IOP Ralph Kenna - Q Theory

 
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Q-scaling, Ϙ, ϙ, ηQ and all that

  In the 1960's four famous scaling relations were discovered in the context of critical phenomena (phase transitions of higher order characterised by divergent correlation length and thermodynamic functions at a critical temperature Tc). These are

ν d = 2 - α     (hyperscaling - developed by Ben Widom, in 1965)

2β + γ = 2 - α     (discovered by John Essam and Michael Fisher in 1963)

β(δ - 1) = γ     (also by Ben Widom, 1964)

ν(2-η) = γ     (also by Michael Fisher, 1964).

The hyperscaling relation is special in that it involves the space (or space-time) dimensionality d. While the other three scaling relations hold for all dimensionalities, hyperscaling only holds at and below the upper critical dimension dc, which for most models is four. This can be understood heuristically by recalling that the critical exponents coming from mean field theory (MFT) obey hyperscaling independent of dimensionality. Now, MFT neglects fluctuations, and according to the Ginzburg criterion it should hold above the upper critical dimension. It gives α=0, &nu=1/2;. Since these are dimension independent and d is not, hyperscaling cannot hold. Recently we have extended hyperscaling to beyond the upper critical dimension.
Date Logo News Item Notes
  IOP
The image on the left is of a configuration for the Ising model with long-range interactions (sigma = 0.1) on a 2D 96x96 lattice (above the upper critical dimension) with periodic<\i> boundary conditions at the critical point.
The image on the right is of a configuration for the Ising model with long-range interactions (sigma = 0.1) on a 2D 96x96 lattice (above the upper critical dimension) with periodic<\i> boundary conditions at the pseudocritical point.
IOP
 
  IOP
The image on the left is of a configuration for the Ising model with long-range interactions (sigma = 0.1) on a 2D 96x96 lattice (above the upper critical dimension) with free<\i> boundary conditions at the critical point.
The image on the right is of a configuration for the Ising model with long-range interactions (sigma = 0.1) on a 2D 96x96 lattice (above the upper critical dimension) with free<\i> boundary conditions at the pseudocritical point.
IOP
 
   
 

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