Potts model
The Potts model is a mathematical model and can be considered as a generalization of the Ising model used in statistical physics. If a system consists of a countable set of entities (eg atoms), which are all different states (eg spins) can accept and between which there is an interaction, the Potts model can be used to analyze the system. The basics are from graph theory and probability theory. It is applied, among others, in physics (statistical mechanics), computer science ( signal processing ) and biology ( neural networks ). The model was named after Renfrey Potts, which defined the model in 1951 in his dissertation.
- 3.1 Energy of a node configuration
- 3.2 partition
- 3.3 Potts measure
- 3.4 Free Energy
Definition
The Potts model consists of a d- dimensional grid graphs, a set of nodes assignments (node configurations ) and a Hamiltonian on this set. Each node is a member of the set
Occupied. This can be interpreted as points on the 2-dimensional unit circle and hot spins. The Hamiltonian is given by
Is summed over all neighboring nodes for. The coupling constant describes the interaction between the neighboring nodes. As an alternative to the planar Potts model just described, the equivalent standard Potts model ( or simply: Potts model) is generally used. The nodes are occupied by elements of the set. The Hamiltonian is given by
Where the Kronecker delta is. That is, if two adjacent nodes have different spins the corresponding addend disappears. The negative sign is a convention of, motivated by the Ising model.
Relationship to other statistical models
Public version
On the grid graph with the set of nodes assignments a more general version of the Potts model can be defined:
In contrast to the original model, the interaction between the neighboring nodes varied. In addition, the operator is given by the term
Supplemented. It can be understood as an external field, which acts on the original system.
The Ising model as a special case
Substituting follows from the Potts model, the Ising model.
The XY model as a special case
For obtaining the XY - model, which in turn can be understood as a special case of the n-vector model with. Considering the planar Potts model, the state space of the spins is not a finite subset of the unit circle, but the entire 2-dimensional unit circle.
Properties
Energy of a node configuration
Partition function
The partition function is one of the most important variables of the Potts model; they normalized the Potts - level and serves as a basis for the calculation of the free energy:
Is summed over all nodes configurations. Since the cardinality of all node configurations - depending on the number of nodes - is growing exponentially, there are approximation algorithms that simulate the partition function. It is usually around Monte Carlo method.
Potts measure
The Potts measure is a probability measure and belongs to the class of Boltzmann distributions: