Authors : Adama Gaye; Omolola Dorcas, Atanda

Volume/Issue : Volume 10 - 2025, Issue 5 - May

Google Scholar : https://tinyurl.com/ycfaprh2

Scribd : https://tinyurl.com/3r5jujfe

DOI : https://doi.org/10.38124/ijisrt/25may414

Note : A published paper may take 4-5 working days from the publication date to appear in PlumX Metrics, Semantic Scholar, and ResearchGate.

Abstract : This study presents a comprehensive Monte Carlo simulation of the two-dimensional (2-D) Ising model using the Metropolis algorithm to investigate critical phenomena and thermodynamic behavior in spin-lattice systems. The model, implemented in MATLAB with periodic boundary conditions, explores equilibrium properties such as magnetization, internal energy, specific heat, and susceptibility across a range of temperatures. By employing various initial spin configurations—ordered and random—the simulations demonstrate the system's ergodicity and convergence to thermal equilibrium. Key results include a sharp decline in magnetization and a pronounced peak in specific heat near the critical temperature, consistent with second-order phase transition behavior. The simulation captures microscopic domain evolution, highlighting the transition from ferromagnetic to paramagnetic phases as thermal fluctuations increase. The study further evaluates algorithmic efficiency, discusses the impact of lattice size on statistical accuracy, and proposes improvements using advanced cluster algorithms and parallel computing frameworks. The findings validate theoretical predictions from Onsager's solution and underscore the versatility of Monte Carlo techniques in modeling collective behavior in magnetic systems. The simulation framework offers a robust foundation for analyzing critical dynamics and extends its relevance to broader applications in material science, computational physics, and complex systems modeling.

Keywords : Monte Carlo, Modeling, 2-D, Ising Systems, Metropolis Algorithm, Simulation Techniques, Thermodynamic Behavior, Magnetization Patterns.

References :

1. Anyibama, B. J., Orjinta, K. K., Omisogbon, T. O., Atalor, S. I., Daniels, E. O., Fadipe, E. & Galadima, D. A. (2025). Modern Agricultural Technologies for Sustainable Food Production: A Comprehensive Review of Technological Innovations and Their Impact on Global Food Systems. International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 Volume 10, Issue 2 https://doi.org/10.5281/zenodo.14964384
2. Azonuche, T. I., & Enyejo, J. O. (2024). Evaluating the Impact of Agile Scaling Frameworks on Productivity and Quality in Large-Scale Fintech Software Development. International Journal of Scientific Research and Modern Technology, 3(6), 57–69. https://doi.org/10.38124/ijsrmt.v3i6.449
3. Azonuche, T. I., & Enyejo, J. O. (2024). Exploring AI-Powered Sprint Planning Optimization Using Machine Learning for Dynamic Backlog Prioritization and Risk Mitigation. International Journal of Scientific Research and Modern Technology, 3(8), 40–57. https://doi.org/10.38124/ijsrmt.v3i8.448.
4. Baxter, R. J. (2016). Exactly solved models in statistical mechanics. Dover Publications.
5. Binder, K. (1981). Finite size scaling analysis of Ising model block distribution functions. Zeitschrift für Physik B Condensed Matter, 43(2), 119–140. https://doi.org/10.1007/BF01293604
6. Binder, K., & Heermann, D. W. (2010). Monte Carlo simulation in statistical physics: An introduction (5th ed.). Springer. https://doi.org/10.1007/978-3-642-03163-2
7. Binder, K., & Luijten, E. (2001). Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models. Physics Reports, 344(4–6), 179–253. https://doi.org/10.1016/S0370-1573(00)00127-4
8. Cardy, J. (1996). Scaling and renormalization in statistical physics. Cambridge University Press.
9. Domb, C. (1996). The critical point: A historical introduction to the modern theory of critical phenomena. CRC Press.
10. Ferrenberg, A. M., Xu, J., & Landau, D. P. (2018). Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model. Physical Review E, 97(4), 043301. https://doi.org/10.1103/PhysRevE.97.043301
11. Goldenfeld, N. (2018). Lectures on phase transitions and the renormalization group. CRC Press.
12. Higham, D. J., & Higham, N. J. (2017). MATLAB guide (3rd ed.). SIAM.
13. Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik, 31(1), 253–258. https://doi.org/10.1007/BF02980577
14. Kalos, M. H., & Whitlock, P. A. (2008). Monte Carlo methods (2nd ed.). Wiley-VCH. https://doi.org/10.1002/9783527632535
15. Katzgraber, H. G. (2010). Introduction to Monte Carlo methods. In J. Moreno (Ed.), New optimization algorithms in physics (pp. 1–20). Wiley-VCH. https://doi.org/10.1002/9783527632580.ch1
16. Landau, D. P., & Binder, K. (2021). A guide to Monte Carlo simulations in statistical physics (5th ed.). Cambridge University Press.
17. McCoy, B. M., & Wu, T. T. (2014). The two-dimensional Ising model. Courier Corporation.
18. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087–1092. https://doi.org/10.1063/1.1699114
19. Newman, M. E. J., & Barkema, G. T. (1999). Monte Carlo methods in statistical physics. Oxford University Press.
20. Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review, 65(3–4), 117–149. https://doi.org/10.1103/PhysRev.65.117
21. Pelissetto, A., & Vicari, E. (2002). Critical phenomena and renormalization-group theory. Physics Reports, 368(6), 549–727. https://doi.org/10.1016/S0370-1573(02)00219-3
22. Preis, T., Virnau, P., Paul, W., & Schneider, J. J. (2009). GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model. Journal of Computational Physics, 228(12), 4468–4477. https://doi.org/10.1016/j.jcp.2009.03.018
23. Sachdev, S. (2011). Quantum phase transitions (2nd ed.). Cambridge University Press.
24. Swendsen, R. H., & Wang, J. S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58(2), 86–88. https://doi.org/10.1103/PhysRevLett.58.86
25. Vojta, T. (2003). Disorder-induced rounding of certain quantum phase transitions. Physical Review Letters, 90(10), 107202. https://doi.org/10.1103/PhysRevLett.90.107202
26. Wang, J. S., & Swendsen, R. H. (1990). Low-temperature properties of the Ising model from Monte Carlo simulations. Physical Review B, 42(4), 2465–2474. https://doi.org/10.1103/PhysRevB.42.2465
27. Wolff, U. (1989). Collective Monte Carlo updating for spin systems. Physical Review Letters, 62(4), 361–364. https://doi.org/10.1103/PhysRevLett.62.361
28. Yeomans, J. M. (1992). Statistical mechanics of phase transitions. Oxford University Press