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

Abstract

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 $N=l{\times}l$ spins and the asymptotic Onsager solution ($N\to\infty$). 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.