of the model above and is defined by a simpler Hamiltonian: where is the Kronecker delta, which equals one whenever and zero otherwise. The standard Potts model is equivalent to the Ising model and the 2-state vector Potts model, with . The standard Potts model is equivalent to the three-state vector Potts model, with . A common generalization is to introduce an external "magnetic field" term , and moving the parameters inside the sums and allowing them to vary across the model : where the inverse temperature, the Boltzmann constant and the temperature. Different papers may adopt slightly different conventions, which can alter and the associated partition function by additive or multiplicative constants. Physical properties Phase transitions Despite its simplicity as a model of a physical system, the Potts model is useful as a model system for the study of phase transitions. For example, for the standard ferromagnetic Potts model in , a phase transition exists for all real values , with the critical point at . The phase transition is continuous for and discontinuous for . For the clock model, there is evidence that the corresponding phase transitions are infinite order BKT transitions, and a continuous phase transition is observed when . Further use is found through the model's relation to percolation problems and the Tutte and chromatic polynomials found in combinatorics. For integer values of , the model displays the phenomenon of 'interfacial adsorption' with intriguing critical wetting properties when fixing opposite boundaries in two different states . Relation with the random cluster model The Potts model has a close relation to the Fortuin-Kasteleyn random cluster model, another model in statistical mechanics. Understanding this relationship has helped develop efficient Markov chain Monte Carlo methods for numerical exploration of the model at small , and led to the rigorous proof of the critical temperature of the model. At the level of the partition function , the relation amounts to transforming the sum over spin configurations into a sum over edge configurations i.e. sets of nearest neighbor pairs of the same color. The transformation is done using the identity with . This leads to rewriting the partition function as where the clusters are the connected components of the union of closed segments . This is proportional to the partition function of the random cluster model with the open edge probability . An advantage of the random cluster formulation is that