Image:Ising-tartan.png

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1 Overview
2 Image details
3 As an invariant measure
4 Licensing

[edit] Overview

This tartan-like graph shows the Ising model probability density $P (σ)$ for the two-sided lattice using the dyadic mapping.

That is, a lattice configuration of length $2 N$

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

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 $σ$ is computed using the classical Hamiltonian,

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

Here, $τ$ 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 (σ) = J σ 0 σ 1 + B σ 0

Here, the constant $J$ is the interaction strength between two neighboring spins $σ 0$ and $σ 1$ , 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 $Ω = {σ}$ form a canonical ensemble, with each different configuration occurring with a probability $P (σ)$ 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

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

[edit] Image details

The image here shows $P (σ) = P (x, y)$ 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 $P (σ) = P (x, y)$ are zero, blue are small values, with yellow and red being progressively larger values.

[edit] As an invariant measure

This fractal tartan is invariant under the Baker's map. The shift operator $τ$ 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 $H (σ)$ associated with the state $σ$ is invariant under the action of $τ$ :

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

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.

[edit] Licensing

Created by Linas Vepstas User:Linas on 24 September 2006

I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Subject to disclaimers.

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	Date/Time	Dimensions	User	Comment
current	16:14, 24 September 2006	1,024×1,024 (5 KB)	Linas (Talk \| contribs)	(Created by Linas Vepstas User:Linas on 24 September 2006)