Potts model: Difference between revisions

Revision as of 10:59, 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, 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.

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 models exhibit 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 $C(T)=(\partial E/\partial T)$ diverges at the transition temperature. The critical behavior 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 } corresponds to different critical universality classes.

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 (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 } shows a discontinuity at the transition temperature).

References

Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society 48 pp. 106−109 (1952)
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)

@@ Line 8: / Line 8: @@
 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]].
-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]].
+Considering a symmetric situation (i.e. equal chemical potential for all the species):
+<math> \mu_1 = \mu_2 = \cdots = \mu_q </math>;
+the Potts models exhibit 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 behavior of
+different values of <math> q </math> corresponds to different critical universality classes.
+For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are first order (<math> E </math>
+shows a discontinuity at the transition temperature).
 ==See also==
 *[[Ashkin-Teller model]]

Potts model: Difference between revisions

Revision as of 10:59, 7 July 2008

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