is that capacitors, inductors, and resistors are only approximately the ideal classical things treated in the simple circuit theory you are discussing. In reality they are complex multiparticle quantum devices. Designing the latter so that they approximately yield the ideal behavior desired of them is a complex quantum-mechanical task. On this level, there is no difference between the three.
Quantum systems with friction are not created by quantizing ideal classical systems, but by approximating quantum field theoretic conservative models, restricted to a small set of relevant variables. This entails dissipation of energy into degrees of freedom unmodelled in this small set, which is called friction. How this is restriction is actually done is briefly explained in the sections ”Open quantum systems”, ”Interaction with a heat bath”, and ”Dissipative dynamics and Lagrangians” of my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html
For a systematic treatment, see Grabert’s book cited there.
Re: This Week’s Finds in Mathematical Physics (Week 296)
You write that in the category of circuit diagrams (extended from a semicategory),
I have in mind a circuit diagram made to look like the zig-zag relation. That is, it’s the composite of two other diagrams. The first has a straight short-circuit from bottom to top to the left of a bent short-circuit connecting the top back to itself. The second has a bent short-circuit from the bottom back to itself, and this is to the left of a straight short-circuit from the bottom to the top.
Neither of these two diagrams is an identity – indeed, they don’t even seem go from one object back to itself – and yet their composite is the short-circuit. What have I misunderstood?