Potts model: Difference between revisions

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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. For a general discussion on Potts models see Refs. 2 and 3.
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>.  
In practice one has a lattice system. The sites of the lattice can be occupied by
In practice one has a lattice system. The sites of the lattice can be occupied by
particles of different ''species'', <math> S=1,2, \cdots, q </math>.
particles of different ''species'', <math> S=1,2, \cdots, q </math>.
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where <math> K </math> is the coupling constant, <math> \langle ij \rangle </math> indicates
where <math> K </math> is the coupling constant, <math> \langle ij \rangle </math> indicates
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]].
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]].
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 transitions==
Considering a symmetric situation (i.e. equal [[chemical potential]] for all the species):
 
:<math> \mu_1 = \mu_2 = \cdots = \mu_q </math>;
 
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
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 transitions |first order]] (<math> E </math>  shows a discontinuity at the transition temperature).
==See also==
==See also==
*[[Ashkin-Teller model]]
*[[Ashkin-Teller model]]
*[[Kac model]]
*[[Kac model]]
==References==
==References==
#Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106−109 (1952)
<references/>
#[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]
'''Related reading'''
#[http://dx.doi.org/10.1103/RevModPhys.55.315  F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]
*[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)]
[[category:models]]
[[category:models]]

Latest revision as of 12:22, 11 November 2009

The Potts model, proposed by Renfrey B. Potts in 1952 [1][2], is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs [3][4]. 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 transitions[edit]

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

;

the Potts model exhibits 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 behaviour 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[edit]

References[edit]

Related reading