File:Ising-tartan.png

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Description
English: This tartan-like graph shows the Ising model probability density P(\sigma) for the two-sided lattice using the dyadic mapping.

That is, a lattice configuration of length 2N

\sigma=(\sigma_{-N-1},\cdots,\sigma_{-2},\sigma_{-1},\sigma_{0},\sigma_{1},\sigma_{2},\cdots,\sigma_{N})

is understood to consist of a sequence of "spins" \sigma_k=\pm 1. This sequence may be represented by two real numbers 0\le x,y\le 1 with

x(\sigma)=\sum_{k=0}^{N}\left(\frac{\sigma_{k}+1}{2}\right)2^{-(k+1)}

and

y(\sigma)=\sum_{k=0}^{N}\left(\frac{\sigma_{-k-1}+1}{2}\right)2^{-(k+1)}

The energy of a given configuration \sigma is computed using the classical Hamiltonian,

H(\sigma)=\sum_{k=-N}^N V(\tau^k \sigma)

Here, \tau is the shift operator, acting on the lattice by shifting all spins over by one position:

\tau(\cdots,\sigma_{0},\sigma_{1},\sigma_{2},\cdots) = (\cdots,\sigma_{-1},\sigma_{0},\sigma_{1},\cdots)

The interaction potential V is given by the Ising model interaction

V(\sigma)=J\sigma_0\sigma_1 + B\sigma_0

Here, the constant J is the interaction strength between two neighboring spins \sigma_0 and \sigma_1, while the constant B may be interpreted as the strength of the interaction between the magnetic field and the magnetic moment of the spin.

The set of all possible configurations \Omega=\{\sigma\} form a canonical ensemble, with each different configuration occurring with a probability P(\sigma) given by the Boltzmann distribution

P(\sigma)=\frac{1}{Z(T)} e^{-H(\sigma)/k_B T}

where k_B is Boltzmann's constant, T is the temperature, and Z(T) is the partition function. The partition function is defined to be such that the sum over all probabilities adds up to one; that is, so that

Z(T)=\sum_{\sigma\in\Omega} e^{-H(\sigma)/k_B T}

Image details

The image here shows P(\sigma)=P(x,y) for the Ising model, with J=0.3, B=0 and temperature T=1/k_B. The lattice is finite sized, with N=10, so that all 1024\times 1024=2^N \times 2^N lattice configurations are represented, each configuration denoted by one pixel. The color choices here are such that black represents values where P(\sigma)=P(x,y) are zero, blue are small values, with yellow and red being progressively larger values.

As an invariant measure

This fractal tartan is invariant under the Baker's map. The shift operator \tau on the lattice has an action on the unit square with the following representation:

\tau(x,y)=\left(\frac{x+\left\lfloor 2y\right\rfloor }{2}\,,\,2y-\left\lfloor 2y\right\rfloor \right)

This map (up to a reflection/rotation around the 45-degree axis) is essentially the Baker's map or equivalently the Horseshoe map. As the article on the Horseshoe map explains, the invariant sets have such a tartan pattern (an appropriately deformed Sierpinski carpet). In this case, the invariance arises from the translation invariance of the Gibbs states of the Ising model: that is, the energy H(\sigma) associated with the state \sigma is invariant under the action of \tau:

H\left(\tau^k\sigma\right) = H(\sigma)

for all integers k. Similarly, the probability density is invariant as well:

P\left(\tau^k\sigma\right) = P(\sigma)

The naive classical treatment given here suffers from conceptual difficulties in the N\to\infty limit. These problems can be remedied by using a more appropriate topology on the set of states that make up the configuration space. This topology is the cylinder set topology, and using it allows one to construct a sigma algebra and thus a measure on the set of states. With this topology, the probability density can be understood to be a translation-invariant measure on the topology. Indeed, there is a certain sense in which the seemingly fractal patterns generated by the iterated Baker's map or horseshoe map can be understood with a conventional and well-behaved topology on a lattice model.

Created by Linas Vepstas User:Linas on 24 September 2006
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  • 2006-09-24 16:14 Linas 1024×1024× (5013 bytes) Created by Linas Vepstas [[User:Linas]] on 24 September 2006

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