Potts model

The Potts model was proposed by Renfrey B. Potts in 1952 . The Potts model is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs . In practice one has a lattice system. The sites of the lattice can be occupied by particles of different species, $S=1,2,\cdots ,q$.

The energy of the system, $E$, is defined as:

$E=-K\sum _{\langle ij\rangle }\delta (S_{i},S_{j})$

where $K$ is the coupling constant, $\langle ij\rangle$ indicates that the sum is performed exclusively over pairs of nearest neighbour sites, and $\delta (S_{i},S_{j})$ is the Kronecker delta. Note that the particular case $q=2$ is equivalent to the Ising model.

Phase transitions

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

$\mu _{1}=\mu _{2}=\cdots =\mu _{q}$;

the Potts model exhibits order-disorder phase transitions. For space dimensionality $d=2$, 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 $d=3$, 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).

See also

• Ashkin-Teller model
• Kac model

References

Related reading

• Nathan Duff and Baron Peters "Nucleation in a Potts lattice gas model of crystallization from solution", Journal of Chemical Physics 131 184101 (2009)