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 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 Martian 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
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
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
where L0 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 larger 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
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/
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 BThis 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 BThis is called "position 1", since we started with "position 0" and then did add one! But if the referee says 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 BThis 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. But, it's perfectly legal in this game.
Anyway, after a while the ropes will be in "position p/q" for some complicated rational number p/q. The'll 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 the players make 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. John Leech, Pergamon Press, Oxford, 1970, 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 + dwith 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.
Usually people describe the scheme in a different way, which is actually equivalent to what you said. Denote the 10 digit ISBN number as Ni, for 1≤i≤10. Compute10 Mod( ∑ (11-i)Ni , 11) (*) i=1which should give you zero. That is take the first digit, multiply it by 10, the second by 9, etc. Add them up and compute the sum mod 11. You will always get zero.
Some fun things you might like to add:
- If you make a mistake writing down a single digit in the ISBN then the equation (*) will not equal zero.
- The equation (*) may fail to give you zero mod 11 if you make a mistake with two of the digits, but it will never fail if you interchange two adjacent digits.
Self-organized criticality (SOC) does in fact involve special settings of the parameters in a model.The Wikipedia article is:
SOC occurs in sandpile models because one adds the sand extremely slowly, i.e., one grain at a time. Otherwise a critical state is not obtained. This makes SOC be a special example of dynamical critical phenomena in the case that the flux variable (here the rate of sand addition) is set to ε+, i.e., an infinitesimal value greater than zero. This formulation allows SOC to be studied using quantum field theory.
Of course the model is built around an underlying instability, namely the fact that sand piles which are too steep will fall down. Also, one must remove by idealization most sorts of friction between sand grains which will otherwise blur out the transition. So SOC is no magic prescription for generating scale invariant phenomena. SOC systems are "special" in the way that equilibrium critical points are "special". As you note, theories of this kind are easily oversold among those eager to believe in magic formulas.
Also, you may have been confusing SOC with a favorite concept of the chaos/dynamical systems people, namely the "edge of chaos". It went through several incarnations. Each one tried to formally specify the domain, intermediate between order and chaos, in which complex systems were most "interesting".
Wikipedia summarizes this pretty well. I append the listing for convenience.
All My Best,
19) Edge of chaos, Wikipedia, http://en.wikipedia.org/wiki/Edge_of_chaos
© 2006 John Baez