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.
Keywords2D Isinig model spin systems Onsager solution finite-size lattice free boundary conditions computer simulations critical temperature magnetization analytic expressions
Unable to display preview. Download preview PDF.
- 2.Stanley, H., Introduction to Phase Transitions and Critical Phenomena, Oxford: Clarendon, 1971.Google Scholar
- 3.Onsager, L., Crystal statistics, I. A two-dimensional model with an order–disorder transition, Phys. Rev., 1944, vol. 65, no. 3–4, pp. 117–149.Google Scholar
- 5.Dixon, J.M., Tuszynski, J.A., and Clarkson, P., From Nonlinearity to Coherence: Universal Features of Nonlinear Behavior in Many-Body Physics, Oxford: Clarendon Press, 1997.Google Scholar
- 7.Kubo, R., An analytic method in statistical mechanics, Busserion Kenkyu (Japan), 1943, vol. 1, pp. 1–13.Google Scholar
- 20.Amit, D., Gutfreund, H., and Sompolinsky, H., Statistical mechanics of neural networks near saturation, Adv. Phys., 1987, vol. 173, pp. 30–67.Google Scholar
- 21.van Hemmen, J.L. and Kuhn, R., Collective phenomena in neural networks, in Models of Neural Networks, Domany, E., van Hemmen, J.L., and Shulten, K., Eds., Berlin: Springer-Verlag, 1992, pp. 1–105.Google Scholar
- 32.Kasteleyn, P., Dimer statistics and phase transitions, J. Math. Phys., 1963, vol. 4, no.2.Google Scholar
- 33.Fisher, M., On the dimer solution of planar Ising models, J. Math. Phys., 1966, vol. 7, no.11.Google Scholar
- 34.Karandashev, Ya.M. and Malsagov, M.Yu., Polynomial algorithm for exact calculation of partition function for binary spin model on planar graphs, Opt. Mem. Neural Networks, 2017, vol. 26, no. 2. https://arxiv.org/abs/1611.00922.Google Scholar
- 35.Schraudolph, N. and Kamenetsky, D., Efficient exact inference in planar Ising models, Proc. 21st Int. Conf. on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 8–10, 2008, Red Hook, NY: Curran Associates, 2008., 2008. https://arxiv.org/abs/0810.4401.Google Scholar
- 37.Kryzhanovsky, B.V., The spectral density of a spin system calculated for solvable models. http://arxiv.org/abs/1704.01351.Google Scholar