In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, the Potts model, a generalization of the
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
, is a model of interacting
spins
The spins (as in having "the spins")Diane Marie Leiva. ''The Florida State University College of Education''Women's Voices on College Drinking: The First-Year College Experience"/ref> is an adverse reaction of intoxication that causes a state of ...
on a
crystalline lattice
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
. By studying the Potts model, one may gain insight into the behaviour of
ferromagnet
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
s and certain other phenomena of
solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
. 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
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantity, conserved qua ...
, and that it has a rich mathematical formulation that has been studied extensively.
The model is named after
Renfrey Potts
Renfrey Burnard (Ren) Potts AO (1925–2005) was an Australian mathematician and is notable for the Potts model and his achievements in: operations research, especially networks; transportation science, car-following and road traffic; Ising-ty ...
, who described the model near the end of his 1951 Ph.D. thesis. The model was related to the "planar Potts" or "
clock model", which was suggested to him by his advisor,
Cyril Domb
Cyril Domb FRS (9 December 1920 – 15 February 2012) was a British-Israeli theoretical physicist, best known for his lecturing and writing on the theory of phase transitions and critical phenomena of fluids. He was also known in the Orthodox ...
. The four-state Potts model is sometimes known as the Ashkin–Teller model, after
Julius Ashkin
Julius Ashkin (August 23, 1920 – June 4, 1982) was a leader in experimental and theoretical physics known for furthering the evolution of particle physics from nuclear physics. As a theoretical physicist he made contributions in the fields of st ...
and
Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the Teller–Ulam design), although he did not care fo ...
, who considered an equivalent model in 1943.
The Potts model is related to, and generalized by, several other models, including the
XY model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for .
Definition
G ...
, the
Heisenberg model and the
N-vector model
In statistical mechanics, the ''n''-vector model or O(''n'') model is a simple system of interacting spins on a crystalline lattice. It was developed by H. Eugene Stanley as a generalization of the Ising model, XY model and Heisenberg model. ...
. The infinite-range Potts model is known as the
Kac model KAC or Kac may refer to:
Organizations
* Kenitra Athletic Club, a football club in Kenitra, Morocco
* Knight's Armament Company, US
* Korea Airports Corporation
* Kosciusko Alpine Club, an Australian ski club
People
* Eduardo Kac (born 1960), ...
. When the spins are taken to interact in a
non-Abelian manner, the model is related to the
flux tube model
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
, which is used to discuss
confinement
Confinement may refer to
* With respect to humans:
** An old-fashioned or archaic synonym for childbirth
** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods
** Civil confi ...
in
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
. Generalizations of the Potts model have also been used to model
grain growth
In materials science, grain growth is the increase in size of grains (crystallites) in a material at high temperature. This occurs when recovery and recrystallisation are complete and further reduction in the internal energy can only be achieved ...
in metals and
coarsening in
foam
Foams are materials formed by trapping pockets of gas in a liquid or solid.
A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the reg ...
s. A further generalization of these methods by
James Glazier
James is a common English language surname and given name:
*James (name), the typically masculine first name James
* James (surname), various people with the last name James
James or James City may also refer to:
People
* King James (disambiguati ...
and
Francois Graner, known as the
cellular Potts model
In computational biology, a Cellular Potts model (CPM, also known as the Glazier-Graner-Hogeweg model) is a computational model of cells and tissues. It is used to simulate individual and collective cell behavior, tissue morphogenesis and cancer de ...
, has been used to simulate static and kinetic phenomena in foam and biological
morphogenesis
Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of devel ...
.
Definition
The Potts model consists of ''spins'' that are placed on a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
; the lattice is usually taken to be a two-dimensional rectangular
Euclidean lattice, but is often generalized to other dimensions and lattice structures.
Originally, Domb suggested that the spin takes one of
possible values , distributed uniformly about the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, at angles
:
where
and that the interaction
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
is given by
:
with the sum running over the nearest neighbor pairs
over all lattice sites, and
is a coupling constant, determining the interaction strength. This model is now known as the vector Potts model or the clock model. Potts provided the location in two dimensions of the phase transition for
. In the limit
, this becomes the
XY model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for .
Definition
G ...
.
What is now known as the standard Potts model was suggested by Potts in the course of his study of the model above and is defined by a simpler Hamiltonian:
:
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
, which equals one whenever
and zero otherwise.
The
standard Potts model is equivalent to the
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
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
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).J. Meixner (1975) "Coldness and Tempe ...
'',
the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and
the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
.
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 transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
s. 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 transition
BKT or bkt can refer to:
* Bak kut teh, a Malaysian and Singaporean dish of pork ribs in broth
* Balkrishna Industries, an Indian tire manufacturer
* Banka Kombëtare Tregtare, a commercial bank in Albania
* Bayesian knowledge tracing, an algori ...
s,
and a continuous phase transition is observed when
.
Further use is found through the model's relation to
percolation
Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials.
It is described by Darcy's law.
Broader applicatio ...
problems and the
Tutte
William Thomas Tutte OC FRS FRSC (; 14 May 1917 – 2 May 2002) was an English and Canadian codebreaker and mathematician. During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a majo ...
and
chromatic polynomial
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to s ...
s found in combinatorics. For integer values of
, the model displays the phenomenon of 'interfacial adsorption' with intriguing critical
wetting
Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. This happens in presence of a gaseous phase or another liquid phase not miscible with th ...
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 In statistical mechanics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model, Potts model, and percolation model. It is used to study random combinatorial structures, elec ...
, another model in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. Understanding this relationship has helped develop efficient
Markov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
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