April 16, 2010

This Week's Finds in Mathematical Physics (Week 295)

John Baez

This week I'll talk about the principle of least power, and Poincaré duality for electrical circuits, and a generalization of Hamiltonian mechanics that people have introduced for dissipative systems. But first....

photo by Brynjar Gaudi / AP

Now and then the world does something that forcefully reminds us of its power. As you probably know, the Eyjafjallajökull volcano in Iceland is emitting a plume of glass dust which has brought air traffic to a halt over much of Europe. This dust is formed as lava hits cold water and shatters. When sucked into a jet engine, it can heat up to about 1400 degree Celsius and re-melt. And when it cools again, it can stick onto the turbine blades.

This is not good. In 1982, a British Airways Boeing 747 flew through an ash cloud created by a volcano in Indonesia. All four engines cut out. The plane descended from 11,000 meters to 3,700 meters before the engines could be restarted. Whee!

Here's a picture of the Eyjafjallajökull plume, taken yesterday by NASA's "Aqua" satellite:

1) NASA, Ash plume from Eyjafjallajokull Volcano over the North Atlantic (afternoon overpass), http://rapidfire.sci.gsfc.nasa.gov/gallery/?2010105-0415

Here's what the volcano looked like back in March:

This photo was taken by someone named Bjarni T. He has a great photo gallery here:

2) Bjarni T, 2010 Eruptions of Eyjafjalljökull, http://www.fotopedia.com/en/2010_eruptions_of_Eyjafjallaj%C3%B6kull/slideshow/sort/MostVotedFirst/status/default/photos

And here's what the volcano looked like on April 15th:

Starting around 1821, the same volcano erupted and put out ash for about 6 months. What will it do this time? Nobody seems to know. If it goes on long enough, will people invent some sort of ash filter for jet engines?

Oh well. Back to electrical circuits...

I want to explain the "principle of minimum power" and how we can use it to understand electrical circuits built from linear resistors. In future Weeks this will lead us to some symplectic geometry, complex analysis and loop groups. But I want to start with some very basic stuff! I want to illustrate the principle of minimum power by using it to solve two basic problems: resistors in series and resistors in parallel. But first I should work out the answers to these problems using a more standard textbook approach - just in case you haven't seen this stuff already.

In the textbook approach, we'll use Kirchoff's voltage and current laws over and over again. I explained these laws in "week293" and "week294" - so if necessary, you can either review what I said there, or just nod and act like you understand what I'm doing.

First, suppose we have two resistors "in series". This means they're stuck together end to end, like this:

        |
        |
      -----
     | R1  |
      -----
        |
        |
      -----
     | R2  |
      -----
        |
        |

To answer this, fix the voltage across the whole circuit, say V. By Kirchoff's voltage law, this is the sum of the voltages across the individual resistors, say V₁ and V₂:

        | 
        |
      -----
     | R1  | V1
      -----
        |                        V = V1 + V2
        |        
      -----
     | R2  | V2
      -----
        |
        | 

        | 
        |
      -----
   I | R1  | V1
      -----
        |                     V = V1 + V2
        | 
      -----
   I | R2  | V2
      -----
        |
        |

I R₁ = V₁

and

I R₂ = V₂

Adding these two equations we get:

I (R₁ + R₂) = V

This looks like Ohm's law again, but now for a resistor with resistance R₁ + R₂.

The moral: two resistors in series act like a single resistor whose resistance is the sum of theirs!

Next, suppose we have two resistors "in parallel". This means they're stuck together side by side, like this:

             /\
            /  \
           /    \
          /      \
         /        \
        /          \
       /            \
     ----          ----
    | R1 |        | R2 |            
     ----          ----
       \            /
        \          /
         \        /
          \      /
           \    /
            \  /
             \/ 

To answer this, fix the current through the whole circuit, say I. By Kirchoff's current law, this is the sum of the currents through the individual resistors, say I₁ and I₂:

             /\
            /  \
           /    \
          /      \
         /        \
        /          \
       /            \
     ----          ----
 I1 | R1 |     I2 | R2 |            I = I1 + I2
     ----          ----
       \            /
        \          /
         \        /
          \      /
           \    /
            \  /
             \/ 

             /\
            /  \
           /    \
          /      \
         /        \
        /          \
       /            \
     ----          ----
 I1 | R1 | V   I2 | R2 | V          I = I1 + I2
     ----          ----
       \            /
        \          /
         \        /
          \      /
           \    /
            \  /
             \/ 

I₁ = V / R₁

and

I₂ = V / R₂

Adding these two equations we get:

I = V (1/R₁ + 1/R₂)

In our previous problem we were adding up resistances. Now we're adding up reciprocals of resistances. Luckily, there's a name for the reciprocal of a resistance: it's called an "admittance".

The moral: two resistors in parallel act like a single resistor whose admittance is the sum of theirs!

And there's also another moral. If you compare this problem to the previous one, you'll see that everything was almost exactly the same! In fact, I repeated a lot of sentences almost word for word. I just switched certain concepts, which come in pairs:

• current and voltage
• series and parallel
• resistance and admittance

In fact, switching concepts like this is an example of Poincaré duality for electrical circuits, as mentioned in "week291".

You may know Poincaré duality for graphs drawn on a sphere: you get a new graph from an old one by:

• drawing a new vertex in the middle of each old face,
• replacing each edge with a new one that crosses the old one, and
• drawing a new face centered at each old vertex.

This works fine for "closed" planar circuits - but for circuits with input and output wires, like we've got here, we need Poincaré duality for graphs drawn on a closed disk! This is should probably be called "Poincaré-Lefschetz duality".

Instead of giving you a long-winded description of how this works, let me just illustrate it. We start with two resistors in series. This is a graph with two edges and three vertices drawn on something that's topologically a closed disk. Let's draw it on a rectangle:

 ......x...... 
 .     |     .      
 .     |     .      
 .     |     .
 .     o     .
 .     |     .
 .     |     .      
 .     |     .
 ......x...... 

Now let's superimpose the Poincaré dual graph:

 ......x...... 
 .  ___|___  .      
 . /   |   \ .      
 ./    |    \.
 x     o     x
 .\    |    /.
 . \___|___/ .
 .     |     .      
 ......x...... 

 ............. 
 .  _______  .      
 . /       \ .      
 ./         \.
 x           x
 .\         /.
 . \_______/ .
 .           .      
 ............. 

Poincaré duality is cool. But now let's solve the same problems - resistors in series and resistors in parallel - using the "principle of least power". Here's what the principle says. Suppose we have any circuit made of resistors and we fix boundary conditions at the wires leading in and out. Then the circuit will do whatever it takes to minimize the amount of power it uses - that is, turns into heat.

What do I mean by "boundary conditions"? Well, first of all, I'm thinking of an electrical circuit as a graph with resistors on its edges, and with some special vertices that we think of as inputs and outputs:

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

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

But what do I mean by "boundary conditions"? Well, one sort of boundary condition is to fix the "electrostatic potential" at the input and output vertices of our graph. Remember from last week that the electrostatic potential is a function φ on the vertices of our graph. So, we'll specify the value of this function at the input and output vertices. Then we'll compute its values at all the other vertices using the principle of minimum power.

To do this, we need to remember some stuff from "week293" and "week294". First, for any edge

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

V(e) = φ(y) - φ(x)

Second, since we have a circuit made of linear resistors, the current I(e) through that edge obeys Ohm's law:

V(e) = I(e) R(e)

where R(e) is the resistance.

Third, the power consumed by that edge will be

P(e) = V(e) I(e)

The principle of minimum power says: fix φ at the input and output vertices. Then, to find φ at the other vertices, just minimize the total power:

P = ∑_e P(e)

Using all the equations I've lined up, we see that the total power is indeed a function of φ, since:

P(e) = (φ(y) - φ(x))² / R(e)

The total power is a quadratic function in a bunch of variables, so it's easy to minimize.

Let's actually do this for two resistors in series:

     φ0 x
        |
        | R1
        |        
     φ1 o
        |
        | R2
        |
     φ2 x

P = (φ₁ - φ₀)²/R₁ + (φ₂ - φ₁)²/R₂

So, we differentiate P with respect to φ₁ and set the derivative to zero:

2(φ₁ - φ₀)/R₁ - 2(φ₂ - φ₁)/R₂ = 0

This implies that

V₁ / R₁ = V₂ / R₂

where V₁ and V₂ are the voltages across our two resistors.

By Ohm's law, voltage divided by resistance is current. So, we get

I₁ = I₂

where I₁ and I₂ are the currents through our two resistors. Hey - the current flowing through the first resistor equals the current flowing through the second one! That's no surprise: it's a special case of Kirchoff's current law! But the cool part is that we derived Kirchhoff's current law from the principle of minimum power. This works quite generally, not just in this baby example.

Since the currents I₁ and I₂ are equal, let's call them both I. Then we're back to the textbook approach to this problem. Ohm's law says

I R₁ = V₁

and

I R₂ = V₂

Adding these equations, we see that when you put resistors in series, their resistances add.

Okay, now let's try two resistors in parallel:

        x φ0
       / \
      /   \
     /     \
    /R1     \R2
    \       /
     \     /
      \   /
       \ /
        x φ1

Why was this problem more boring than two resistors in series? Shouldn't they be very similar? After all, they're Poincaré duals of each other!

Well, yeah. But the problem is, we're not using the Poincaré dual boundary conditions. For the resistors in series we had a graph with a vertex in the middle:

 ......x...... 
 .     |     .      
 .     |     .      
 .     |     .
 .     o     .
 .     |     .
 .     |     .      
 .     |     .
 ......x...... 

 ............. 
 .  _______  .      
 . /       \ .      
 ./         \.
 x           x
 .\         /.
 . \_______/ .
 .           .      
 ............. 

All this could be further illuminated using the chain complex approach I outlined in "week293". Let me just sketch how that goes. We can associate a cochain complex to our circuit:

       d           d
C0 --------> C1 ---------> C2

V = dφ

is a 1-cochain. As we've seen, the total power is

P = ∑_e V(e)² / R(e)

We can write this in a slicker way using an inner product on the space of 1-cochains:

P = <V, V>

The principle of minimum power says we should find the electrostatic potential φ that minimizes the total power subject to some boundary conditions. So, we're trying to minimize

P = <dφ, dφ>

while holding φ fixed at some "input and output vertices". If you know some mathematical physics you'll see this is just a discretized version of the minimum principle that gives Laplace's equation!

There's also a dual version of this whole story. Our circuit also gives a chain complex:

        δ           δ
C0 <-------- C1 <-------- C2

I = δJ

In these terms, the total power is

P = ∑_e R(e) I(e)²

We can write this in a slicker way using an inner product on the space of 1-chains:

P = <I, I>

In fact I already talked about this inner product in "week293".

In these terms, the principle of minimum power says we should find the mesh current that minimizes the total power subject to some boundary conditions. So, now we're trying to minimize

P = <δJ, δJ>

while holding J fixed along certain "input and output meshes".

In short, everything works the same way in the two dual formulations. In fact, we can reinterpret our chain complex as a cochain complex just by turning it around! This:

        δ           δ
C0 <-------- C1 <-------- C2

effortlessly becomes this:

        δ           δ
C2 --------> C1 --------> C0

And we didn't even need our graph to be planar! The only point in having the graph be planar is that this gives us a specific choice of meshes. Otherwise, we must choose them ourselves.

Finally, I want to mention an interesting book on nonequilibrium thermodynamics. The "principle of minimum power" is also known as the "principle of least entropy production". I'm very curious about this principle and how it relates to the more familiar "principle of least action" in classical mechanics. This book seems to be pointing towards a unification of the two:

3) Hans Christian Öttinger, Beyond Equilibrium Thermodynamics, Wiley, 2005.

I thank Arnold Neumaier for pointing it out! It considers a fascinating generalization of Hamiltonian mechanics that applies to systems with dissipation: for example, electrical circuits with resistors, or mechanical systems with friction.

In ordinary Hamiltonian mechanics the space of states is a manifold and time evolution is a flow on this manifold determined by a smooth function called the Hamiltonian, which describes the energy of any state. In this generalization the space of states is still a manifold, but now time evolution is determined by two smooth functions: the energy and the entropy! In ordinary Hamiltonian mechanics, energy is automatically conserved. In this generalization that's also true, but energy can go into the form of heat... and entropy automatically increases!

Mathematically, the idea goes like this. We start with a Poisson manifold, but in addition to the skew-symmetric Poisson bracket {F,G} of smooth functions on some manifold, we also have a symmetric bilinear bracket [F,G] obeying the Leibniz law

[F,GH] = [F,G]H + G[F,H]

and this positivity condition:

[F,F] ≥ 0

The time evolution of any function is given by a generalization of Hamilton's equations:

dF/dt = {H,F} + [S,F]

where H is a function called the "energy" or "Hamiltonian", and S is a function called the entropy. The first term on the right is the usual one. The new second term describes dissipation: as we shall see, it pushes the state towards increasing entropy.

If we require that

[H,F] = {S,F} = 0

for every function F, then we get conservation of energy, as usual in Hamiltonian mechanics:

dH/dt = {H,H} + [S,H] = 0

But we also get the second law of thermodynamics:

dS/dt = {H,S} + [S,S] ≥ 0

Entropy always increases!

Öttinger calls this framework "GENERIC" - an annoying acronym for "General Equation for the NonEquilibrium Reversible-Irreversible Coupling". There are lots of papers about it. But I'm wondering if any geometers have looked into it!

If we didn't need the equations [H,F] = {S,F} = 0, we could easily get the necessary brackets starting with a Kähler manifold. The imaginary part of the Kähler structure is a symplectic structure, say ω, so we can define

{F,G} = ω(dF,dG)

as usual to get Poisson brackets. The real part of the Kähler structure is a Riemannian structure, say g, so we can define

[F,G] = g(dF,dG)

This satisfies

[F,GH] = [F,G]H + F[G,H]

and

[F,F] ≥ 0

Don't be fooled: this stuff is not rocket science. In particular, the inequality above has a simple meaning: when we move in the direction of the gradient of F, the function F increases. So adding the second term to Hamilton's equations has the effect of pushing the system towards increasing entropy.

Note that I'm being a tad unorthodox by letting ω and g eat cotangent vectors instead of tangent vectors - but that's no big deal. The big deal is this: if we start with a Kähler manifold and define brackets this way, we don't get [H,F] = 0 or {S,F} = 0 for all functions F unless H and S are constant! That's no good for applications to physics. To get around this problem, we would need to consider some sort of degenerate Kähler structure - one where ω and g are degenerate bilinear forms on the cotangent space.

Has anyone thought about such things? They remind me a little of "Dirac structures" and "generalized complex geometry" - but I don't know enough about those subjects to know if they're relevant here.

This GENERIC framework suggests that energy and entropy should be viewed as two parts of a single entity - maybe even its real and imaginary parts! And that in turn reminds me of other strange things, like the idea of using complex-valued Hamiltonians to describe dissipative systems, or the idea of "inverse temperature as imaginary time". I can't tell yet if there's a big idea lurking here, or just a mess....

Addendum: I thank Eric Forgy, Tom Leinster, Gunnar Magnusson and Esa Peuha for catching typos. Also, Esa Peuha noticed that I was cutting corners in my definition of "admittance" as the inverse of "resistance". Admittance is the inverse of resistance in circuits made of linear resistors, which are the circuits I was talking about. But he notes:

In Week 295, you claim that admittance is the inverse of resistance, but that's not true; admittance is the inverse of impedance. Of course, resistance and impedance are the same thing for circuits containing only resistors, but not in the presence of capacitors and inductors. Usually it's said that the inverse of resistance is conductance (and the inverse of reactance is susceptance), but that's not quite right: resistance and reactance are the real and imaginary parts of impedance, and conductance and susceptance are the real and imaginary parts of admittance, so resistance, reactance, conductance and susceptance don't usually have physically meaningful inverses.

[H,F] = {S,F} = 0

for all functions F. To derive the few results I describe, it's enough to have

[H,S] = {S,H} = 0

It seems that Öttinger assumes the stronger formulation but only uses the weaker one - see the text before equation (1.22) in his book. This changes the story considerably!

Eugene Lerman pointed out that everything I said about Kähler manifolds would work equally well for almost Kähler manifolds: nothing I said required that the complex structure relating the symplectic structure and the Riemannian metric be integrable. So, you could say I'm looking to fill in the missing item in this analogy:

But, I'd be equally happy to hear from you if you know the missing item in this analogy:

For more discussion, visit the n-Category Café.

I would rather discover a single fact, even a small one, than debate the great issues at length without discovering anything new at all. - Galileo Galilei

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