Visualization of the translation-invariant probability measure of the one-dimensional Ising model
English: This tartan-like graph shows the Ising model probability density P ( σ ) {\displaystyle P(\sigma )} for the two-sided lattice using the dyadic mapping. That is, a lattice configuration of length 2 N {\displaystyle 2N} σ = ( σ − N − 1 , ⋯ , σ − 2 , σ − 1 , σ 0 , σ 1 , σ 2 , ⋯ , σ N ) {\displaystyle \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" σ k = ± 1 {\displaystyle \sigma _{k}=\pm 1} . This sequence may be represented by two real numbers 0 ≤ x , y ≤ 1 {\displaystyle 0\leq x,y\leq 1} with x ( σ ) = ∑ k = 0 N ( σ k + 1 2 ) 2 − ( k + 1 ) {\displaystyle x(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{k}+1}{2}}\right)2^{-(k+1)}} and y ( σ ) = ∑ k = 0 N ( σ − k − 1 + 1 2 ) 2 − ( k + 1 ) {\displaystyle y(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{-k-1}+1}{2}}\right)2^{-(k+1)}} The energy of a given configuration σ {\displaystyle \sigma } is computed using the classical Hamiltonian, H ( σ ) = ∑ k = − N N V ( τ k σ ) {\displaystyle H(\sigma )=\sum _{k=-N}^{N}V(\tau ^{k}\sigma )} Here, τ {\displaystyle \tau } is the shift operator, acting on the lattice by shifting all spins over by one position: τ ( ⋯ , σ 0 , σ 1 , σ 2 , ⋯ ) = ( ⋯ , σ − 1 , σ 0 , σ 1 , ⋯ ) {\displaystyle \tau (\cdots ,\sigma _{0},\sigma _{1},\sigma _{2},\cdots )=(\cdots ,\sigma _{-1},\sigma _{0},\sigma _{1},\cdots )} The interaction potential V {\displaystyle V} is given by the Ising model interaction V ( σ ) = J σ 0 σ 1 + B σ 0 {\displaystyle V(\sigma )=J\sigma _{0}\sigma _{1}+B\sigma _{0}} Here, the constant J {\displaystyle J} is the interaction strength between two neighboring spins σ 0 {\displaystyle \sigma _{0}} and σ 1 {\displaystyle \sigma _{1}} , while the constant B {\displaystyle 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 } {\displaystyle \Omega =\{\sigma \}} form a canonical ensemble, with each different configuration occurring with a probability P ( σ ) {\displaystyle P(\sigma )} given by the Boltzmann distribution P ( σ ) = 1 Z ( T ) e − H ( σ ) / k B T {\displaystyle P(\sigma )={\frac {1}{Z(T)}}e^{-H(\sigma )/k_{B}T}} where k B {\displaystyle k_{B}} is Boltzmann's constant, T {\displaystyle T} is the temperature, and Z ( T ) {\displaystyle 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 ) = ∑ σ ∈ Ω e − H ( σ ) / k B T {\displaystyle Z(T)=\sum _{\sigma \in \Omega }e^{-H(\sigma )/k_{B}T}} Contents 1 Image details 2 As an invariant measure 3 Licensing 4 Original upload log Image details The image here shows P ( σ ) = P ( x , y ) {\displaystyle P(\sigma )=P(x,y)} for the Ising model, with J = 0.3 {\displaystyle J=0.3} , B = 0 {\displaystyle B=0} and temperature T = 1 / k B {\displaystyle T=1/k_{B}} . The lattice is finite sized, with N = 10 {\displaystyle N=10} , so that all 1024 × 1024 = 2 N × 2 N {\displaystyle 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 ) {\displaystyle 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 τ {\displaystyle \tau } on the lattice has an action on the unit square with the following representation: τ ( x , y ) = ( x + ⌊ 2 y ⌋ 2 , 2 y − ⌊ 2 y ⌋ ) {\displaystyle \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 ) {\displaystyle H(\sigma )} associated with the state σ {\displaystyle \sigma } is invariant under the action of τ {\displaystyle \tau } : H ( τ k σ ) = H ( σ ) {\displaystyle H\left(\tau ^{k}\sigma \right)=H(\sigma )} for all integers k {\displaystyle k} . Similarly, the probability density is invariant as well: P ( τ k σ ) = P ( σ ) {\displaystyle P\left(\tau ^{k}\sigma \right)=P(\sigma )} The naive classical treatment given here suffers from conceptual difficulties in the N → ∞ {\displaystyle 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