Dependence of Critical Parameters of 2D Ising Model on Lattice Size

For the 2D Ising model, we analyzed dependences of thermodynamic characteristics on number of spins by means of computer simulations. We compared experimental data obtained using the Fisher-Kasteleyn algorithm on a square lattice with N = l × l spins and the asymptotic Onsager solution (N → ∞). We derived empirical expressions for critical parameters as functions of N and generalized the Onsager solution on the case of a finite-size lattice. Our analytical expressions for the free energy and its derivatives (the internal energy, the energy dispersion and the heat capacity) describe accurately the results of computer simulations. We showed that when N increased the heat capacity in the critical point increased as lnN. We specified restrictions on the accuracy of the critical temperature due to finite size of our system. Also in the finite-dimensional case, we obtained expressions describing temperature dependences of the magnetization and the correlation length. They are in a good qualitative agreement with the results of computer simulations by means of the dynamic Metropolis Monte Carlo method. © 2018, Allerton Press, Inc

Dependence of critical temperature on dispersion of connections in 2d grid

The calculation of probabilities in many problems of computer vision and machine learning is reduced to the finding of a normalizing constant (partition function). In the paper we evaluate the normalizing constant for a two-dimensional nearest-neighboring Ising model with almost constant average interaction between neighbors with a little noise. The two-dimensional Ising model is a perfect object for investigation. Firstly, a plane grid can be regarded as a set of image pixels. Secondly, the statistical physics offers an exact analytical solution obtained by Onsager for identical grid elements. We carry out numerical experiments to compute the normalizing constant for the case in which the noise in grid elements grows smoothly, analyze the results and compare them with Onsager’s solution. © Springer International Publishing AG, part of Springer Nature 2018

Spectral Characteristics of a Finite 2D Ising Model

Abstract: The paper gives the results of a numerical simulation of a two-dimensional Ising model built on finite lattices of dimension L = 50, 100, …, 500. Approximate analytical formulae for the spectral energy density are offered. Derived from Onsager’s solution with consideration of the finite size of the system, the formulae agree well with the simulation results. © 2018, Allerton Press, Inc