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| In [[statistical mechanics]], the '''Potts model''', a generalization of the [[Ising model]], is a model of interacting [[spin (physics)|spins]] on a [[crystalline lattice]]. By studying the Potts model, one may gain insight into the behaviour of [[ferromagnet]]s 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.
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| The model is named after [[Renfrey Potts]], 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]]. The four-state planar Potts model is sometimes known as the '''Ashkin–Teller model''', after [[Julius Ashkin]] and [[Edward Teller]], who considered an equivalent model in 1943.
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| The Potts model is related to, and generalized by, several other models, including the [[XY model]], the [[Heisenberg model (classical)|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 group|non-Abelian]] manner, the model is related to the [[flux tube model]], which is used to discuss confinement in [[quantum chromodynamics]]. Generalizations of the Potts model have also been used to model [[grain growth]] in metals and [[coarsening]] in [[foam]]s. A further generalization of these methods by [[James Glazier]] and [[Francois Graner]], known as the [[cellular Potts model]], has been used to simulate static and kinetic phenomena in foam and biological [[morphogenesis]].
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| ==Physical description==
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| The Potts model consists of ''spins'' that are placed on a [[lattice (group)|lattice]]; the lattice is usually taken to be a two-dimensional rectangular [[Euclidean space|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 the [[circle]], at angles
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| :<math>\theta_n = \frac{2\pi n}{q},</math>
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| where ''n'' = 1, ..., ''q'' and that the interaction [[Hamiltonian mechanics|Hamiltonian]] be given by
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| :<math>H_c = J_c\sum_{(i,j)} \cos \left( \theta_{s_i} - \theta_{s_j} \right)</math> | |
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| with the sum running over the nearest neighbor pairs (''i'', ''j'') over all lattice sites. The site ''colors'' ''s<sub>i</sub>'' take on values in {1, ..., ''q''}. Here, ''J<sub>c</sub>'' 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'' → ∞, this becomes the [[XY model]].
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| 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:
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| :<math>H_p = -J_p \sum_{(i,j)}\delta(s_i,s_j) \,</math>
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| where δ(''s<sub>i</sub>'', ''s<sub>j</sub>'') is the [[Kronecker delta]], which equals one whenever ''s<sub>i</sub>'' = ''s<sub>j</sub>'' and zero otherwise.
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| The ''q''=2 standard Potts model is equivalent to the [[Ising model]] and the 2-state vector Potts model, with ''J<sub>p</sub>'' = −2''J<sub>c</sub>''. The ''q'' = 3 standard Potts model is equivalent to the three-state vector Potts model, with ''J<sub>p</sub>'' = −(3/2)''J<sub>c</sub>''.
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| 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:
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| :<math>\beta H_g = - \beta \sum_{(i,j)}J_{ij} \delta(s_i,s_j) - \sum_i h_i s_i \,</math>
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| 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.
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| Different papers may adopt slightly different conventions, which can alter ''H'' and the associated [[Partition function (statistical mechanics)|partition function]] by additive or multiplicative constants.
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| ==Discussion==
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| 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 ''q'' ≤ 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.
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| The model has a close relation to the Fortuin-[[Pieter Kasteleyn|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''.
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| For integer values of ''q'', ''q'' ≥ 3, the model displays the phenomenon of 'interfacial adsorption' with intriguing critical [[wetting]] properties when fixing opposite boundaries in two different states.
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| ==Measure theoretic description==
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| 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 of [[transfer operator]]s. (However, [[Ernst Ising]] used combinatorial methods to solve the [[Ising model]], which is the "ancestor" of the Potts model, in his 1924 PhD thesis). This section develops the mathematical formalism, based on [[measure theory]], behind this solution.
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| 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 (classical)|Heisenberg model]] and the [[N-vector model]].
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| ===Topology of the space of states===
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| Let ''Q'' = {1, ..., ''q''} be a finite set of symbols, and let
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| :<math>Q^\mathbf{Z}=\{ s=(\ldots,s_{-1},s_0,s_1,\ldots) : s_k \in Q \; \forall k \in \mathbf{Z} \}</math>
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| 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, a [[subshift of finite type]], may be used. Shifts get this name because there exists a natural operator on this space, the [[shift operator]] τ : ''Q''<sup>'''Z'''</sup> → ''Q''<sup>'''Z'''</sup>, acting as
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| :<math>\tau (s_k) = s_{k+1}</math>
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| This set has a natural [[product topology]]; the [[base (topology)|base]] for this topology are the [[cylinder set]]s
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| :<math>C_m[\xi_0, \ldots, \xi_k]= \{s \in Q^\mathbf{Z} : s_m = \xi_0, \ldots ,s_{m+k} = \xi_k \}</math>
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| that is, the set of all possible strings where ''k''+1 spins match up exactly to a given, specific set of values ξ<sub>0</sub>, ..., ξ<sub>''k''</sub>. Explicit representations for the cylinder sets can be gotten by noting that the string of values corresponds to a [[p-adic number|''q''-adic number]], and thus, intuitively, the product topology resembles that of the [[real number]] line.
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| ===Interaction energy===
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| The interaction between the spins is then given by a [[continuous function (topology)|continuous function]] ''V'' : ''Q''<sup>'''Z'''</sup> → '''R''' on this topology. ''Any'' continuous function will do; for example
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| :<math>V(s) = -J\delta(s_0,s_1)</math>
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| will be seen to describe the interaction between nearest neighbors. Of course, different functions give different interactions; so a function of ''s''<sub>0</sub>, ''s''<sub>1</sub> and ''s''<sub>2</sub> 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 ''s'' ∈ ''Q''<sup>'''Z'''</sup>, 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''<sub>0</sub> and ''s''<sub>1</sub>. 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.
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| Define the function ''H<sub>n</sub>'' : ''Q''<sup>'''Z'''</sup> → '''R''' as
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| :<math>H_n(s)= \sum_{k=0}^n V(\tau^k s)</math>
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| This function can be seen to consist of two parts: the self-energy of a configuration [''s''<sub>0</sub>, ''s''<sub>1</sub>, ..., ''s<sub>n</sub>''] of spins, plus the interaction energy of this set and all the other spins in the lattice. The ''n'' → ∞ limit of this function is the Hamiltonian of the system; for finite ''n'', these are sometimes called the '''finite state Hamiltonians'''.
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| ===Partition function and measure===
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| The corresponding finite-state [[partition function (statistical mechanics)|partition function]] is given by
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| :<math>Z_n(V) = \sum_{s_0,\ldots,s_n \in Q} \exp(-\beta H_n(C_0[s_0,s_1,\ldots,s_n]))</math>
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| with ''C''<sub>0</sub> 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 (mathematics)|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
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| :<math>\mu (C_k[s_0,s_1,\ldots,s_n]) = \frac{1}{Z_n(V)} \exp(-\beta H_n (C_k[s_0,s_1,\ldots,s_n]))</math>
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| 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 the [[configuration space]] ''Q''<sup>'''Z'''</sup>. By endowing the configuration space with a probability measure built from a Hamiltonian in this way, the configuration space turns into a [[canonical ensemble]].
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| Most thermodynamic properties can be expressed directly in terms of the partition function. Thus, for example, the [[Helmholtz free energy]] is given by
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| :<math>A_n(V)=-kT \log Z_n(V)</math>
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| Another important related quantity is the [[topological pressure]], defined as
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| :<math>P(V) = \lim_{n\to\infty} \frac{1}{n} \log Z_n(V)</math>
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| which will show up as the logarithm of the leading eigenvalue of the [[transfer operator]] of the solution.
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| ===Free field solution===
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| The simplest model is the model where there is no interaction at all, and so ''V'' = ''c'' and ''H<sub>n</sub>'' = ''c'' (with ''c'' constant and independent of any spin configuration). The partition function becomes
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| :<math>Z_n(c) = e^{-c\beta} \sum_{s_0,\ldots,s_n \in Q} 1</math>
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| 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
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| :<math>Z_n(c) = e^{-c\beta} q^{n+1}</math>
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| 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
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| :<math>Z_n(c) = e^{-c\beta} |\mbox{Fix}\, \tau^n| = e^{-c\beta} \mbox{Tr} A^n</math>
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| where card is the [[cardinality]] or count of a set, and Fix is the set of [[Fixed point (mathematics)|fixed points]] of the iterated shift function:
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| :<math>\mbox{Fix}\, \tau^n = \{ s \in Q^\mathbf{Z} : \tau^n s = s \}</math>
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| The ''q'' × ''q'' matrix ''A'' is the [[adjacency matrix]] specifying which neighboring spin values are allowed.
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| ===Interacting model===
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| The simplest case of the interacting model is the [[Ising model]], where the spin can only take on one of two values, ''s<sub>n</sub>'' ∈ {−1, 1} and only nearest neighbor spins interact. The interaction potential is given by
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| :<math>V(\sigma) = -J_p s_0 s_1\,</math>
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| This potential can be captured in a 2 × 2 matrix with matrix elements
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| :<math>M_{\sigma \sigma'} = \exp \left( \beta J_p \sigma \sigma' \right)</math>
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| with the index σ, σ′ ∈ {−1, 1}. The partition function is then given by
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| :<math>Z_n(V) = \mbox{Tr}\, M^n</math>
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| 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.
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| The goal of solving a model such as the Potts model is to give an exact [[closed-form expression]] for the partition function (which we've done) and an expression for the [[Gibbs state]]s or [[equilibrium state]]s in the limit of ''n'' → ∞, the [[thermodynamic limit]].
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| == The Potts model in signal and image processing ==
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| The Potts model has applications in signal reconstruction. Assume that we are given noisy observation of a piecewise constant signal ''g'' in '''R'''<sup>''n''</sup>. To recover ''g'' from the noisy observation vector ''f'' in '''R'''<sup>''n''</sup>, on seeks a minimizer of the corresponding inverse problem, the ''L<sup>p</sup>''-Potts functional ''P''<sub>γ</sub>(''u'') which is defined by
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| :<math> P_\gamma(u) = \gamma \| \nabla u \|_0 + \| u-f\|_p^p = \gamma \# \{ i : u_i \neq u_{i+1} \} + \sum_{i=1}^n |u_i - f_i|^p</math>
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| The jump penalty <math>\| \nabla u \|_0</math> forces piecewise constant solutions and the data term <math>\| u-f\|_p^p</math> couples the minimizing candidate ''u'' to the data ''f''. The parameter γ > 0 controls the tradeoff between regularity and data fidelity. There are fast algorithms for the exact minimization of the ''L''<sup>1</sup> and the ''L''<sup>2</sup>-Potts functional (Friedrich, Kempe, Liebscher, Winkler, 2008).
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| In image processing, the Potts functional is related to the segmentation problem. However, in two dimensions the problem is NP-hard (Boykov, Veksler, Zabih, 2001).
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| ==References==
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| *{{cite journal |first=Julius |last=Ashkin |first2=Edward |last2=Teller |year=1943 |title=Statistics of Two-Dimensional Lattices With Four Components |journal=[[Physical Review|Phys. Rev.]] |volume=64 |issue=5–6 |pages=178–184 |doi=10.1103/PhysRev.64.178 |bibcode = 1943PhRv...64..178A }}
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| *{{cite journal |first=François |last=Graner |first2=James A. |last2=Glazier |year=1992 |title=Simulation of Biological Cell Sorting Using a Two-Dimensional Extended Potts Model |journal=[[Physical Review Letters|Phys. Rev. Lett.]] |volume=69 |issue=13 |pages=2013–2016 |doi=10.1103/PhysRevLett.69.2013 |bibcode = 1992PhRvL..69.2013G }}
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| *{{cite journal |first=Renfrey B. |last=Potts |year=1952 |title=Some Generalized Order-Disorder Transformations |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society|Mathematical Proceedings]] |volume=48 |issue=1 |pages=106–109 |doi=10.1017/S0305004100027419 |bibcode = 1952PCPS...48..106P }}
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| *{{cite journal |first=Fa-Yueh |last=Wu |year=1982 |title=The Potts model |journal=[[Reviews of Modern Physics|Rev. Mod. Phys.]] |volume=54 |issue=1 |pages=235–268 |doi=10.1103/RevModPhys.54.235 |bibcode = 1982RvMP...54..235W }}
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| *{{cite journal |first=F. |last=Friedrich |first2=al. |last2=et. |year=2008 |title=Complexity Penalized M-Estimation |journal=Journal of Computational and Graphical Statistics |volume=64 |issue=5–6 |pages=201–224 }}
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| *{{cite journal |first=Y. |last=Boykov |first2=al. |last2=et. |year=2001 |title=Fast approximate energy minimization via graph cuts |journal=Pattern Analysis and Machine Intelligence, IEEE Transactions on |pages=1222–1239 }}
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| *{{cite journal |first=Walter |last=[[Walter Selke|Selke]] |first2=David A. |last2=[[David Alan Huse|Huse]] |year=1983 |title= Interfacial adsorption in planar Potts models|journal= Zeitschrift für Physik B |volume=50 |issue=2 |pages=113–116|doi=10.1007/BF01304093|bibcode = 1983ZPhyB..50..113S }}
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| == External links ==
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| *{{cite web |title=Code for efficiently computing Tutte, Chromatic and Flow Polynomials |first=Gary |last=Haggard |first2=David J. |last2=Pearce |first3=Gordon |last3=Royle |url=http://www.mcs.vuw.ac.nz/~djp/tutte/ }}
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| {{Stochastic processes}}
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| [[Category:Spin models]]
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| [[Category:Exactly solvable models]]
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| [[Category:Statistical mechanics]]
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| [[Category:Lattice models]]
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