Duda Jarek 10 Posted January 3 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 ( https://en.wikipedia.org/wiki/Hammersley%E2%80%93Clifford_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 ( https://en.wikipedia.org/wiki/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 patterns e.g. in 2D Ising model: Pr(u) = (psi_u)^2 Pr(u,v) = 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: 0 Share this post Link to post Share on other sites