Potts model: Difference between revisions

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m (some details about phase behavior)
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Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models | Ising model]].
Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models | Ising model]].


===Phase transtions===
Considering a symmetric situation (i.e. equal chemical potential for all the species):
Considering a symmetric situation (i.e. equal chemical potential for all the species):


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values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the heat capacity
values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the heat capacity
<math> C(T) = (\partial E/\partial T) </math> diverges at the transition temperature. The critical behavior of
<math> C(T) = (\partial E/\partial T) </math> diverges at the transition temperature. The critical behavior of
different values of <math> q </math> corresponds to different critical universality classes.
different values of <math> q </math> belong to (or define) different universality classes of criticality


For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are first order (<math> E </math>  
For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are first order (<math> E </math>  

Revision as of 11:04, 7 July 2008

The Potts model was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs. 2 and 3. In practice one has a lattice system. The sites of the lattice can be occupied by particles of different species, .

The energy of the system, , is defined as:

where is the coupling constant, indicates that the sum is performed exclusively over pairs of nearest neighbour sites, and is the Kronecker delta. Note that the particular case is equivalent to the Ising model.

Phase transtions

Considering a symmetric situation (i.e. equal chemical potential for all the species):

;

the Potts models exhibit order-disorder phase transitions. For space dimensionality , and low values of the transitions are continuous ( is a continuous function), but the heat capacity diverges at the transition temperature. The critical behavior of different values of belong to (or define) different universality classes of criticality

For space dimensionality , the transitions for are first order ( shows a discontinuity at the transition temperature).

See also

References

  1. Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society 48 pp. 106−109 (1952)
  2. F. Y. Wu "The Potts model", Reviews of Modern Physics 54 pp. 235-268 (1982)
  3. F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics 55 p. 315 (1983)