Renormalization Group

Full Title or Meme

Renormalization Groups are used to solve a difficult class of problems where fluctuations persist out to macroscopic wavelengths, and fluctuations on all intermediate length scales are important but cannot be solved by averages. This is most commonly seen in weather pattern which break flows into swirls and branches at very large scales as well as micro bursts at much smaller scales down to millimeters. At smaller scales viscosity damps the eddies and turbulent fluctuationfd are not seen until atomic scales are reached.

Context

The Renormalization Group approach is a strategy for dealing with problems involving many length scales. The strategy is to tackle the problem in steps, one step for each length scale. In the case of critical phenomena, the problem, technically, is to carry out statistical averages over thermal fluctuations on all size scales. The renormalization group approach is to integrate out the fluctuations in sequence starting with fluctuations on an atomic scale and then moving to successively larger scales until fluctuations on all scales have been averaged out, To illustrate the renormalization group ideas the case of’ critical phenomena will be discussed in more detail. First the mean field theory of Landau will be described, and important questions defined. The renormalization group will be presented as an improvement to Landau’s theory. The Curie point of a ferromagnet will be used as a specific example of a critical point. Below the Curie temperature, an ideal ferromagnet exhibits spontaneous magnetization in the absence of’ an external magnetic field; the direction of’ the magnetization depends on the history of the magnet. Above the Curie temperature Tc, there is no spontaneous magnetization.

Solutions

The primary solution described here is Phase Transitions, especially phase transitions in iron.

Magnetism IS caused at the atomic level by unpaired electrons with magnetic moments, and in a ferromagnet, a pair of nearby electrons with moments aligned has a lower energy than if the moments are anti-aligned.10 At high temperatures, thermal fluctuations prevent magnetic order. As the temperature is reduced towards the Curie temperature, alignment of one moment causes preferential alignment out to a considerable distance called the correlation length ξ.

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