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March 18, 2006

This Week's Finds in Mathematical Physics (Week 228)

John Baez

Last week I showed you some pretty pictures of dunes on Mars.

This week I'll talk about dunes called "barchans" and their relation

to self-organized criticality. Then I'll say a bit about Lauscher

and Reuter's work on quantum gravity... and then I'll beg for help

on a topology problem involving so-called "rational tangles".

But first, a demonstration of my psychic powers.

Take any book off the shelf and look at its 10-digit ISBN number.

Multiply the first digit by 1, the second digit by 2, the third

digit by 3 and so on... up to the NEXT TO LAST DIGIT. Add them up.

Then take this sum and see what it equals mod 11. At the end of

this article, I'll say what you got.

Okay. Here's a photo of the icy dunes of northern Mars. I love it

because it shows that Mars is a lively place with wind and water:

1) North polar sand sea, Mars Odyssey Mission, THEMIS (Thermal

emission imaging system), http://themis.mars.asu.edu/features/polardunes

These dunes, occupying a region the size of Texas, have been sculpted by

wind into long lines with crests 500 meters apart. Their hollows are

covered with frost, which appears bluish-white in this infrared photograph.

The big white spot near the bottom is a hill 100 meters high.

Where the dunes become sparser - for example, near that icy hill - they

break apart into "barchans". These are crescent-shaped formations

whose horns point downwind. Barchans are also found on the deserts of

Earth, and surely on many other planets across the Universe. They are

one of several basic dune patterns, an inevitable consequence of the

laws of nature under fairly common conditions.

The upwind slope of a barchan is gentle, while the downwind slope is

between 32 and 34 degrees. This is the "angle of repose" for sand - the

maximum angle it can tolerate before it starts slipping down:

2) Wikipedia, Barchan, http://en.wikipedia.org/wiki/Barchan

Wind-blown sand accumulates on the front of the barchan, and then

slides down the "slip face" on the back.

Barchans gradually migrate in the direction of the wind at speeds of

about 1-20 meters per year, with small barchans moving faster than big

ones. In fact, when they collide, the smaller barchans pass right

through the big ones! So, they act like solitons in some ways.

It would be great to see one of these frosty barchans close up.

We almost can do it now! The European Space Agency's orbiter

called Mars Express took this wonderful closeup, already shown

in "week211":

3) ESA/DLR/FU Berlin (G. Neukum),

Glacial, volcanic and fluvial activity on Mars: latest images,

http://www.esa.int/SPECIALS/Mars_Express/SEMLF6D3M5E_1.html

However, this is not a barchan - it's a lot bigger. On top of

the picture we see dunes, but then there's a cliff almost 2 kilometers

high leading down into what may be a volcanic caldera. The white stuff

is ice, while the dark stuff could be volcanic ash.

It's actually a bit surprising that there's enough wind on Mars to

create dunes. After all, the air pressure there is about 1% what it

is here on Earth! But in fact the wind speed on Mars often exceeds

200 kilometers per hour, with gusts up to 600 kilometers per hour.

There are dust storms on Mars so big they were first seen from

telescopes on Earth long ago. So, wind is a big factor in Martian

geology:

4) NASA, Mars exploration program: dust storms,

http://mars.jpl.nasa.gov/gallery/duststorms/

The Mars rover Spirit even got its solar panels cleaned by

some dust devils, and it took some movies of them:

5) NASA, Exploration rover mission: dust devils at Gusev, Sol 525,

http://marsrovers.nasa.gov/gallery/press/spirit/20050819a.html [Broken]

Turning to mathematical physics per se, I can't resist pointing out

that sand piles became very fashionable in this subject a while back.

Why? Well, for this I need to explain "self-organized criticality".

First, note that when a pile of sand is exactly at its angle of repose,

it will suffer lots of little landslides - and a few of these will become

big.

The theory of "critical phenomena" suggests that in this situation,

the probability that a landslide grows to size L should satisfy a

power law. In other, it should be proportional to

1/L^c

for some number c called the "critical exponent". At least, this

type of behavior is seen in many other situations where a physical

system is on the brink of some drastic change - or more precisely,

a "critical point".

When a system is not at a critical ponit, we typically see exponential

laws, where the probability of a disturbance of size L is proportional

to

exp(-L/L_0)

where L_0 is a fixed length scale. This means that our system will

look qualitatively different depending how much we zoom in with our

microscope. At length scales shorter than L_c, disturbances are

really common, while at large length scales they're incredibly rare.

When a system *is* at a critical point, it's self-similar: you can

zoom in or zoom out, and it looks qualitatively the same! It has

no specific length scale. This is what the power law says.

Here's a good place to learn the basics of power laws and self-similarity:

6) Manfred R. Schroeder, Fractals, Chaos, Power Laws, W. H. Freeman,

New York, 1992.

What makes sand dunes interesting is that as they seem to *enjoy*

living on the brink of danger. As the wind blows, they heap up until

their slip face is right at the angle of repose... ready for landslides!

This is the idea of "self-organized criticality": some physical systems

seem to spontaneously bring themselves towards critical points, without

any need for us to tune their parameters to special values.

The paper that introduced this idea came out in 1987:

7) Per Bak, Chao Tang and Kurt Wiesenfeld, Self-organized criticality:

an explanation of 1/f noise, Phys. Rev. Lett. 59 (1987) 381-384.

They came up with a simple model of a sand pile that exhibits

self-organized criticality. In the words of Jos Thijssen:

Bak and co-workers modelled the sand pile as a regular array

of columns consisting of cubic sand grains. Addition of new

grains is simply performed by selecting a column at random and

increasing its height by one. If the column then exceeds its

neighbours in height more than some threshold, it will "collapse":

it will lose some grains which are distributed evenly over its

nearest neighbours. As this collapse alters the height differences

involving those neighbours, there is the possibility that they

collapse in turn. A cascade process sets in until all height

differences are below the threshold. The size of such an avalanche

is defined as the number of sand grains sliding as a result of

a single grain of sand being added to the pile.

What is so interesting about the sand pile model? It turns out

that the sides of the sand pile acquire a specific slope, which is

such that the distribution of avalanches as function of size scales

as a power law. Power laws indicate the absence of scale and indeed

avalanches on all scales are sustained for the equilibrium slope.

If the slope is changed artificially from its equilibrium value,

the distribution is no longer a power law, but it will have an

intrinsic scale (e.g. exponential). Power laws and absence of scale

are the signature of a system being critical. Because the sand pile

tends to adjust the slope of its sides until the power law scaling

sets in, the criticality is called "self-organised".

If your computer runs Java applets, you can play with Thijssen's

simulation sand pile and see the avalanches yourself:

8) Jos Thijssen, The sand pile model and self organised criticality,

http://www.tn.tudelft.nl/tn/People/Staff/Thijssen/sandexpl.html [Broken]

And here's a cellular automaton sand pile you can play with:

9) Albert Schueller, Cellular automaton sand pile model,

http://schuelaw.whitman.edu/JavaApplets/SandPileApplet/ [Broken]

This one is only 2-dimensional, so the avalanches are less dramatic,

but you can have some fun using the mouse to build structures that

impede the motion of sand.

Like a speck of sand landing at the right place at the right time,

the original paper of Bak et al started a huge landslide of work on

self-organized criticality, some of which has been popularized here:

10) Per Bak, How Nature Works: The Science of Self-Organized Criticality,

Copernicus, New York, 1996.

As you can guess from the title "How Nature Works", some people got a

little carried away with the importance of self-organized criticality.

Then there was a kind of backlash, just as happened with fractals,

chaos, and catastrophe theory. These are all perfectly respectable and

interesting topics in mathematical physics that suffered from being

oversold. People are always eager to find the secret key that will

unlock all the mysteries of the universe. So, if some new idea seems

very general, people will run around trying to unlock all the mysteries

of the universe with it - and become sorely disappointed when it only

unlocks *some*.

I'd be interested to see how well mathematical physicists can model

actual sand dunes. These display an interesting complexity of behavior,

as the pictures here show:

11) US Army Corps of Engineers, Dunes,

http://www.tec.army.mil/research/products/desert_guide/lsmsheet/lsdune.htm

I've only looked at a few papers on the subject, all dealing with

barchans:

12) V. Schwaemmle and H. J. Herrmann, Solitary wave behaviour of sand

dunes, Nature 426 (Dec. 11, 2003), 619-620.

13) Klaus Kroy, Gerd Sauermann, and Hans J. Hermann, Minimal model for

sand dunes, Phys. Rev. Lett. 88 (2002), 054301. Also available at

cond-mat/0101380.

14) H. Elbelrhiti, P. Claudin, and B. Andreotti, Field evidence for

surface-wave-induced instability of sand dunes, Nature 437 (Sep. 29, 2005),

720-723.

The first paper describes how barchans pass through each other like

solitons, simulating them by an equation that's described in the second

one. (By the way, the term "minimal model" in the title of the second

paper is not being used in the sense familiar in conformal field theory!)

The third paper reports the results of a 3-year field study: in reality,

barchans are not stable, and big ones (called "megabarchans") can break

apart into smaller "elementary barchans".

If you're more interested in Mars than the mathematical physics of sand

dunes, you'll be happy to hear that Google has just moved to drastically

expand its customer base by introducing "Google Mars":

15) Google Mars, http://www.google.com/mars/

Using this you can explore many features of Mars, including its dunes.

I'm getting a little tired out, but there's one thing I've been

meaning to mention for a while. It's actually related to renormalization,

which is secretly the same subject as this "critical point" business

I just mentioned. But, it's not about sand piles - it's about quantum

gravity!

In "week222" I spoke about the work of Lauscher and Reuter, who claim to

have found evidence for an ultraviolet fixed point in quantum gravity

without matter. In other words, as you zoom in closer and closer, they

claim quantum gravity without matter acts more and more like some fixed

theory. This would be big news: it would suggest that gravity without

matter is a sensible theory, contrary to what everyone in string theory

says!

Not surprisingly, the string theorist Jacques Distler examined Lauscher

and Reuter's work with a critical eye. And, he wrote up a nice

explanation of the problems with their work:

16) Jacques Distler, Unpleasantness,

http://golem.ph.utexas.edu/~distler/blog/archives/000648.html

Briefly, the problem is that Lauscher and Reuter make a drastic

approximation. They start with the "exact renormalization group

equation", which is a beautiful thing: it says how a Lagrangian

for a field theory at one length scale gives rise to an effective

Lagrangian for the same theory at a larger length scale. However,

then they truncate the incredibly complicated formula for a fully

general Lagrangian, restricting to Lagrangians with only an

Einstein-Hilbert term and a cosmological constant. Like Distler,

I see no reason to think this approximation is valid. So, their

claimed ultraviolet fixed point could be an artifact of their method.

Whether it's worth going further and checking this by considering

a slightly less brutal approximation, using Lagrangians with a few

more terms, is a matter of taste. Distler doesn't think so. I

hope Lauscher and Reuter do. If they don't, we may never know for

sure what happens. I think it's actually rather amazing that they

get an fixed point with their brutal approximation, instead of

coupling constants that run to infinity or zero, which is what

I would have naively expected. But who knows? Maybe this is

easily understood if you think hard enough.

Today I was also going to talk about the 3-strand braid group, the

group PSL(2,Z), and rational tangles, but now I don't have the energy.

So instead, I'll just put out a request for help!

There's a wonderful game invented by John Conway called "rational

tangles". Here's how it works. It involves two players and a referee.

The players, call them A and B, start by facing each other and holding

ropes in each hand connecting them together like this:

A A

| |

| |

B B

This is called "position 0". The referee then cries out either "add one!"

or "take the negative reciprocal!". If the referee yells "add one!",

player B has to switch which hand he's using to hold which rope, making

sure to pass the right one over the left, like this:

A A

\ /

/

/ \

B B

This is called "position 1", since we started with "position 0" and

then did "add one!". But if the referee had said "take the inverse

reciprocal!" both players must cooperate to move all four ends of

the ropes a quarter-turn clockwise, like this:

A A

\_/

_

/ \

B B

This is called "position -1/0", since we started with 0 and then

did "take the negative reciprocal!".

The referee keeps crying "add one!" or "take the negative reciprocal!"

in whatever order she feels like, and players A and B keep doing the

same sort of thing: either player B switches the ropes right over left,

or both players rotate the whole tangle a quarter-turn clockwise. It's

actually best if the referee doesn't start with "take the negative

reciprocal!", since some people refuse to divide by zero, for religious

reasons.

Anyway, after a while the ropes are all tangled up and the rope is

in "position p/q" for some complicated rational number p/q. It will

be all tangled up - but in a special way, called a "rational tangle".

Then the players have to *undo* the tangling and get back to "position 0".

They may not remember the exact sequence of moves that got them into

the mess they are in. In fact the game is much more fun if they *don't*

remember. It's best to do it at a party, possibly after a few drinks.

Luckily, any sequence of "add one!" and "take the negative reciprocal!"

moves that carry their number back to 0, will carry their tangle back

to "position 0". So they just need to figure out how to get their

number back to 0, and the tangle will automatically untangle itself.

That's the cool part! It's a highly nonobvious theorem due to Conway.

I'm vaguely aware of a few proofs of this fact. As far as I know,

Conway's original proof uses the Alexander-Conway polynomial:

16) John Horton Conway, An enumeration of knots and links and some

of their algebraic properties, in Computational Problems of Abstract

Algebra, ed. D. Welsh, Pergamon Press, New York, 1970, pp. 329-358.

There's also a proof by Goldman and Kauffman using the Jones polynomial:

17) Jay R. Goldman and Louis H. Kauffman, Rational tangles, Advances in

Applied Mathematics 18 (1997), 300-332. Also available at

http://www.math.uic.edu/~kauffman/RTang.pdf

There are also two proofs in here:

18) Louis H. Kauffman and Sofia Lambropoulou, On the classification of

rational tangles, available as math.GT/0311499.

But here's what I want to know: is there a proof that makes

extensive use of the group PSL(2,Z) and its relation to topology?

After all, the basic operations on rational tangles are "adding

one" and "negative reciprocal", and these generate all the

fractional linear transformations

az + b

z |-> -------

cz + d

with a,b,c,d integer and ad-bc = 1. The group of these transformations

is PSL(2,Z). It acts on rational tangles, and Conway's theorem says

this action is isomorphic to the obvious action of PSL(2,Z) as fractional

linear transformations of the "rational projective line", meaning the

rational numbers together with a point at infinity. Since PSL(2,Z) has

lots of relations to topology, there should be some proof of Conway's

theorem that *uses* these relations to get the job done.

Does anybody know one?

Finally, the answer to the psychic powers puzzle: if you did the

calculation right, you got the last digit of the book's ISBN number -

unless your answer was 10, in which case the ISBN number should end

in the letter X.

This trick is called a "check sum" or "check digit": it's a way to spot

errors. The Universal Product Code, used in those bar codes you see

everywhere, also has a check digit. So do credit cards.

-----------------------------------------------------------------------

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http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

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# This Week's Finds in Mathematical Physics (Week 228)

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