- Potts model
In

statistical mechanics , the**Potts model**, a generalization of theIsing model , is a model of interacting spins on acrystalline lattice . By studying the Potts model, one may gain insight into the behaviour offerromagnet s and certain other phenomena ofsolid 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 isexactly solvable , and that it has a rich mathematical formulation that has been studied extensively.The model is named after

Renfrey B. Potts who described the model near the end of his 1952 Ph.D. thesis. The model was related to the "planar Potts" or "clock model", which was suggested to him by his advisorCyril Domb . The Potts model is sometimes known as theAshkin-Teller model (afterJulius Ashkin andEdward Teller ), as they considered a four component version in 1943.The Potts model is related to, and generalized by, several other models, including the

XY model , theHeisenberg model and theN-vector model . The infinite-range Potts model is known as theKac model . When the spins are taken to interact in anon-Abelian manner, the model is related to theflux tube model , which is used to discuss confinement inquantum chromodynamics . Generalizations of the Potts model have also been used to modelgrain growth in metals andcoarsening infoam s. A further generalization of these methods byJames Glazier andFrancois Graner , known as theCellular Potts Model has been used to simulate static and kinetic phenomena in foam and biologicalmorphogenesis .**Physical description**The Potts model consists of "spins" that are placed on a lattice; the lattice is usually taken to be a two-dimensional rectangular

Euclidean lattice, but is often generalized to other dimensions or other lattices. Domb originally suggested that the spin take one of "q" possible values, distributed uniformly about thecircle , at angles:$heta\_n\; =\; frac\{2pi\; n\}\{q\}$

and that the interaction Hamiltonian be given by

:$H\_c\; =\; J\_csum\_\{(i,j)\}\; cos\; left(\; heta\_\{s\_i\}\; -\; heta\_\{s\_j\}\; ight)$

with the sum running over the nearest neighbor pairs $(i,j)$ over all lattice sites. The site "colors" $s\_i$ take on values, ranging from $1ldots\; q$. Here, $J\_c$ is a coupling constant, determining the interaction strength. This model is now known as the

**vector Potts model**or the**clock model**. Potts provided a solution for two dimensions, for "q"=2,3 and 4. In the limit as q approaches infinity, this becomes theXY model .What is now known as the standard

**Potts model**was suggested by Potts in the course of the solution above, and uses a simpler Hamiltonian, given by::$H\_p\; =\; -J\_p\; sum\_\{(i,j)\}delta(s\_i,s\_j)\; ,$

where $delta(s\_i,s\_j)$ is the

Kronecker delta , which equals one whenever $s\_i=s\_j$ and zero otherwise.The "q"=2 standard Potts model is equivalent to the 2D

Ising model and the 2-state vector Potts model, with $J\_p=-2J\_c$. The "q"=3 standard Potts model is equivalent to the three-state vector Potts model, with $J\_p=-3J\_c/2$.A common generalization is to introduce an external "magnetic field" term $h$, and moving the parameters inside the sums and allowing them to vary across the model:

:$eta\; H\_g\; =\; -\; eta\; sum\_\{(i,j)\}J\_\{ij\}\; delta(s\_i,s\_j)\; -\; sum\_i\; h\_i\; s\_i\; ,$

where $eta=1/kT$ the "inverse temperature", "k" the

Boltzmann constant and "T" thetemperature . The summation may run over more distant neighbors on the lattice, or may in fact be an infinite-range force.Different papers may adopt slightly different conventions, which can alter $H$ and the associated partition function by additive or multiplicative constants.

**Discussion**Despite its simplicity as a model of a physical system, the Potts model is useful as a model system for the study of

phase transition s. For example, two dimensional lattices with $J>0$ exhibit a first order transition if $q>4$. When $qleq\; 4$ a continuous transition is observed, as in the Ising model where $q=2$. Further use is found through the model's relation to percolation problems and the Tutte and chromatic polynomials found in combinatorics.The model has a close relation to the

Fortuin-Kasteleyn random cluster model , another model in statistical mechanics. Understanding this relationship has helped develop efficientMarkov chain Monte Carlo methods for numerical exploration of the model at small $q$.**Measure theoretic description**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. In particular, it can be solved exactly using the techniques oftransfer operator s. (However,Ernst Ising used combinatorial methods to solve theIsing model , which is the "ancestor" of the Potts model, in his 1925 PhD thesis). This section develops the mathematical formalism, based onmeasure 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. Some of the formalism is also broad enough to handle related models, such as the

XY model , theHeisenberg model and theN-vector model .**Topology of the space of states**Let $Q=\{1,2,cdots,q\}$ be a finite set of symbols, and let

:$Q^mathbb\{Z\}=\{\; s=(ldots,s\_\{-1\},s\_0,s\_1,ldots)\; :\; s\_k\; in\; Q\; ;\; forall\; k\; in\; mathbb\{Z\}\; \}$

be the set of all bi-infinite strings of values from the set "Q". This set is called a

full shift . For defining the Potts model, either this whole space, or a certain subset of it, asubshift of finite type , may be used. Shifts get this name because there exists a natural operator on this space, theshift operator $au:Q^mathbb\{Z\}\; o\; Q^mathbb\{Z\}$, acting as:$au\; (s\_k)\; =\; s\_\{k+1\}$

This set has a natural

product topology ; the base for this topology are thecylinder set s:$C\_m\; [xi\_0,\; ldots,\; xi\_k]\; =\; \{s\; in\; Q^mathbb\{Z\}\; :s\_m\; =\; xi\_0,\; ldots\; ,s\_\{m+k\}\; =\; xi\_k\; \}$

that is, the set of all possible strings where "k"+1 spins match up exactly to a given, specific set of values $xi\_0,\; ldots,\; xi\_k$. Explicit representations for the cylinder sets can be gotten by noting that the string of values corresponds to a "q"-adic number, and thus, intuitively, the product topology resembles that of the

real number line.**Interaction energy**The interaction between the spins is then given by a continuous function $V:Q^mathbb\{Z\}\; omathbb\{R\}$ on this topology. "Any" continuous function will do; for example

:$V(s)\; =\; -Jdelta(s\_0,s\_1)$

will be seen to describe the interaction between nearest neighbors. Of course, different functions give different interactions; so a function of $s\_0$, $s\_1$ and $s\_2$ will describe a next-nearest neighbor interaction. 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 $sin\; Q^mathbb\{Z\}$, that is, an infinite string of spins. In the above example, the function "V" just picked out two spins out of the infinite string: the values $s\_0$ and $s\_1$. In general, the function "V" may depend on some or all of the spins; currently, only those that depend on a finite number are exactly solvable.

Define the function $H\_n:Q^mathbb\{Z\}\; omathbb\{R\}$ as

:$H\_n(s)=\; sum\_\{k=0\}^n\; V(\; au^k\; s)$

This function can be seen to consist of two parts: the self-energy of a configuration $[s\_0,\; s\_1,\; ldots,s\_n]$ of spins, plus the interaction energy of this set and all the other spins in the lattice. The $n\; oinfty$ limit of this function is the Hamiltonian of the system; for finite "n", these are sometimes called the

**finite state Hamiltonians**.**Partition function and measure**The corresponding finite-state partition function is given by

:$Z\_n(V)\; =\; sum\_\{s\_0,ldots,s\_n\; in\; Q\}\; exp\; -eta\; H\_n(C\_0\; [s\_0,s\_1,ldots,s\_n]\; )$

with $C\_0$ being the cylinder sets defined above. Here, β=1/"kT", where "k" is

Boltzmann's constant , and "T" is thetemperature . It is very common in mathematical treatments to set β=1, as it is easily regained by rescaling the interaction energy. This partition function is written as a function of the interaction "V" to emphasize that it is only a function of the interaction, and not of any specific configuration of spins. 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:$mu\; (C\_k\; [s\_0,s\_1,ldots,s\_n]\; )\; =\; frac\{1\}\{Z\_n(V)\}\; exp\; -eta\; H\_n\; (C\_k\; [s\_0,s\_1,ldots,s\_n]\; )$

One can then extend by countable additivity to the full σ-algebra. This measure is a

probability measure ; it gives the likelihood of a given configuration occurring in theconfiguration space $Q^mathbb\{Z\}$. By endowing the configuration space with a probability measure built from a Hamiltonian in this way, the configuration space turns into acanonical ensemble .Most thermodynamic properties can be expressed directly in terms of the partition function. Thus, for example, the

Helmholtz free energy is given by:$A\_n(V)=-kT\; log\; Z\_n(V)$

Another important related quantity is the

topological pressure , defined as:$P(V)\; =\; lim\_\{n\; oinfty\}\; frac\{1\}\{n\}\; log\; Z\_n(V)$

which will show up as the logarithm of the leading eigenvalue of the

transfer operator of the solution.**Free field solution**The simplest model is the model where there is no interaction at all, and so $V=0$ and $H\_n=0$. The partition function becomes

:$Z\_n(0)\; =\; e^\{-eta\}\; sum\_\{s\_0,ldots,s\_n\; in\; Q\}\; 1$

If all states are allowed, that is, the underlying set of states is given by a

full shift , then the sum may be trivially evaluated as:$Z\_n(0)\; =\; e^\{-eta\}\; q^\{n+1\}$

If neighboring spins are only allowed in certain specific configurations, then the state space is given by a

subshift of finite type . The partition function may then be written as:$Z\_n(0)\; =\; e^\{-eta\}\; mbox\{card\}\; ,mbox\{Fix\},\; au^n\; =\; e^\{-eta\}\; mbox\{Tr\}\; A^n$

where card is the

cardinality or count of a set, and Fix is the set offixed point s of the iterated shift function::$mbox\{Fix\},\; au^n\; =\; \{\; s\; in\; Q^mathbb\{Z\}\; :\; au^n\; s\; =\; s\; \}$

The $q\; imes\; q$ matrix $A$ is the

adjacency matrix specifying which neighboring spin values are allowed.**Interacting model**The simplest case of the interacting model is the

Ising model , where the spin can only take on one of two values, $s\_n\; in\; \{-1,+1\}$ and only nearest neighbor spins interact. The interaction potential is given by:$V(sigma)\; =\; -J\_p\; s\_0\; s\_1,$

This potential can be captured in a $2\; imes\; 2$ matrix with matrix elements

:$M\_\{sigma\; sigma\text{'}\}\; =\; exp\; left(\; eta\; J\_p\; sigma\; sigma\text{'}\; ight)$

with the index $sigma,sigma\text{'}\; in\; \{-1,+1\}$. The partition function is then given by

:$Z\_n(V)\; =\; mbox\{Tr\},\; M^n$

The general solution for an arbitrary number of spins, and an arbitrary finite-range interaction, is given by the same general form. In this case, the precise expression for the matrix "M" is a bit more complex.

The goal of solving a model such as the Potts model is to give the an exact

closed-form expression for the partition function (which we've done) and an expression for theGibbs state s orequilibrium state s in the limit of $n\; oinfty$, thethermodynamic limit .**References*** Julius Ashkin, Edward Teller (1943); "Statistics of Two-Dimensional Lattices With Four Components", Physical Review, 64, pp. 178–184

* Renfrey B. Potts, (1952); "Some Generalized Order-Disorder Transformations", Proceedings of the Cambridge Philosophical Society, Vol. 48, pp. 106−109

* " [*http://biocomplexity.indiana.edu/jglazier/papers.php?action=browse&cat=03a François Graner and James A. Glazier (1992); "Simulation of Biological Cell Sorting Using a Two-Dimensional Extended Potts Model", Physical Review Letters 69, 2013-2016*] "

* [*http://link.aps.org/abstract/RMP/v54/p235 Fred Y. Wu (1982); "The Potts model", Reviews of Modern Physics, Vo. 54, pp. 235–268*]

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