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Thread: Hammerslay-Clifford theorem and maximal entropy random walk for Ising-like models?

  1. #1 Hammerslay-Clifford theorem and maximal entropy random walk for Ising-like models? 
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    Jul 2008
    It seems that condensed matter people usually just brute force use Monte-Carlo, but there are some subtle mathematical tools which might be worth considering, for example here is fresh paper with Mathematica implementation calculating in seconds parameters for Ising-like models with many digits of accuracy, also probability distribution of patterns or allowing to generate new uncorrelated field with single scan.

    1) Hammersley-Clifford theorem ( ) saying that Gibbs fields are equivalent with Markov fields, which allow to simplify models, e.g. through local Markov condition:
    Pr(value in node | values in remaining nodes) = Pr(value in node | values in its neighbors)

    2) Maximal entropy random walk ( ) provides probability distribution of patterns for Boltzmann ensemble of infinite sequences. Applying it to transition matrix (M_uv = exp(-beta E_uv)), while there are usually used its eigenvalues, here from its dominant eigenvector we get probability distribution of single or two neighboring values e.g. patterns in 2D Ising model:

    Pr(u) = (psi_u)^2 as in Born rule
    Pr(uv) = psi_u (M_uv / lambda) psi_v

    Is there a literature applying any of them for Ising-like models?
    Beside Monte-Carlo and molecular dynamics simulations, what interesting mathematical tools are used in modern condensed matter physics?

    Diagram from the paper with concepts and plots of errors:

    Last edited by Jarek Duda; January 5th, 2020 at 04:56 AM.
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