Quantization and Cohomology (Week 13)

February 6, 2007

Quantization and Cohomology (Week 13)

Posted by John Baez

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This week in our course on Quantization and Cohomology, we saw how statistical mechanics involves a number system $ℝ^{T}$ that depends on the temperature $T$ . In the ‘chilly limit’ $T \to 0$ , this reduces to the number system $ℝ^{\min}$ suitable for classical statics, where energy is minimized:

Week 13 (Feb. 6) - Statistical mechanics and deformation of rigs. Statistical mechanics (or better, ‘thermal statics’) as matrix mechanics over a rig ℝ T that depends on the temperature T. As T → 0, the rig ℝ T reduces to ℝ min and thermal statics reduces to classical statics, just as quantum dynamics reduces to classical dynamics as Planck’s constant approaches zero. Tropical mathematics, idempotent analysis and Maslov dequantization.
- Supplementary reading: G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a brief introduction.

Last week’s notes are here.

Posted at February 6, 2007 8:51 PM UTC

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9 Comments & 2 Trackbacks

Read the post Quantization and Cohomology (Week 12)
Weblog: The n-Category Café
Excerpt: Classical mechanics, quantum mechanics and statistical mechanics as 'matrix mechanics' over various rigs (rings without nnegatives).
Tracked: February 6, 2007 9:06 PM

Re: Quantization and Cohomology (Week 13)

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So to re-ask my questions:

Where does quantum thermodynamics fit in? Is it that there’s a statistical mechanics for each complex number, the real part corresponding to temperature, and the complex part the value of $ℏ$ ? Or aren’t the parameters between which Wick rotation acts able to be related like that?

Posted by: David Corfield on February 6, 2007 10:12 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 13)

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Yes, you’ve asked these questions before — and I wanted to avoid answering them until I thought hard about them and discussed them in the course. Alas, I think you’ll have figured out the answer before I get around to doing this!

For now I’ll just drop this hint: temperature lives on the Riemann sphere.

(In the quoted text I was trying to treat temperature as imaginary reciprocal time, via the similarity between $\exp (- i t H / ℏ)$ and $\exp (- H / k T)$ . Right now I’m trying to treat temperature as an imaginary Planck’s constant. Both ideas are fruitful and widely used in physics — but the second is far more relevant to this course.)

Posted by: John Baez on February 7, 2007 3:58 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 13)

So a Wick rotated quantum system in which either time has been speeded up, or the Planck constant has been increased, is like a hotter statistical system?

And you can’t relate the Planck constant and time this side of full blown quantum gravity?

Posted by: David Corfield on February 7, 2007 9:40 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 13)

Is it really fair to say that ‘statistical mechanics’ is actually only a form of statics? After all, some people work on non-equilibrium statistical mechanics. It seems to me that only equilibrium statistical mechanics is a form of statics. (Unfortunately, that’s all that they usually teach one ….)

Posted by: Toby Bartels on February 7, 2007 7:44 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 13)

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Toby wrote:

Is it really fair to say that ‘statistical mechanics’ is actually only a form of statics?

No — only the portion that people understand well enough to teach kids.

It seems to me that only equilibrium statistical mechanics is a form of statics.

Right, that’s the portion I was talking about.

Posted by: John Baez on February 7, 2007 9:49 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 13)

Is there a way to conceptualise the equilibrium/nonequilibrium distinction to resemble the statics/dynamics distinction we’ve talked about before? Remember static solutions occurred where extremal paths happened to be constant paths. So can one think of nonequilibrium statistical mechanics as about extremal paths through a space of probability distributions, to include the constant path solutions, i.e., the equilibrium states?

This is all to do with Jaynes’ maximum caliber principle. Jaynes

noted a tantalising analogy between the caliber in NESM and the Lagrangian in mechanics. (R. Dewar, Maximum Entropy Production and Non-equilibrium Statistical Mechanics, p. 47)

Dewar refers us to: Jaynes ET (1985b) Macroscopic prediction, in H. Haken (ed.) Complex systems – operational approaches in neurobiology. Springer-Verlag, Berlin, pp 254–269, where we read:

The caliber of a space-time process thus appears as the fundamental quantity that “presides over” the theory of irreversible processes in much the same way that the Lagrangian presides over mechanics. That is, in ordinary mechanics we learn first that a variational principle (minimum potential energy) determines the conditions of stable equilibrium; then we learn how to generalize this to the Lagrangian, whose variational properties generate the equations of motion. In close analogy, Gibbs showed that a variational principle (maximum entropy) determines the states of stable thermal equilibrium; now we have learned how to generalize this to the caliber, whose variational properties generate the “equations of motion” of irreversible processes.

But these equations of motion are not, except in a certain approximation, differential equations. In general they turn out to be nonlinear integral equations. Close to equilibrium they become linear integral equations, which contain the conventional Kubo relations for linear transport phenomena as special cases. Or, in a short-memory approximation, they reduce to generalized Fokker-Planck-Onsager equations showing how the approach to equilibrium is both steered and stabilized by the entropy function. (p. 11)

Posted by: David Corfield on February 8, 2007 11:37 AM | Permalink | Reply to this

Read the post Quantization and Cohomology (Week 14)
Weblog: The n-Category Café
Excerpt: An example of path integral quantization.
Tracked: February 15, 2007 10:51 PM

Re: Quantization and Cohomology (Week 13)

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I think there ought to be a correction in the formula for deforming the rig. It disappears in the $T \to 0$ limit, but it’s important for $T \to ∞$ . Right now the deformed addition looks like

(1)

x \oplus_{T} y = - T \ln (\exp (- x / T) + \exp (- y / T));

however

(2)

\lim_{T \to ∞} x \oplus_{T} y = - T (\ln (2) - (x + y) / 2 T)

so I think we should redefine it to

(3)

x \oplus_{T} y = T [\ln (2) - \ln (\exp (- x / T) + \exp (- y / T))] .

That way, the $T \to ∞$ limit gives a rig where multiplication is + and addition is avg. Note that multiplication still distributes over addition!

(4)

a + (b avg c) = a / 2 + a / 2 + b / 2 + c / 2 = (a + b) avg (a + c)

Posted by: Mike Stay on July 30, 2007 6:15 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 13)

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The way I do it in the week 13 notes, the addition and multiplication in the rig $ℝ^{T}$ come from pulling back the usual addition and multiplication of nonnegative real numbers along the ‘Boltzmann map’: that is, the function

$E \mapsto \exp (- E / kT)$

that sends energies to relative probabilities. The idea is that the usual rules for adding and multiplying (‘or-ing’ and ‘and-ing’) relative probabilities become temperature-dependent operations on energies.

Since we’re just pulling back a rig structure along a bijection, the new operations we get are guaranteed to give a rig.

Given all this, your modified addition looks ad hoc and worrisome to me. I especially don’t like the fact that your proposed $T \to ∞$ limit of addition, namely ‘averaging’, is not associative! I bet this means the operation in (3) is not associative at finite $T$ , either. And, I don’t know a multiplication that distributes over this addition at finite $T$ .

On the other hand, I’d never noticed fact (2) before, so it could take a while to appreciate the good aspects of what you’re saying!

It’s possible the $\ln (2)$ in formula (2) is actually trying to tell us a lot of profound things! Or at least a little bit of information.

Posted by: John Baez on July 31, 2007 3:35 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 13)

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$\dots$ `averaging’ is not associative!

Yes, I noticed this a few hours later, but didn’t have the chance to make another post. You’re right, it’s not associative for any positive T. It’s associative for the $T \to 0$ limit, because that’s still min.

And, I don’t know a multiplication that distributes over this addition at finite $T$ .

Plus is distributive for all $T$ , since for $T > 0$

(1)

- T \ln a + (- T \ln x \oplus - T \ln y)

(2)

= - T \ln a - T \ln \frac{x + y}{2}

(3)

= - T \ln \frac{ax + ay}{2}

(4)

= - T \ln ax \oplus - T \ln ay

(5)

= ((- T \ln a) + (- T \ln x)) \oplus ((- T \ln a) + (- T \ln x)),

and for $T = 0$ it distributes over min.

a little bit of information

Heh!

Posted by: Mike Stay on July 31, 2007 7:18 PM | Permalink | Reply to this

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