In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. 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. Jump to: navigation, search Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... The Ising model, named after the physicist Ernst Ising, is a model in statistical mechanics. ... In physics, spin is an intrinsic angular momentum associated with microscopic particles. ... Rose des Sables (Sand Rose), formed of gypsum crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... A ferromagnet is a piece of ferromagnetic material, in which the microscopic magnetized regions, called domains, have been aligned by an external magnetic field (e. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ...

The model is named after R. B. Potts who described the model near the end of his 1952 PhD thesis. The model was related to the "planar Potts" or "clock model", which was suggested to him by his PhD advisor C. Domb. The Potts model is sometimes known as the Ashkin-Teller model, as they considered a four component version in 1943. Jump to: navigation, search 1952 was a leap year starting on Tuesday (link will take you to calendar). ... Jump to: navigation, search 1943 is a common year starting on Friday. ...

The Potts model is related to, and generalized by, several other models, including the XY model, the Heisenberg model and the N-vector model. The infinite-range Potts model is known as the Kac model. When the spins are taken to interact in a non-Abelian manner, the model is related to the flux tube model, which is used to discuss confinement in quantum chromodynamics. Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ... The Heisenberg model is the case of the n-vector model, one of the models used in Statistical Physics in order to model ferromagnetism and other phenomena. ... The n-vector model or O(n) model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ... Jump to: navigation, search Quantum chromodynamics (QCD) is the theory describing one of the fundamental forces, the strong interaction. ...

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 one of q possible values, distributed uniformly about the circle, at angles See lattice for other meanings of this term, both within and without mathematics. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... Jump to: navigation, search In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ...

and that the interaction Hamiltonian be given by Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

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 . 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 the XY model. Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ...

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:

where δ(s_i,s_j) is the Kronecker delta, which equals one whenever s_i = s_j and zero otherwise. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 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. The Ising model, named after the physicist Ernst Ising, is a model in statistical mechanics. ...

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:

where β = 1 / kT the inverse temperature, k the Boltzmann constant and T the temperature. The summation may run over more distant neighbors on the lattice, or may in fact be an infinite-range force. Jump to: navigation, search The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ...

Different papers may adopt slightly different conventions, which can alter H and the associated partition function by additive or multiplicative constants. In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

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 transitions. For example 2D dimensional lattices with J > 0 exhibit a first order transition if q > 4. When a continuous transition is observed, as in the Ising model where q = 2. Further use is found through the models relation to percolation problems and the Tutte and chromatic polynomials found in combinatorics. Jump to: navigation, search In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ...

The 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 q. Markov chain Monte Carlo (MCMC) methods, sometimes called random walk Monte Carlo methods, are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. ...

Measure theoretic description

The one dimensional Potts model mey 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 of transfer operators. Indeed, Ernst Ising used transfer operator methods to solve the closely related Ising model in his 1925 PhD thesis. This section develops the mathematical formalism, based on measure theory, behind this solution. In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ... In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. ... Ernst Ising (born May 10, 1900, Cologne Germany – May 11, 1998, Peoria, Illinois) was a German physicist, who is best remembered for the development of the Ising model of ferromagnetism. ... The Ising model, named after the physicist Ernst Ising, is a model in statistical mechanics. ... Jump to: navigation, search 1925 was a common year starting on Thursday (link will take you to calendar). ... In mathematics, a measure is a function that assigns a number, e. ...

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, the Heisenberg model and the N-vector model. Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ... The Heisenberg model is the case of the n-vector model, one of the models used in Statistical Physics in order to model ferromagnetism and other phenomena. ... The n-vector model or O(n) model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ...

Topology of the space of states

Let be a finite set of symbols, and let

be the set of all bi-infinite strings of values from the set Q. This set is called a full shift. For the defining the Potts model, either this whole space, or a certain subset of it, a subshift of finite type, may be used. Shifts get this name because there exists a natural operator on this space, the shift operator , acting as In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ... In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ... In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...

τ(s_k) = s_{k + 1}

This set has a natural product topology; the base for this topology are the cylinder sets In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ...

that is, the set of all possible strings where k+1 spins match up exactly to a given, specific set of values . 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. The p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

Interaction energy

The interaction between the spins is then given by a continuous function on this topology. Any continuous function will do; for example In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ...

V(s) = − Jδ(s₀,s₁)

will be seen to describe the interaction between nearest neighbors. Of course, different functions give different interactions; so a function of s₀, s₁ and s₂ 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 , 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 s₀ and s₁. 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 as

This function can be seen to consist of two parts: the self-energy of a configuration of spins, plus the interaction energy of this set and all the other spins in the lattice. The 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 In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

with C₀ being the cylinder sets defined above. Here, β=1/kT, where k is Boltzmann's constant, and T is the temperature. 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 topology. The measure of a cylinder set is given by The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... In mathematics, a measure is a function that assigns a number, e. ...

This measure is a probability measure; it gives the likelyhood of a given configuration occuring in the configuration space . By endowing the configuration space with a probability measure built from a Hamiltonian in this way, the configuration space turns into a canonical ensemble. In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ...

Most thermodyanmic properties can be expressed directly in terms of the partition function. Thus, for example, the Helmholtz free energy is given by This page develops the Helmholtz free energy from the point of view of thermal and statistical physics. ...

A_n(V) = − kTlogZ_n(V)

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. In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. ...

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

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 In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ...

Z_n(0) = e ^{− β}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 In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ...

where card is the cardinality or count of a set, and Fix is the set of fixed points of the iterated shift function: In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ...

The matrix A is the adjacency matrix specifying which neighboring spin values are allowed. In mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry aij is the number of edges from vi to vj in . ...

Interacting model

The simplest case of the interacting model is the Ising model, where there the spin can only take on one of two values, and only nearest neighbor spins interact. The interaction potential is given by The Ising model, named after the physicist Ernst Ising, is a model in statistical mechanics. ...

This potential can be captured in a matrix with matrix elements

with the index . The partition function is then given by

The general solution for an arbitrary number of spins, and an arbitary 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.

To do

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 the Gibbs states or equilibrium states in the limit of , the thermodynamic limit. Jump to: navigation, search A Gibbs state in probability theory and statistical mechanics is an equilibrium probability distribution which remains invariant under future evolution of the system (for example, a stationary or steady-state distribution of a Markov chain, such as that achieved by running a Markov Chain Monte Carlo... (LTE is an acronym for the progressive-instrumental rock band  Liquid Tension Experiment) In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann distribution. ... In physics, the thermodynamic limit is the statistical mechanical limit described by a system in which the number of particles approaches infinity. ...

ToDo: (article under development):

• Show graph of energy states of the Ising model.
• Show the general form of the solution for a finite-range interaction.
• Show that the infinite-range force (Kac model) is the trace of a transfer operator.
• Show that the largest eigenvalue of the transfer operator, per the Ruelle-Perron-Frobenius theorem, is the state giving the thermodynamic equilibrium of the system.

In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. ... In mathematics, the Perron-Frobenius theorem is a theorem in matrix theory about the eigenvalues and eigenvectors of a real positive n×n matrix: Let A = (aij) be a real n×n matrix with positive entries . ...

References

• R. Potts, (1952) Proceedings of the Cambridge Philosophical Society 48, pp106.
• Peter Meyer, Computational Studies of Pure and Dilute Spin Models, (2000) (Given as thesis for Masters of Philosophy). (See, in particular, Appendix A: Calculation of the Metropolis and the Glauber Transition Probabilities for the Ising Model and for the q-state Potts Model).