Potts model: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
m (Added a new reference)
m (Added DOI to a paper.)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
The '''Potts model''' was proposed by Renfrey B. Potts in 1952 <ref>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>. 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 <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>.  
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>.

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]

  1. ↑ Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society 48 pp. 106-109 (1952)
  2. ↑ Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available pdf)
  3. ↑ F. Y. Wu "The Potts model", Reviews of Modern Physics 54 pp. 235-268 (1982)
  4. ↑ F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics 55 p. 315 (1983)

Related reading

Retrieved from "http://www.sklogwiki.org/SklogWiki/index.php?title=Potts_model&oldid=9248"