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Some of the applications of random matrices is to find the spectra of heavy atoms in nuclear physics which are usually difficult to find otherwise.

How can starting from randomness of some kind, such as random matrices, lead to prediction of that sort?

Any idea why this work?

Random matrices are used in string theory too, may be some string theorist here can shed some light on this topic


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If you answer this question, you'll be famous. The reason that the normal distribution works so well for many applications is the central limit theorem ... a lot of small effects tend to add up to a normal distribution. Possibly, something similar is going on here. – Peter Shor Aug 8 '11 at 1:10
I'm curious about what sort of algorithm uses random matrices to calculate spectral lines... – Colin K Aug 8 '11 at 5:41
@Colin, it's not so much an algorithm, just an observation that the distribution of energy eigenvalues for many heavy nuclei happen to fall upon the distribution of eigenvalues of a certain class of random matrices (gaussian um... something (orthogonal, unitary, symplectic - I forget which) ensemble). While perhaps not physically satisfying, I think Peter's comment is the best answer to this question. – wsc Aug 8 '11 at 14:09
@wsc: It's not just heavy nuclei and random matrices; there are a number of other random processes which give rise to this same probability distribution of eigenvalues, and as far as I know, nobody has yet found a satisfactory explanation for why this happens. – Peter Shor Aug 8 '11 at 14:29
@Revo: String theorists would be the last people to know why this happens; no knock on stringy guys, but deriving the properties of even light nuclei from just quarks and gluons is already an immensely heady task. (not to mention that calculating the spectra of heavy nuclei from nucleons and pions is also a difficult task, thus the immense beauty of their distribution being reproduced by the GUE.) – wsc Aug 9 '11 at 3:27

1 Answer 1

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This is because the classical system consisting of nucleons interacting with a realistic pair potential is chaotic. A classically chaotic system careens between unstable periodic orbits, which in a path-integralish view ( gutzwiller trace formula) tells you that nearby energy levels are concentrated on completely different periodic orbits, so that they have mixed with each other strongly if you consider the unperturbed wavefunctions to be homogenous ( Heller scarring) The statistics of random matrix eigenvalues is based on the principle of heavy generic mixing between energy levels, leading to level repulsion. By contrast, classically integrable systems have energy levels which are regularly spaced, coming as they do from semiclassically setting the action variables to be integers. Each action variable gives a smooth level spacing, and the spacings add up linearly, to produce a uniformish distribution of energy levels without level repulsion.

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Could you please provide some references? – Revo Aug 18 '11 at 2:08
This is just folklore from the 90s. That an integrable system has no level repulsion is an ancient result, probably known to Bohr/Sommerfeld/Einstein. There's a nice review of the Gutzwiller trace formula here: Here is a review of scarring, focusing on the predicted corrections to random matrix The basic answer I give is more elementary than the corrections, and is found in the early pages of this book: – Ron Maimon Aug 18 '11 at 6:07

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