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| The '''Potts model''', proposed by Renfrey B. Potts in 1952 <ref>[http://dx.doi.org/10.1017/S0305004100027419 Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106-109 (1952)]</ref><ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>, is a generalisation of the [[Ising Models | Ising model]] to more than two components. For a general discussion on Potts models see Refs <ref>[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]</ref><ref>[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]</ref>. | | {{Stub-general}} |
| In practice one has a lattice system. The sites of the lattice can be occupied by
| | The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the [[Ising Models | Ising model]] to more than two components. |
| particles of different ''species'', <math> S=1,2, \cdots, q </math>.
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| The energy of the system, <math> E </math>, is defined as:
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| :<math> E = - K \sum_{ \langle ij \rangle } \delta (S_i,S_j) </math>
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| where <math> K </math> is the coupling constant, <math> \langle ij \rangle </math> indicates
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| that the sum is performed exclusively over pairs of nearest neighbour sites, and <math> \delta(S_i,S_j) </math> is the [[Kronecker delta|Kronecker delta]].
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| Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models | Ising model]].
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| ==Phase transitions==
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| Considering a symmetric situation (i.e. equal [[chemical potential]] for all the species):
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| :<math> \mu_1 = \mu_2 = \cdots = \mu_q </math>;
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| the Potts model exhibits order-disorder [[phase transitions]]. For space dimensionality <math> d=2 </math>, and low 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 behaviour of
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| different values of <math> q </math> belong to (or define) different [[universality classes]] of criticality
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| For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are [[First-order transitions |first order]] (<math> E </math> shows a discontinuity at the transition temperature).
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| ==See also== | | ==See also== |
| *[[Ashkin-Teller model]] | | *[[Ashkin-Teller model]] |
| *[[Kac model]] | | *[[Kac model]] |
| ==References== | | ==References== |
| <references/>
| | #Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106−109 (1952) |
| '''Related reading''' | | #[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)] |
| *[http://dx.doi.org/10.1063/1.3250934 Nathan Duff and Baron Peters "Nucleation in a Potts lattice gas model of crystallization from solution", Journal of Chemical Physics '''131''' 184101 (2009)]
| | #[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)] |
| [[category:models]] | | [[category:models]] |