In Mysterious Pattern, Math and Nature Converge (2013)(quantamagazine.org) |
In Mysterious Pattern, Math and Nature Converge (2013)(quantamagazine.org) |
http://nautil.us/issue/20/creativity/the-strange-inevitabili...
"Avalanches in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales. The frequency of network outages on the Internet, as a function of size and duration. The frequency of citations of journal articles, considered in the network of all citations amongst all papers, as a function of the number of citations in a given paper. The formation and propagation of cracks and tears in materials ranging from steel to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale. The electrical breakdown of dielectrics, which resemble cracks and tears. The percolation of fluids through disordered media, such as petroleum through fractured rock beds, or water through filter paper, such as in chromatography. Power-law scaling connects the rate of flow to the distribution of fractures. The diffusion of molecules in solution, and the phenomenon of diffusion-limited aggregation. The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks)."
In phase transition theories, you've got two different states (like liquid water and water vapor), and when you vary some high-level parameters (like pressure and temperature) you can go from one of these states to the other. This means that the only "interesting" physics (in the sense of "distinct from the well-understood liquid/gas behavior" can only happen right as you get near that transition, so that liquids are seamlessly becoming gases which are seamlessly becoming liquids again. The "transition" allows a lot of behavior not seen elsewhere, in fact all of the behavior "in the middle" between the two regimes. Because you've got this considerable mixing of the two states, often "zooming in" is the same as, say, adjusting the proportion of liquid to gas -- so since it all happens in the same space, you look for these "scale invariant" theories that are the same upon zooming in. Those theories then can't depend on too much particulars, but just depend on various symmetries, so they become "universal". See http://en.wikipedia.org/wiki/Critical_exponent .
Random matrix theory is similar, but for a different reason. The issue is, if you have a very complicated system that you can represent as a huge matrix, often the eigenvalues of the matrix tell you something concrete and physical. The example given above was the eigenvalues of a Hamiltonian matrix in quantum mechanics, which gives you an "energy spectrum" (discrete energies that the system can be at, so that it can e.g. absorb a photon of energy B - A to transition from a state of energy A to one of energy B).
Why would a Hamiltonian be random? You have to imagine a big molecule with lots of parts, not entirely under the control of the experimentalist. Maybe you've got a carbon nanotube hanging over a trench that you've etched under it, but the etching has caused other atoms to be stuck to the tube in unpredictable ways, and maybe the "islands" on either side couple to the nanotube in complicated ways.
Wigner discovered that very often these random variations break the symmetry between certain levels, so that where you once had 3 states at the same energy, now it's like those "energy eigenvalues" have "repelled" each other. He realized that the right way to start to think of the problem involved taking a matrix and adding random elements to it; this led to a nice set of models where you take it to the extreme and just randomize the entire matrix and look at its eigenvalue "density" rather than the exact levels. Wigner in particular discovered that this density function tends to look like a semicircle.
The essential similarity between these two is, you get to some point where "there's nothing more to say". With random matrices, when you specify how you're building the matrix and what sorts of properties it has to have, then you get the eigenvalue spectrum, and there's no other details to fixate upon. Similarly when you have a phase transition, the "we're taking on all the states in between gas and liquid" status of what you're doing means that the particulars of those states cannot matter. So in both cases it becomes just, "what's the configuration, what symmetries does it have" that determines how macroscopic parameters (whether critical exponents or eigenvalue densities) ultimately behave.
I think I get the scale invariant theory concept from a mathematical perspective, but I don't see why this would be the case: "Because you've got this considerable mixing of the two states, often "zooming in" is the same as, say, adjusting the proportion of liquid to gas"
Regarding random matrices, my understanding now is that given a random matrix of sufficient size, under a suitable definition of "random," if we look at the eigenvalue density function it's always going to be (roughly?) the same--we at least know it will be semi-circular. Further, there exist physical systems that can be modeled by random matrices; and there's a mapping between eigenvalues of the matrix and certain physical characteristics of the system. So, knowing the density function is always the same for these random matrices, we can assume certain shared characteristics of any systems that can be modeled by a random matrix.
In random matrices and phase transitions we would like to know how certain macroscopic parameters will behave, given some data like a matrix, or state of a phase transition. But, in both cases our starting data contains a lot of essentially irrelevant data that these theories prescribe a method for filtering out, since it assures us that knowledge of the symmetries involved are all that will matter.
Am I close? :)
These types of systems are basically anything from the 3-body problem (3 particles interacting via gravity), to the system of all the particles in this universe. And everything that falls in between.
http://arxiv.org/abs/math-ph/0603038
It's moderately technical, but really interesting.
It's basically the observation that large systems converge to exhibit rather simple behaviors. It generalizes other reoccurring patterns such as power laws, Euler's number, the normal distribution and the fibonacci sequence. For some reason nature is surprisingly frugal regarding the forms and behaviors that she allows. I think the simpest explanation for that comes from the anthropic principle and the multiverse hypothesis which state that all possible physical laws are realized in different universes and we happen to be in the one that has the necessary conditions to bring us into existence (universality being perhaps one of them).
This helps, because I don't need to re-invent logarithms and e to study animal population growth. And of course, different rules lead to different math (be it power laws, the Normal distribution, universiality, Phi, whatever).
This is interesting philosophically. HOWEVER, these models are still simplifications of reality. What happens with slight departures in animal population trajectories can be very different than perturbations in interest rates, so it would be a mistake to say "these things are truly connected" beyond the practicality of using similar math.
I maintain a large (publicly available) ecological dataset, and my data have been drawn into several meta-analyses of this type. Often the idea is to simply see if my data empirically fit the "right" distribution. And they fit it and say "wow, it's all connected".
But then I look at the Bus data in this example (the +'s that represent actual data). I'm guessing I could fit a lognormal distribution, a Gamma distribution, or the Dyson distribution that they actuall use, and the data wouldn't be enough to distinguish between them.
Now, all of these distributions result from "simple" rules, but they are three very different sets of simple rules. For the Bus data, the "repelling" by little slips of paper makes sense as the mechanism, so it's a good hypothesis.
But then to flip that around, and say "since this distribution fits the ecological data, the underlying mechanism must be Bus Repelling" is wholly unjustified (as there are other possible fits). And there's a lot of junk science that does that.