Tuesday, May 19, 2009

A simple example of universality

I have been unable to find on-line a copy of Ken Wilson's 1978 Scientific American article describing the scaling, the renormalisation group, and phase transitions since I like to refer my undergraduate class to it. (Anyone know how to get an electronic copy, or something comparable in clarity and level?)
In the search, I found the following article in the American Journal of Physics. I found the following paragraph helpful and insightful:
The simplest nontrivial example of universality is given by the law of large numbers (the central limit theorem) which is crucial in statistical mechanics. In systems where it can be applied, all the details of the underlying probability distribution of the constituents of the system are irrelevant for the cooperative phenomena which are governed by a Gaussian probability distribution. This drastic reduction of complexity is precisely what is necessary for physics because it lets us build effective theories in which only a few couplings are kept. Renormalizability in statistical field theory is one of the nontrivial generalizations of the central limit theorem.
The image below is taken from a nice site which contains an interactive simulation of the central limit theorem.

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