The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model

Shams, Mehdi; Mirzaie, Mohammad Ali

doi:10.22034/cmde.2025.63161.2814

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	The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model
Computational Methods for Differential Equations
مقاله 29، دوره 14، شماره 1، فروردین 2026، صفحه 421-469 اصل مقاله (1.99 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2025.63161.2814
نویسندگان
Mehdi Shams^* ؛ Mohammad Ali Mirzaie
Department of Mathematics, University of Kashan, Kashan, Iran.
چکیده
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.
کلیدواژه‌ها
The Kolmogorov forward equation؛ Stationary distribution؛ Limiting distribution؛ Entropy؛ Kullback-Leibler divergence

مراجع
[1] N. Abbas, K. U. Rehman, W. Shatanawi, and A. A. Al-Eid, Theoretical study of non-Newtonian micropolar nanofluid flow over an exponentially stretching surface with free stream velocity, Advances in Mechanical Engineering, 14(7) (2022), 1-9. [2] N. Abbas, W. Shatanawi, K. U. Rehman, and T. A. M. Shatnawi, Velocity and thermal slips impact on boundary layer flow of micropolar nanofluid over a vertical nonlinear stretched Riga sheet, Nanoengineering and Nanosystems, 238(3-4) (2024), 107-117. [3] E. Aurell, G. Del Ferraro, E. Domınguez and R. Mulet, Cavity master equation for the continuous time dynamics of discrete-spin models, Physical Review E, 95(5) (2017), 052119. [4] D. Beysens, Brownian motion in strongly fluctuating liquid, Thermodynamics of Interfaces and Fluid mechanics, 3 (2019), 1-8. [5] K. Binder, D. Stauffer and H. Müller-Krumbhaar, Calculation of Dynamic Critical Properties from a Cluster Reaction Theory, Physical Review B, 10(9) (1974), 3853. [6] M. J. Catanzaro, V. Y. Chernyak, and J. R. Klein, A higher Boltzmann distribution, Journal of Applied and Computational Topology, 1(2) (2017), 215-240. [7] E. Cinlar, Introduction to stochastic processes, Dover Publications, 2013. [8] D. Dhar, Stochastic evolution in Ising models, Stochastic Processes Formalism and Applications: Proceedings of the Winter School Held at the University of Hyderabad, India, Springer, (1982), 300-313. [9] R. P. Dobrow, Introduction to stochastic processes with R, John Wiley & Sons, 2016. [10] W. Ebeling and I. S. Sokolov, Statistical Thermodynamics and Stochastic: Theory of Nonequilibrium Systems, World Scientific; New Jersey, 34 (2005). [11] P. A. Flinn, Monte Carlo calculation of phase separation in a two-dimensional Ising system, Journal of Statistical Physics, 10 (1974), 89-97. [12] R. Folk, The Survival of Ernst Ising and the Struggle to Solve His Model, Chapter submitted to the book: Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory. Ed. by Yu. Holovatch, World Scientific, Singapore, 7 (2023), 1-77. [13] Z. Gao and X. Wang, The Application of Ising Model in Simulating Neural Networks, 2nd International Conference on Mechatronics, IoT and Industrial Informatics, Melbourne, Australia, (2024), 836-846. [14] H. Ge and H. Qian, Physical origins of entropy production, free energy dissipation, and their mathematical representations, Physical Review E, 81(5) (2010), 051133. [15] G. H. Gilmer, Ising Model Simulations of Crystal Growth, In: Vanselow, R., Howe, R. (eds), Chemistry and Physics of Solid Surfaces V, (1984), 297-316. [16] R. J. Glauber, Time-Dependent Statistics of the Ising Model, Journal of Mathematical Physics, 4 (1963), 294-307. [17] A. Gut, Probability: A Graduate Course 2nd edition, Springer New York, 2014. [18] G. Haag, Derivation of the Chapman–Kolmogorov Equation and the Master Equation, Modelling with the Master Equation, Springer International Publishing AG, (2017), 39-61. [19] P. G. Hoel, S. C. Port, and C. J. Stone, Introduction to Stochastic Processes, Waveland Pr Inc, 1986. [20] J. Honerkamp, Statistical Physics. An Advanced Approach with Applications, Springer, Berlin, 1998. [21] H. Hooyberghs, S. Van Lombeek, C. Giuraniuc, B. Van Schaeybroeck, and J. O. Indekeu, Ising model for distribution networks, Philosophical Magazine, 92(1-3) (2012), 168-191. [22] A. Hosseiny, M. Bahrami, A. Palestrini, and M. Gallegat, Metastable Features of Economic Networks and Responses to Exogenous Shocks, PLoS ONE, 11(e016036) (2016), 1-22. [23] M. N. Khan, S. Nadeem, N. Abbas, and A. M. Zidan, Heat and mass transfer investigation of a chemically reactive Burgers nanofluid with an induced magnetic field over an exponentially stretching surface, Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, SAGE Publications 235(6) (2021). [24] S. Kiesewetter and P. D. Drummond, Coherent Ising machine with quantum feedback: The total and conditional master equation methods, Physical Review A, 106(2) (2022), 022409. [25] S. Kullback and R. A. Leibler, On Information and Sufficiency, Annals of Mathematical Statistics, 22 (1951), 79-86. [26] Z. Lei, Irreversible Markov Chains for Particle Systems and Spin Models: Mixing and Dynamical Scaling. PhD thesis, Universite Paris sciences et lettres, (2018). [27] S. Madhira and S. Deshmukh, Introduction to Stochastic Processes Using R, Springer, 2023. [28] E.V. Murashkin and Y. N. Radayev, Generalization of the algebraic Hamilton–Cayley theory, Mechanics of Solids., 56(6) (2021), 996-1003. [29] S. Nadeem, A. Amin, and N. Abbas, On the stagnation point flow of nanomaterial with base viscoelastic micropolar fluid over a stretching surface, Alexandria Engineering Journal, 59(9) (2020), 1751-1760. [30] M. Niss, History of the Lenz-Ising model 1920-1950: from ferromagnetic to cooperative phenomena, Archive for History of Exact Sciences, 59(3) (2005), 267-318. [31] J. A. Pachter, Y. Jen. Yang, and K. A. Dill, Entropy, irreversibility and inference at the foundations of statistical physics, Nature Reviews Physics, (2024), 1-12. [32] A. Papale and A. Rosa, The Ising model in swollen vs. compact polymers: Mean-field approach and computer simulations, The European Physical Journal E, 41(12) (2018), 141. [33] R. B. Potts, Some generalized order-disorder transformations, Mathematical Proceedings of the Cambridge Philosophical Society, 48(1) (1952), 106-109. [34] S. M. Ross, Stochastic Processes 2nd edition, Wiley, 1995. [35] S. M. Ross and E. A. Peköz, A second course in probability, Cambridge University Press, 2023. [36] B. Senoglu and B. Surucu, Goodness-of-fit tests based on Kullback-Leibler information, IEEE Transactions on Reliability, 53(3) (2004), 357-361. [37] C. A. Shannon, Mathematical Theory of Communication, Bell System Technical Journal, 27 (1948), 379-423. [38] T. Shoji, K. Aihara, and Y. Yamamoto, Quantum model for coherent Ising machines: Stochastic differential equations with replicator dynamics, Physical Review A, 96(5) (2017), 053833. [39] S. P. Singh, The Ising Model: Brief Introduction and Its Application, In Solid State Physics—Metastable, Spintronics Materials and Mechanics of Deformable Bodies—Recent Progress; Sivasankaran, S., Nayak, P.K., Gu¨nay, E., Eds.; IntechOpen: London, UK, (2020). [40] G. Slade, Probabilistic Models of Critical Phenomena, The Princeton Companion to Mathematics, Princeton University Press, (2008). [41] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry 3rd edition, North Holland Publ, Amsterdam, 2007. [42] L. Zdeborova and F. Krzakala, Statistical physics of inference: Thresholds and algorithms, Advances in Physics, 65(5) (2016), 453-552.
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The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model