Critical exponent: Difference between revisions

sectioning
(sectioning)
This problem does not appear in [[dimensional regularization|3.99 dimensions]], though. <!-- :) -->
 
== Mean field theory ==
:<math>\alpha \equiv \alpha'</math>
:<math>\gamma \equiv \gamma'</math>
:<math>\nu \equiv \nu'</math>
 
Thus, the exponents above and below the critical temperature, repectively, have identical values. This is understandable, since the respective scaling functions, <math> f_\pm(k\xi ,\dots)</math>, originally defined for <math>k\xi \ll 1</math>, should become identical if extrapolated to <math> k\xi \gg 1\,.</math>
 
The classical ([[Landau theory]] aka [[mean field theory]]) values for a scalar field are
One of the major discoveries in the study of critical points is that mean field theory is completely wrong in the vicinity of critical points in two and three dimensions. In four dimensions, we have logarithmic corrections.
 
== Experimental values ==
 
The most accurately measured value of <math>\alpha</math> is &minus;0.0127 for the phase transition of superfluid helium (the so-called [[lambda-transition]]). The value was measured in a satellite to minimize pressure differences in the sample ([http://xxx.arxiv.org/abs/cond-mat/0310163 see here]). This result agrees with theoretical prediction obtained by [[variational perturbation theory]] ([http://prola.aps.org/abstract/PRD/v60/i8/e085001 see here] or [http://www.physik.fu-berlin.de/~kleinert/kleiner_re279/seventh.html here]).
 
== Scaling relations ==
 
:<math>\alpha \equiv \alpha'</math>
:<math>\gamma \equiv \gamma'</math>
:<math>\nu \equiv \nu'</math>
 
Thus, the exponents above and below the critical temperature, repectively, have identical values. This is understandable, since the respective scaling functions, <math> f_\pm(k\xi ,\dots)</math>, originally defined for <math>k\xi \ll 1</math>, should become identical if extrapolated to <math> k\xi \gg 1\,.</math>
 
Critical exponents are denoted by Greek letters. They fall into [[universality classes]] and obey [[scaling relation]]s such as
 
and a lot of similar relations, which implies that there are only two independent exponents, e.g., <math>\,\nu</math> and {{nowrap|<math>\eta\,</math>.}} All this follows from the theory of the [[renormalization group]].
 
== Anisotropy
 
There are some [[anisotropic]] systems where the correlation length is direction dependent.