The n-Category Café

John wrote:

homotopy theorists feel perfectly fine about studying simplicial sets rather than topological spaces. The reason…

for us oldsters is the adjoint pair: realization denoted | | and Sing

This preceded Quillen but I would guess that it inspired him

This time we’ll meet differential graded Lie algebras, and see how they yield a massive generalization of Lie theory.

Really? (-:O

“The basic inspiration is that electical circuit diagrams look sort of like Feynman diagrams” should be “The basic inspiration is that electrical circuit diagrams look sort of like Feynman diagrams”

Otherwise, wonderful on first reading!

Maxwell was also heavily influenced by fluid dynamics. One of the more difficult concepts to understand is electric displacement. Electric displacement fields are used to attribute charge to an object. The conceptually difficult part, and one that caused some consternation for physicists in the 19th century, was the acceptance of an internal degree of freedom (e.g. what is meant by displacement when there is no actual physical displacement).

Max Born makes a good go at trying to explain this concept in his book on Einstein’s Theory of Relativity (start around pg 171…the book can be found on google books)

In any case, displacement implies a measurement of degrees of freedom. So if you have voltage, which is analogous to temperature, given as Energy/unit charge (eg energy per degree of freedom), then entropy is Energy/Volt. Your household electric system is designed to maintain a constant voltage, so as you draw power out your socket, you are actually being provided with the ability to reduce entropy in your home (remember Energy = Power * time). In other words, as you draw power, you are working to maintain a constant entropy in you home.

Another way to think about it is in terms of current. Current is a measurement of charges/unit time; so you can imagine that as current flows into your house, the number of degrees of freedom are increasing, and the new degrees of freedom have more energy associated with them. This causes the energy distribution across all the degrees of freedom in your house to be less uniform; meaning that the entropy has decreased relative to from what it was before the current flowed.

All very interesting stuff.

John asked about Martian Longitude; and high-up in a google search for “martian longitude” we get this handy site. (In my firefox it doesn’t come out very tidy; the “caption” is squished in a column about 18ex wide on the left edge.)

I have a question about ‘power’ and ‘conservation of energy’

The power drawn by an electrical circuit is ‘voltage times current’, or in the more general notation I introduced here, ‘effort times flow’, or $\dot{p} \dot{q}$. I’ll use that more general notation.

For a capacitor we have

$$q = C \dot{p}$$

so we can define an `internal energy’ for the capacitor

$$H(p,q) = q^2 / 2 C$$

such that

$$\frac{d H}{d t} = \dot{p} \dot{q}$$

According to some electrical engineering books I’m reading, this equation is the reason we say a capacitor conserves energy: the power we pour into it equals the rate of increase of its internal energy.

We can also define an internal energy for an inductor, which again satisfies

$$\frac{d H}{d t} = \dot{p} \dot{q}$$

Capacitors and inductors are analogous to springs and masses in mechanics, so everything I’ve said has an analogue for those too. A spring stores internal energy that’s a function of $q$ (potential energy); a mass stores internal energy that’s a function of $p$ (kinetic energy).

But I’m quite puzzled by the general status of this equation:

$$\frac{d H}{d t} = \dot{p} \dot{q}$$

It seems to be hinting at a generalization of Hamiltonian mechanics for ‘open systems’, where we no longer have

$$\frac{d H}{d t} = 0$$

as we do in a closed system, but instead our system can gain or lose energy via interaction with its environment.

In ordinary Hamiltonian mechanics, Hamilton’s equations

$$\dot{p} = \frac{\partial H}{\partial q}$$

$$\dot{q} = -\frac{\partial H}{\partial p}$$

automatically give

$$\frac{d H}{d t} = 0$$

I think I’ve seen a generalization of Hamiltonian mechanics to open systems where we can also have an ‘external force’. And I’ve seen Hamiltonian mechanics for systems that have a time-dependent Hamiltonian. But I don’t see the equation

$$\frac{d H}{d t} = \dot{p} \dot{q}$$

arising naturally from these formalisms! So what’s the realm of validity of this equation? Or is it just a special case of something better?

Thanks for this! A great way to start the year :)

You discuss two things close to my heart: algebraic topology as a driver of improvements in computational physics and the “cochain problem” involving the failure of cup product to be graded commutative “on the nose”. Of course, my interest in both stem from the fact I think the two are related.

I didn’t quite understand the bit about Eilenberg-Zilber map (EZ) and Alexander-Whitney map (AW). Are these related to the de Rham map (R) and the Whitney map (W)?

I wish I knew the correct “maths” way to express this, but given a manifold $M$ and differential forms $\Omega(M)$ on $M$, we can construct a simplicial complex $S$ with cochains $C^*(S)$. The de Rham map $R$ takes forms on $M$ and turns them into cochains on $S$

$$R:\Omega(M)\to C^*(S).$$

The Whitney map (W) takes cochains on $S$ and turns them into (Whitney) forms on $M$

$$W:C^*(S)\to\Omega(M).$$

We have

$$R\circ W = Id_{C^*(S)}$$

and

$$W\circ R \sim Id_{\Omega(M)},$$

which sounds similar to EZ and AW.

As you pointed out, the wedge product in $\Omega(M)$ is graded commutative and the cup product in $C^*(S)$ is not graded commutative “on the nose” so all these maps do not quite fit together perfectly, e.g.

$$W(a\smile b) \ne W(a)\wedge W(b)$$

and

$$R(\alpha\wedge\beta) \ne R(\alpha)\smile R(\beta).$$

Some people have proposed a modified cup product for computational physics via

$$a\tilde\smile b := R(W(a)\wedge W(b)).$$

I don’t think this helps much with the “cochain problem”. In particular, this modified cup product is not even associative!

I have proposed an alternative, which at first will seem radical but I actually think might help with things like the “cochain problem”. That is to introduce a modified wedge product

$$\alpha\tilde\wedge\beta := W(R(\alpha)\smile R(\beta)).$$

How dare we mess with the continuum :)

This modified wedge product has the (uber) nice property that it is an algebra homomorphism “on the nose”, i.e.

$$R(\alpha\tilde\wedge\beta) = R(\alpha)\smile R(\beta).$$

The undesirable property of this modified wedge product is that it will depend on $S$, but that dependence disappears when you pass to cohomology, homotopy, etc, which is what the mathematicians really care about.

My gut tells me that having a true algebra morphism like this will give you true functors and the category theoretic analysis will be much cleaner. But that is just a hunch.

Also note that in a suitable limit of refinements of $S$, i.e. a kind of “continuum limit” we have

$$\alpha\wedge\beta = \lim_{continuum} \alpha\tilde\wedge\beta.$$

Phew, back online again after a massive failure by my internet provider.

Preece was another electrical engineer, who liked the hydraulic analogy and disliked Heaviside’s fancy math.

Which goes to show how relative these feelings are. Heaviside was a critic of Hamilton’s quaternions and their use in physics, so came into conflict with defenders such as Tait.

Regarding nineteenth century physics analogies, Smith and Norton Wise’s Energy and empire: a biographical study of Lord Kelvin is a must.