March 21, 2010

This Week's Finds in Mathematical Physics (Week 294)

John Baez

Sorry, I've been busy writing papers for the last couple of months. But I'm not done with my story of electrical circuits! It will take a few more episodes for me to get to the really cool part: the symplectic geometry, the complex analysis, and how they fit together using the theory of loop groups. I plan to talk about this in Dennis Sullivan's seminar at the City University of New York later this spring. I haven't written anything about it yet. So, I need to prepare by discussing it here.

You'll understand why I need to prepare if you've heard about Sullivan's seminar. It's a "Russian style" seminar, meaning that it's modeled after Gelfand's famous seminar in Moscow. And what does that mean? Well, Gelfand was famous for asking lots of questions. He wanted to understand all that was said - and he wasn't willing to put up with any nonsense. So, his seminar went on for hours and hours, leaving the speaker exhausted.

Here's a nice description of it:

1) Simon Gindikin, Essay on the Moscow Gelfand Seminar, in Advances in Soviet Mathematics, 16, eds. Sergei Gelfand and Simon Gindikin, 1993. Available at http://www.math.rutgers.edu/home/gelfand/

Let me quote a bit:

The Gelfand seminar was always an important event in the very vivid mathematical life in Moscow, and, doubtless, one of its leading centers. A considerable number of the best Moscow mathematicians participated in it at one time or another. Mathematicians from other cities used all possible pretexts to visit it. I recall how a group of Leningrad students agreed to take turns to come to Moscow on Mondays (the day of the seminar, to which other events were linked), and then would retell their friends what they had heard there. There were several excellent and very popular seminars in Moscow, but nevertheless the Gelfand seminar was always an event.

I would like to point out that, on the other hand, the seminar was very important in Gelfand's own personal mathematical life. Many of us witnessed how strongly his activities were focused on the seminar. When, in the early fifties, at the peak of antisemitism, Gelfand was chased out of Moscow University, he applied all his efforts to seminar. The absence of Gelfand at the seminar, even because of illness, was always something out of the ordinary.

One cannot avoid mentioning that the general attitude to the seminar was far from unanimous. Criticism mainly concerned its style, which was rather unusual for a scientific seminar. It was a kind of a theater with a unique stage director playing the leading role in the performance and organizing the supporting cast, most of whom had the highest qualifications. I use this metaphor with the utmost seriousness, without any intention to mean that the seminar was some sort of a spectacle. Gelfand had chosen the hardest and most dangerous genre: to demonstrate in public how he understood mathematics. It was an open lesson in the grasping of mathematics by one of the most amazing mathematicians of our time. This role could be only be played under the most favorable conditions: the genre dictates the rules of the game, which are not always very convenient for the listeners. This means, for example, that the leader follows only his own intuition in the final choice of the topics of the talks, interrupts them with comments and questions (a privilege not granted to other participants) [....] All this is done with extraordinary generosity, a true passion for mathematics.

Let me recall some of the stage director's strategems. An important feature were improvisations of various kinds. The course of the seminar could change dramatically at any moment. Another important mise en scene involved the "trial listener" game, in which one of the participants (this could be a student as well as a professor) was instructed to keep informing the seminar of his understanding of the talk, and whenever that information was negative, that part of the report would be repeated. A well-qualified trial listener could usually feel when the head of the seminar wanted an occasion for such a repetition. Also, Gelfand himself had the faculty of being "unable to understand" in situations when everyone around was sure that everything is clear. What extraordinary vistas were opened to the listeners, and sometimes even to the mathematician giving the talk, by this ability not to understand. Gelfand liked that old story of the professor complaining about his students: "Fantastically stupid students - five times I repeat proof, already I understand it myself, and still they don't get it."

It has remained beyond my understanding how Gelfand could manage all that physically for so many hours. Formally the seminar was supposed to begin at 6 pm, but usually started with an hour's delays. I am convinced that the free conversations before the actual beginning of the seminar were part of the scenario. The seminar would continue without any break until 10 or 10:30 (I have heard that before my time it was even later). The end of the seminar was in constant conflict with the rules and regulations of Moscow State University. Usually at 10 pm the cleaning woman would make her appearance, wishing to close the proceedings to do her job. After the seminar, people wishing to talk to Gelfand would hang around. The elevator would be turned off, and one would have to find the right staircase, so as not to find oneself stuck in front of a locked door, which meant walking back up to the 14th (where else but in Russia is the locking of doors so popular!). The next riddle was to find the only open exit from the building. Then the last problem (of different levels of difficulty for different participants) - how to get home on public transportation, at that time in the process of closing up. Seeing Gelfand home, the last mathematical conversations would conclude the seminar's ritual. Moscow at night was still safe and life seemed so unbelievably beautiful!

This is a great example of how taking things really seriously, and pursuing them intelligently, with persistent passion, can infuse them with the kind of intensity that leaves echoes resonating decades later.

Sullivan's seminar is also intense, though it plays to a smaller audience. Like Gelfand's, it's set in a tall building: in fact, a 30-story skyscraper called the Graybar Building, right next to Grand Central Station. The first time I was asked to speak there, my talk was supposed to start at 3 pm. But before that, there was an informal "pre-talk" where people discussed math and sat around eating lunch. Someone went down to get sandwiches, and I was asked what kind I wanted. I said I wasn't hungry, but someone who knew better got me one anyway.

My talk started at 3... and it went on until 9! I loved it: here was someone who really wanted to understand my work. None of the usual routine where everyone starts eyeing the clock impatiently as the allotted hour nears its end. It was clear: this seminar would last as long as it took to get the job done. And when we were done, we all went out to dinner... and talked about math.

So, I should get back to my tale of electrical circuits. I'm really just using these as a nice example of physical systems made of components. Part of my goal is to get you interested in "open systems" - systems that interact with their environment. My physics classes emphasized "closed systems", where we assume that we've modelled all the relevant aspects of what's going on, so the interaction with the outside environment is negligible. Why? It lets us use the marvelous techniques of symplectic mechanics - Hamilton's equations, Noether's theorem giving conserved quantities from symmetries, and all that. These techniques don't work for open systems - at least, not until we generalize them. But almost every device we design is an open system, in a crucial way: we do things to it, and it does things for us. So engineers need to think about open systems.

And mathematical physicists should too - because life gets more interesting when you treat every system as having an "interface" through which it interacts with its environment. For starters, this lets you build bigger systems from components by attaching them along their interfaces. We can also formalize the problem of taking a system and decomposing it into smaller subsystems. In engineering this is called "tearing". For example, we can take this electrical circuit:

        |           |
        |           |
      -----         |
     |     |        |
      -----         |
      /   \         |
     /     \        |
  ------------      |
 |            |     |
  ------------      |
    |  |  |         |
    |  |   \_______/
    |  |
    |  |

        |           |
        |           |
      -----         |
     |     |        |
      -----         |
      /   \         |


     /     \        |
  ------------      |
 |            |     |
  ------------      |
    |  |  |         |
    |  |   \_______/
    |  |
    |  |

2) Jan C. Willems, In control, almost from the beginning until the day after tomorrow, European Journal of Control 13 (2007), 71-81. Also available as http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.2.pdf

You don't need to know anything about control theory to enjoy this! Well, it helps to know that "control theory" is the art of getting open systems to do what you want. But it's always fun to begin learning a subject by hearing about its history - especially from somebody who was there.

Here's a passage that connects to the point I was just trying to make:

One can, one should, ask the question if closed systems, as flows on manifolds and

dx/dt = f(x)

form a good mathematical vantage point from which to embark on the study of dynamics. In my opinion they do not, for a number of reasons. First, in a good theory the state x should be derived from a less structured model. A more serious objection is that closed systems are not good concepts to deal with modeling. A model usually consists of a number of interacting subphenomena that need to be modeled one-by-one. In these sub-models, the influence of the other subsystems needs par force to be viewed as external, and in principle free. Tearing leads to models that are open.

If you view a closed system as an interconnection of two systems, these two systems will be open. Systems that take into account unmodeled external influences form therefore a much more logical starting point. Third, many basic laws in physics address open systems. For example, Newton's second law, Maxwell's equations, the gas law, and the first and second laws of thermodynamics. A good setting of dynamics should incorporate these important examples from the beginning. Finally, closed systems put one in the absurd situation that in order to model a system, one ends up having to model also the environment.

These arguments seem obvious and compelling. Twenty five years ago, it was my hope that system theory, with its emphasis on open systems, would by now have been incorporated and accepted as the new starting point for dynamical systems in mathematics. Better, more general, more natural, more apt for modeling, offering interesting new concepts as controllability, observability, dissipativity, model reduction, and with a rich, well developed, domain as linear system theory. It is disappointing that this didn't happen. What seemed like an intellectual imperative did not even begin to happen. Mathematicians and physicists invariably identify dynamical systems with closed systems.

I think this will change. I think we just need to develop the right framework for open systems. Luckily, a lot of this framework is already available: concepts like operads, n-categories and the like give very general ways of describing how to build big things by gluing together little pieces. For example, a trained mathematician will take one look at this:

        |           |
        |           |
      -----         |
     |     |        |
      -----         |
      /   \         |
     /     \        |
  ------------      |
 |            |     |
  ------------      |
    |  |  |         |
    |  |   \_______/
    |  |
    |  |

For a deeper look at Willems' ideas on open systems, try this:

3) Jan C. Willems, The behavioral approach to open and interconnected systems: modeling by tearing, zooming and linking, Control Systems Magazine 27 (2007), 46-99. Also available at http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.1.pdf

In particular, people who doubt that engineers could ever enjoy fancy math like operads and n-categories should check out the box near the end, on "polynomial modules and syzygies".

Now, I've been talking recently about "bond graphs". This is a general framework for physical systems which treats variables as coming in groups of four:

• q - displacement
• p - momentum
• q' - flow
• p' - effort

If we use the example of a massive object that can move back and forth, q and p stand are its position and momentum, while q' and p' are velocity and force. But if we use the example of an electrical circuit, q is charge and p is something fairly obscure called "flux linkage". Then their time derivatives are current, q', and voltage, p'.

In both these examples the quantity q'p' has dimensions of power. Bond graphers consider this very important: the idea is that when we consider mixed systems, like an electrical motor pushing a massive object around, it's power that flows from one part to another.

In "week289", I listed two examples of systems where q'p' does not have dimensions of power: thermal systems and economic systems. People do draw bond graphs of these, but they're considered second-class citizens: they're called "pseudo bond graphs".

Jan Willems has some criticisms of the bond graph methodology, including its obesssion with power - and also its focus on q' and p' at the expense of q and p. I've tried to give q and p more importance in my discussion so far, since for people trained in classical mechanics they're of utmost importance. But for people trained in electrical circuits, it's q' and p' that seem important: they talk about current and voltage all the time, and a bit less about the other other two.

Here's a summary of Jan Willems' criticisms of bond graphs, taken from a little box in the above paper. I'll paraphrase a bit here and there:

The tearing, zooming, and linking methodology for modeling interconnected systems advocated and developed in this article has many things in common with bond graphs. Introduced by Paynter in the 1960s, bond graphs are popular as a methodology for modeling interconnected physical systems, especially in mechanical engineering. For modeling physical systems, bond-graph modeling is a superior alternative to signal-flow diagrams and input/output-based modeling procedures.

Bond graphs view each system interconnection in terms of power and energy. The variables associated with terminals are assumed to consist of an effort and a flow, where the (inner) product of effort and flow is power. Connections are formalized by junctions. Using a combination of junctions and component subsystems, complex physical systems can be modeled in a systematic way. The power interpretation automatically takes care of conservation of energy. The philosophy underlying bond graphs is, as stated by P.J. Gawthrop and G.P. Bevan,

Power is the universal currency of physical systems.

The idea that terminal variables come in pairs, an effort and a flow, with efforts preserved at each interconnection and the sum of flows equated to zero at each interconnection, is appealing and deep. But, in addition to weak mathematical underpinnings and unconventional graph notation with half arrows, bond graphs have some shortcomings as a modeling philosophy, as explained in the section "Bond-Graph Modeling". The main points discussed in that section are the following:

1. The requirement that the product of effort and flow must be power is sometimes not natural, for example, in thermal interconnections.

2. In connecting terminals of mechanical systems, bondgraph modeling equates velocities, and sets the sum of the forces equal to zero. In reality one ought to equate positions, not velocities. Equating velocities instead of positions leads to incomplete models.

3. Interconnections are made by means of terminals, while energy is transferred through ports. Ports involve many terminals simultaneously. The interconnection of two electrical wires involves equating two terminal potentials and putting the sum of two terminal currents to zero. The product of effort, namely, the electrical potential, and flow, namely, the electrical current, for an electrical connection has the dimension of power, but it is not power. Power involves potential differences, while the interconnection constraints involves the terminal potentials themselves. It is not possible to interpret these interconnection constraints as equating the power on both sides of the interconnection point.

4. In many interconnections, it is unnecessary to have to worry about conservation of energy.

Willems has his own methodology, which he explains. I'll need to learn about it!

I'll get into the deeper aspects of electrical circuits next Week. There are just a few leftovers I want to mention now. I told you about five basic 1-ports in "week290": resistors, capacitors, inductors, voltage sources and current sources. Each was defined by a single equation involving q, p, q' and p, and perhaps the time variable t. These five are the most important 1-ports. But there are some weirder ones worth thinking about. Here they are:

1. A "short circuit". A linear resistor has p' = R q'. If the resistance R equals zero, you get a "short circuit". Now the relation between voltage and current becomes:

p' = 0

So, there's always zero voltage across this circuit element - it's a perfect conductor. Or in the language of bond graphs: there's always zero effort across this 1-port.

2. An "open circuit". If you take a linear resistor and say its resistance is infinite, you get an "open circuit". Now the relation between voltage and current becomes:

q' = 0

So, there's always zero current through this circuit element - it's a perfect insulator. Or in the language of bond graphs: there's always zero flow through this 1-port. By the way, the word "open" here has nothing to do with "open system".

The point of these examples is that most linear resistors let us treat current as a function of voltage or voltage as a function of current, since R is neither zero nor infinite. But in the these two limiting cases - the short circuit and the open circuit - that's not true. To fit these cases neatly in a unified framework, we shouldn't think of the relation between current and voltage as defining a function. It's just a relation!

In the world of algebraic geometry, a relation defined by polynomial equations is called a "correspondence". One way to get a correspondence is by taking the graph of a function. But it's important to go beyond functions to correspondences. And my claim is that this is important in electrical circuits, too.

But here are some even weirder one ports:

3. A "nullator". Here we bend the rules for 1-ports and impose two equations:

p' = 0
q' = 0

I don't think you can actually build this thing! The Wikipedia article sounds downright Zen: "In electronics, a nullator is a theoretical linear, time-invariant one-port defined as having zero current and voltage across its terminals. Nullators are strange in the sense that they simultaneously have properties of both a short (zero voltage) and an open circuit (zero current). They are neither current nor voltage sources, yet both at the same time."

4. A "norator". Here we bend the rules in the opposite direction and impose no equations:

Yes, that's a picture of no equations. Truly Zen: what is the sound of no equations clapping? I don't think you can build this thing either! At least, not by itself....

Now, you may wonder why electrical engineers bother talking about things that don't actually exist. That's normally the prerogative of mathematicians. But sometimes if you combine two things that don't exist, you get one that does! This is often how we introduce new kinds of things. For example, i × i = -1 lets us introduce the "imaginary" number i in terms of the "real" number -1.

As far as 1-ports go: if I have one equation too many, and you have one too few, together we're just right. So, there's a 2-port called the "nullor", which is built - theoretically speaking - from a nullator and a norator. Remmber, a 2-port is specified by by two equations involving q₁,q'₁,p₁,p'₁, q₂,q'₂,p₂,p'₂, and perhaps the time variable t. Here are the equations for the nullor:

p'₂ = 0
q'₂ = 0

So, the first wire acts like a norator while the second acts like a nullator. To see why engineers like this gizmo, try this:

4) Wikipedia, Nullor, http://en.wikipedia.org/wiki/Nullor

For more, try these:

5) Herbert J. Carlin, Singular network elements, IEEE Trans. Circuit Theory, March 1965, vol. CT-11, pp. 67-72.

6) P. Kumar and R. Senani, Bibliography on nullors and their applications in circuit analysis, synthesis and design, Analog Integrated Circuits and Signal Processing 33 (2002), 65-76.

Here's the last 1-port I want to mention:

5. The "memristor". This is a 1-port where the momentum p is a function of the displacement q:

p = f(q)

The function f is usually called the "memristance". It was invented and given this name by Leon Chua in 1971. The idea was that it completes a collection of four closely related 1-ports. In "week290" I listed the other three, namely the resistor:

p' = f(q')

the capacitor:

q = f(p')

and the inductor:

p = f(q')

The memristor came later because it's inherently nonlinear. Why? A linear memristor is just a linear resistor, since we can differentiate the linear relationship p = Mq and get p' = Mq'. But if p = f(q) for a nonlinear function f we get something new:

p' = f'(q) q'

So, we see that in general, a memristor acts like a resistor whose resistance is some function of q. But q is the time integral of the current q'. So a nonlinear memristor is like a resistor whose resistance depends on the time integral of the current that has flowed through it! Its resistance depends on its history. So, it has a "memory" - hence the name "memristance".

Memristors have recently been built in a number of ways, which are nicely listed here:

7) Wikipedia, Memristor, http://en.wikipedia.org/wiki/Memristor

Electrical engineering journals are notoriously resistant to the of open access, and I don't think there's a successful equivalent of the "arXiv" in this field. Shame on them! So, you have to nose around to find a freely accessible copy of Chua's original paper on memristors:

8) Leon Chua, Memristor, the missing circuit element, IEEE Transactions on Circuit Theory 18 (1971), 507-519. Also available at http://www.lane.ufpa.br/rodrigo/chua/Memristor_chua_article.pdf

To see what the mechnical or chemical analogue of a memristor is like, try this:

9) G. F. Oster and D. M. Auslander, The memristor: a new bond graph element, Trans. ASME, J. Dynamic Systems, Measurement and Control 94 (1972), 249-252. Also available as http://nature.berkeley.edu/~goster/pdfs/Memristor.pdf

Memristors supposedly have a bunch of interesting applications, but I don't understand them yet. I also don't understand "memcapacitors" and "meminductors". The above PDF file also contains a New Scientist article on the wonders of these.

To wrap up the loose ends, I want to tell you about Tellegen's theorem. Last week I started talking about electrical circuits and chain complexes. I considered circuits built from linear resistors. But now let's talk about completely general electrical circuits.

Last time I said an electrical circuit has "vertices", "edges" and "faces":

               o---------o---------o
               |/////////|/////////| 
               |/////////|/////////| 
               |//FACE///|///FACE//| 
               |/////////|/////////| 
               |/////////|/////////| 
               o---------o---------o

               o---------o---------o
               |         |         | 
               |         |         | 
               |         |         | 
               |         |         | 
               |         |         | 
               o---------o---------o

               o---->----o---->----o
               |         |         | 
               |         |         | 
               V         V         V 
               |         |         | 
               |         |         | 
               o----<----o---->----o

        δ          
C0 <-------- C1 

      e
x --------> y

This gadget

        δ          
C0 <-------- C1 

If we take the duals of the vector spaces involved, our 2-term chain complex turns around and becomes a 2-term "cochain complex":

      d
C0 --------> C1

(df)(e) = f(δe)

for any 0-cochain f and any 1-chain e.

What can we do with such pathetically puny mathematical structures?

First, in any electrical circuit, the current I is a 1-chain. Moreover, Kirchoff's current law says:

δI = 0

meaning the total current flowing into any vertex equals the total current flowing out. Last week I stated this law for closed circuits made of resistors, but it's true for any closed circuit as long as the current isn't changing too rapidly with time. Indeed, we can take it as a mathematical definition of what it means for a circuit to be "closed". By "closed" here, I mean that no current is flowing in from outside.

Second, in any electrical circuit, the voltage V is a 1-cochain. Moreover, Kirchoff's voltage law says:

V = dφ

meaning that we can define a "potential" φ(x) for each vertex x, with the property that for any edge

      e
x --------> y

Third, the power dissipated by the circuit equals

V(I)

Here we are pairing a 1-cochain and a 1-chain to get a number. Again, we talked about this last week, but it's true in general.

But now comes something new!

Let's compute the power V(I) using Kirchoff's voltage law and Kirchoff's current law:

V(I) = (dφ)(I) = φ(δI) = 0

Hey - it's zero!

At first this might seem strange. The power is always zero???

But maybe it isn't so strange if you think about it: it's a version of conservation of energy. In particular, it fails when we consider circuits with current flowing in from outside: then δI doesn't need to be zero. We don't expect energy conservation in its naive form to hold in that case. Instead, we expect a "power balance equation", as explained in "week290".

But maybe it is strange. After all, if you have a circuit built from resistors, why should it conserve energy? Didn't I say resistors were dissipative?

I still don't understand this as well as I'd like. The math seems completely trivial to me, but its meaning for circuits still doesn't seem obvious. Can someone explain it in plain English?

Anyway, this result is called "Tellegen's theorem". Clearly you have to be in the right place at the right time to get your name on a theorem! It doesn't have to be hard. It just has to be new and important. If I'd been there when they first discovered numbers, 2+2=4 would be called "Baez's theorem".

Still, you might be surprised to discover there's a whole book on Tellegen's theorem:

10) Paul Penfield, Jr., Robert Spence and Simon Duinker, Tellegen's Theorem and Electrical Networks, The MIT Press, Cambridge, MA, 1970.

Part of why this result is interesting is that depends on such minimal assumptions. Typically in circuit theory we need to know the voltages V as a function of the currents I, or vice versa, before we can do much. For example, for circuits built from linear resistors, we have a linear map

R: C₁ → C¹

such that

V = RI

This is Ohm's law. But Tellegen's theorem doesn't depend on this, or on any relationship between voltages and currents! Indeed, we can take two different circuits with the same underlying graph, and let V be the voltage of one circuit at one time, and let I be the current of the other circuit at some other time. We still get

V(I) = (dφ)(I) = φ(δI) = 0

so long as Kirchoff's voltage and current laws hold for each circuit!

I'm a bit fascinated by this paper, which you can get online:

11) G.F. Oster and C.A. Desoer, Tellegen's theorem and thermodynamic inequalities, J. Theor. Biol 32 (1971), 219-241. Also available at http://nature.berkeley.edu/~goster/pdfs/Tellagen.pdf

They give a decent description of Tellegen's theorem, and they use it to derive something they call "Prigogine's theorem", which is supposed to be in here:

12) Ilya Prigogine, Thermodynamics of Irreversible Processes, 3rd edition, Wiley, New York, 1968.

I don't understand it well enough to give a beautiful lucid explanation of it. But it's not complicated. It's an inequality that applies to closed circuits built from resistors and capacitors, or analogous systems in chemistry or other subjects.

According to Robert Kotiuga, the chain complex approach to electrical circuits goes back to Weyl:

13) Hermann Weyl, Repartici´on de corriente en una red conductora, Rev. Mat. Hisp. Amer. 5 (1923), 153-164.

He also recommend these references:

14) Paul Slepian, Mathematical Foundations of Network Analysis, Springer, Berlin, 1968.

15) Harley Flanders, Differential Forms with Applications to the Physical Sciences, Dover, New York, 1989, pp. 79-81.

16) Stephen Smale, On the mathematical foundations of electrical network theory, J. Diff. Geom. 7 (1972), 193-210.

17) G. E. Ching, Topological concepts in networks; an application of homology theory to network analysis, Proc. 11th. Midwest Conference on Circuit Theory, University of Notre Dame, 1968, pp. 165-175.

18) J. P. Roth, Existence and uniqueness of solutions to electrical network problems via homology sequences, Mathematical Aspects of Electrical Network Theory, SIAM-AMS Proceedings III, 1971, pp. 113-118.

For a quick discussion of Tellegen's theorem, this is also good:

19) Wikipedia, Tellegen's theorem, http://en.wikipedia.org/wiki/Tellegen%27s_theorem

By the way: if you've been paying careful attention and reading between the lines, you'll note that I've been advocating the study of the category where an object is a bunch of points:

     x        x        x

        x           x
        |           |
        |           |
        |           |
        o           |
       / \          |
      /   \         |
     /     o        |
    |     / \       | 
    |    /   \_____/
    |   |
    |   |
    x   x

I would like to know all possible slick ways of understanding this category, because it underlies fancier categories where the morphisms are electrical circuits, or Feynman diagrams, or other things.

For one thing, this category is "compact closed". In other words, it's a symmetric monoidal category where every object has a dual. Duality lets us take an input and turn it into an output, like this:

        x      
        |      
        |      
        |      
        o      
       / \     
      /   \    
     /     o   
    |     / \  
    |    /   \
    |   |    |
    |   |    |
    x   x    x

And in fact, this category is the free compact closed category on one self-dual object, namely x, and one morphism from the unit object to each tensor power of x. The unit object is drawn as the empty set, while the nth tensor power of x is drawn as a list of n x's. So, for example, when n = 3, we have a morphism that looks like a "trivalent vertex":

       o
      /|\ 
     / | \
    /  |  \
   x   x   x

       x
       |
       |
       o
      / \ 
     /   \
    /     \
   x       x

In this category, a morphism from the unit object to itself is just a finite undirected graph. Or, strictly speaking, it's an isomorphism class of finite undirected graphs!

For electrical circuits it's also nice to equip the edges with orientations, so we can tell whether the current flowing through the edge is positive or negative. At least it might be nicer - everyone seems to do it, but maybe it's bit artificial. Anyway, if we want to do this, we should find a category where the morphisms from the unit object to itself are finite directed graphs.

I think this is the free compact category on one object +, the "positively oriented point" and one morphism from the unit object to any tensor product built by tensoring a bunch of copies of this object + and then a bunch of copies of its dual, -. So, among the generating morphisms in this compact closed category, we'll have four trivalent vertices like this:

  
       o
      /|\ 
     / | \
    |  |  |
    V  V  V
    |  |  |
    +  +  +


       o
      /|\ 
     / | \
    |  |  |
    V  V  ^
    |  |  |
    +  +  -


       o
      /|\ 
     / | \
    |  |  |
    V  ^  ^
    |  |  |
    +  -  -


       o
      /|\ 
     / | \
    |  |  |
    ^  ^  ^
    |  |  |
    -  -  -

I can't help but hope there's a slicker desciption of this category. Anybody know one?

From directed graphs we can get chain complexes, and we've seen how this is important in electrical circuit theory. Can we do something similar to all the morphisms in our category?

Well, we can think of a directed graph as a functor

X: G → Set

where G is category with two objects, "vertex" and "edge", and two morphisms:

source: edge → vertex
target: edge → vertex

together with identity morphisms. We can think of G as the "Platonic idea" of a graph, and actual graphs as embodiments of this idea in the world of sets.

Taking this viewpoint, we can compose a directed graph

X: G → Set

with the "free vector space on a set" functor

F: Set → Vect

and get a gizmo that's like a graph, but with a vector space of vertices and a vector space of edges. A category theorist might call this a "graph object in Vect".

This may sound scary, but it's not. When we perform this process, we're just letting ourselves take formal linear combinations of vertices, and formal linear combinations of edges. So all we really get is a 2-term chain complex.

This sheds some light on how graphs are related to chain complexes. In fact, we can turn this insight into a little theorem: the category of graph objects in Vect is equivalent to the category of 2-term chain complexes. There's a bit to check here!

In short, waving the magic wand of linearity over the concept of "directed graph", we get the concept of "chain complex". So, there should be some way to take compact closed category I just described and wave the magic wand of linearity over that, too. And the result should be a category important in the theory of electrical circuits.

There's a closely related result that's also interesting. Suppose we have a directed graph:

               x---->----x---->----x
               |         |         | 
               |         |         | 
               V         V         V 
               |         |         | 
               |         |         | 
               x----<----x---->----x

But the category of category objects in Vect is also equivalent to the category of 2-term chain complexes! Alissa Crans and I called a category object in Vect a "2-vector space", since we can also think of it as a kind of categorified vector space. See Section 3 here:

20) John Baez and Alissa Crans, Higher-dimensional algebra VI: Lie 2-algebras, Theory and Applications of Categories 12 (2004), 492-528. Available at http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html and also as arXiv:math.QA/0307263.

This idea was known to Grothendieck quite a while ago - read the paper for the history. But anyway, I think it's neat that we can take the bare bones of an electrical circuit:

               o---->----o---->----o
               |         |         | 
               |         |         | 
               V         V         V 
               |         |         | 
               |         |         | 
               o----<----o---->----o

If you made it this far, you deserve a treat:

21) Astronomy Picture of the Day, Cassini spacecraft crosses Saturn's ring plane, http://apod.nasa.gov/apod/ap100215.html

Saturn's rings edge-on, and a couple of moons, photographed by the Cassini probe! Shadows of the rings are visible on the northern hemisphere.

Addendum: Thomas Riepe points out these remarks by Alain Connes:

I soon ran into Dennis Sullivan who used to go up to any newcomers, whatever their field or personality, and ask them questions. He asked questions that you could, superficially, think of as idiotic. But when you started thinking about them, you would soon realize that your answers showed you did not really understand what you were talking about. He has a kind of Socratic power which would push people into a corner, in order to try to understand what they were doing, and so unmask the misunderstandings everyone has. Because everyone talks about things without necessarily having cleaned out all the hidden corners. He has another remarkable quality; he can explain things you don't know in an incredibly clear and lucid manner. It's by discussing with Dennis that I learnt many of the concepts of differential geometry. He explained them by gestures, without a single formula. I was tremendously lucky to meet him, it forced me to realize that the field I was working in was limited, at least when you see it as tightly closed off. These discussions with Dennis pushed me outside my field, through a visual, oral dialogue.

22) An interview with Alain Connes, part II, by Catherine Goldstein and George Skandalis, Newsletter of the European Mathematical Society, March 2008, pp. 29-34. Also available at http://www.ems-ph.org/journals/newsletter/pdf/2008-03-67.pdf

The chemist Jiahao Chen noted some relations between electrical circuits and some aspects of chemistry. I would like to understand these better. He wrote:

I am particularly piqued by your recent expositions on bond graphs, and your most recent exposition on bond graphs have finally seem to have touched base with something I have been trying to understand for a very long time. For my PhD work I worked on understanding the flow of electrical charge of atoms when they are bound together in molecules. It turns out that there is a very clean analogue between atomic voltages (electrical potentials) = dE/dq and what we know in chemistry as electronegativity; also, there is an analogue for atomic capacitance = d²E/dq² and what is known as chemical hardness (in the sense of the hard-soft acid-base principle in general chemistry). It has become clear in recent years that the accurate modeling of such charge transfer processes must necessarily take into account not just the charges on atoms, but the flows between them. Then atoms in molecules can be thought of as being voltage-capacitor pairs connected by some kind of network, exactly like an electrical circuit, and the charges can determined by an equation of the form

bond capacitance × charge transfer variables = pairs of voltage differences

I have described this construction in the following paper:

23) J. Chen, D. Hundertmark and T. J. Martinez, A unified theoretical framework for fluctuating-charge models in atom-space and in bond-space, Journal of Chemical Physics 129 (2008), 214113. DOI: 10.1063/1.3021400. Also available as arXiv:0807.2174.

In this paper, I also reported the discovery that despite there being more charge transfer variables (bond variables) than charge variables (vertex variables), it is always possible to reformulate equations written in terms of charge transfer variables in terms of equations written into charges, and thus there is a non-obvious 1-1 mapping between these two sets of variables. That this is possible is a non-obvious consequence of Kirchhoff's law, because electrostatic processes cannot lead to charge flow in a closed loop, and so combinations of bond variables like (1 → 2) + (2 → 3) + (3 → 1) must lie in the nullspace of the equation. Thus the working equation

capacitance × charge = transformed voltage

can be used instead, where the transformation applied to the voltages is a consequence of the topological relationship between the charge transfer variables and charge variables. This transformation turns out to be exactly the node branch matrix in the Oster and Desoer paper that was mentioned in your column! (p. 222)

I cannot believe that this is merely a coincidence, and certainly your recent exposition on bond graphs seems to be very relevant in a way that could be fruitful to think about. The obvious connection to draw is that the capacitance relation between charges and voltages is exactly one of the four types of 1-ports you have described, except that there are as many charges as there are atoms in the molecule. I don't have a good background in algebraic topology, so I don't entirely follow your discussion on chain complexes. Nevertheless I find this interesting that this stuff is somehow related to mundane chemical concepts like electronegativity and charge capacities of atoms, and I hope you would too.

Thanks,
Jiahao Chen · MIT Chemistry

In addition to trying to explain the difference between acids and bases, this theory involves a distinction between "hard" and "soft" chemical species. "Hard" ones are small and weakly polarizable, while "soft" ones are big and strongly polarizable. The bigger d²E/dq² is, the harder the species is. Mathematically, a hard species is like a spring that's hard to stretch: remember, a spring that's hard to stretch has a big value of d²E/dq² where E is energy and q is how much the spring is stretched.

I thank Kim Sparre for catching a mistake. He also recommended this reference on electrical circuits and bond graphs:

24) Øyvind Bjørke and Ole Immanuel Franksen, editors, System Structures in Engineering - Economic Design and Production, Tapir Publishers, Norway, ca. 1978.

For more discussion, visit the n-Category Café.

© 2010 John Baez
baez@math.removethis.ucr.andthis.edu