Potts model: Difference between revisions
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The '''Potts model''' | 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== | ||
<references/> | |||
'''Related reading''' | |||
*[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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=1,2, \cdots, q } .
The energy of the system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E } , is defined as:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = - K \sum_{ \langle ij \rangle } \delta (S_i,S_j) }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K } is the coupling constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle ij \rangle } indicates that the sum is performed exclusively over pairs of nearest neighbour sites, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(S_i,S_j) } is the Kronecker delta. Note that the particular case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=2 } is equivalent to the Ising model.
Phase transitions[edit]
Considering a symmetric situation (i.e. equal chemical potential for all the species):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_1 = \mu_2 = \cdots = \mu_q } ;
the Potts model exhibits order-disorder phase transitions. For space dimensionality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=2 } , and low values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q } the transitions are continuous (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(T) } is a continuous function), but the heat capacity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(T) = (\partial E/\partial T) } , diverges at the transition temperature. The critical behaviour of different values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q } belong to (or define) different universality classes of criticality For space dimensionality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=3 } , the transitions for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q \ge 3 } are first order ( shows a discontinuity at the transition temperature).
See also[edit]
References[edit]
- ↑ Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society 48 pp. 106-109 (1952)
- ↑ Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available pdf)
- ↑ F. Y. Wu "The Potts model", Reviews of Modern Physics 54 pp. 235-268 (1982)
- ↑ F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics 55 p. 315 (1983)
Related reading