[1] By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid-state physics.
The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is exactly solvable, and that it has a rich mathematical formulation that has been studied extensively.
[4] A further generalization of these methods by James Glazier and Francois Graner, known as the cellular Potts model,[5] has been used to simulate static and kinetic phenomena in foam and biological morphogenesis.
possible values [citation needed], distributed uniformly about the circle, at angles where
and that the interaction Hamiltonian is given by with the sum running over the nearest neighbor pairs
Potts provided the location in two dimensions of the phase transition for
, the model displays the phenomenon of 'interfacial adsorption' [11] with intriguing critical wetting properties when fixing opposite boundaries in two different states [clarification needed].
Understanding this relationship has helped develop efficient Markov chain Monte Carlo methods for numerical exploration of the model at small
The transformation is done using the identity[12] This leads to rewriting the partition function as where the FK 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
The one dimensional Potts model may be expressed in terms of a subshift of finite type, and thus gains access to all of the mathematical techniques associated with this formalism.
This section develops the mathematical formalism, based on measure theory, behind this solution.
While the example below is developed for the one-dimensional case, many of the arguments, and almost all of the notation, generalizes easily to any number of dimensions.
For defining the Potts model, either this whole space, or a certain subset of it, a subshift of finite type, may be used.
Explicit representations for the cylinder sets can be gotten by noting that the string of values corresponds to a q-adic number, however the natural topology of the q-adic numbers is finer than the above product topology.
A function V gives interaction energy between a set of spins; it is not the Hamiltonian, but is used to build it.
The argument to the function V is an element s ∈ QZ, that is, an infinite string of spins.
Define the function Hn : QZ → R as This function can be seen to consist of two parts: the self-energy of a configuration [s0, s1, ..., sn] of spins, plus the interaction energy of this set and all the other spins in the lattice.
The corresponding finite-state partition function is given by with C0 being the cylinder sets defined above.
It is very common in mathematical treatments to set β = 1, as it is easily regained by rescaling the interaction energy.
The partition function, together with the Hamiltonian, are used to define a measure on the Borel σ-algebra in the following way: The measure of a cylinder set, i.e. an element of the base, is given by One can then extend by countable additivity to the full σ-algebra.
Most thermodynamic properties can be expressed directly in terms of the partition function.
Thus, for example, the Helmholtz free energy is given by Another important related quantity is the topological pressure, defined as which will show up as the logarithm of the leading eigenvalue of the transfer operator of the solution.
The partition function may then be written as where card is the cardinality or count of a set, and Fix is the set of fixed points of the iterated shift function: The q × q matrix A is the adjacency matrix specifying which neighboring spin values are allowed.
The simplest case of the interacting model is the Ising model, where the spin can only take on one of two values, sn ∈ {−1, 1} and only nearest neighbor spins interact.
The goal of solving a model such as the Potts model is to give an exact closed-form expression for the partition function and an expression for the Gibbs states or equilibrium states in the limit of n → ∞, the thermodynamic limit.
Assume that we are given noisy observation of a piecewise constant signal g in Rn.
To recover g from the noisy observation vector f in Rn, one seeks a minimizer of the corresponding inverse problem, the Lp-Potts functional Pγ(u), which is defined by The jump penalty
forces piecewise constant solutions and the data term
[13] In image processing, the Potts functional is related to the segmentation problem.