In this paper, one of the applications of the Kolmogorov forward equations in solving the equations in the Ising model will be analyzed. Then, the limiting distribution and the stationary distribution of the corresponding birth and death process will be calculated. The Ising model is one of the famous physical models that is used in other sciences. In this paper, the three special cases of the Ising model, i.e., the square n × n grid, horizontal, and circular systems, will be analyzed and examined. We will show that, in general, Ising models based on the Boltzmann stationary distribution, the maximum likelihood estimator, and a method of moments strategy to estimate the reverse temperature characteristic of the heat baths are the same. The equation related to finding the maximum likelihood estimator and a method of moments strategy cannot be solved analytically, so these estimates will be calculated using the technique of fitting a linear regression model. By the way, it will be observed that the entropy values for the n×n grid system are smaller than the corresponding values in the horizontal and circular systems. Then, a general case for the convexity of the entropy of the Boltzmann probability function will be introduced. Also, considering an Ising model on two and three points, where the points take values independently and follow a Markov chain, the stationary distribution as well as the Boltzmann probability function, its induced probability function, and maximum likelihood estimator for the parameter will be calculated. Finally, we will review the Metropolis-Hastings algorithm and Gibbs sampling to simulate the one-dimensional and two-dimensional Ising model and also the Potts model in the R software. Finally, we will compare the n × n grid, horizontal, and circular systems. The induced probability vectors by system energies will be found for the three systems. The entropy of these three induced probability vectors and the entropy of Boltzmann probability and their two-by-two Kullback-Leibler divergences will be plotted and compared as a function of the parameter.
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Shams, M. and Mirzaie, M. A. (2026). The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model. Computational Methods for Differential Equations, 14(1), 421-469. doi: 10.22034/cmde.2025.63161.2814
MLA
Shams, M. , and Mirzaie, M. A. . "The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model", Computational Methods for Differential Equations, 14, 1, 2026, 421-469. doi: 10.22034/cmde.2025.63161.2814
HARVARD
Shams, M., Mirzaie, M. A. (2026). 'The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model', Computational Methods for Differential Equations, 14(1), pp. 421-469. doi: 10.22034/cmde.2025.63161.2814
CHICAGO
M. Shams and M. A. Mirzaie, "The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model," Computational Methods for Differential Equations, 14 1 (2026): 421-469, doi: 10.22034/cmde.2025.63161.2814
VANCOUVER
Shams, M., Mirzaie, M. A. The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model. Computational Methods for Differential Equations, 2026; 14(1): 421-469. doi: 10.22034/cmde.2025.63161.2814
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