Helmholtz Free Energy in the Two-Dimensional Ising Model with Defects

Abstract

The influence of point defects of the vacancy type on the thermodynamic properties of the two-dimensional Ising model is investigated in the paper. The main attention is paid to the comparison of two approaches to the description of the system: the traditional analysis of the internal energy \(E\) and a more complete analysis of the Helmholtz free energy \(F\) and entropy \(S\). Using Monte Carlo methods—the Metropolis algorithm and the Wang–Landau algorithm—the temperature dependences of the changes in the indicated quantities upon the introduction of defects are calculated. It is established that the interaction of defects has the character of attraction with an effective radius of \(d\approx 3\). It is shown that in the phase transition region, the free energy behaves as a continuous monotonically decreasing function of temperature, whereas the internal energy exhibits a singularity caused by the behaviour of the entropy. The possible dynamics of defects is investigated by comparing the energy of different configurations; a tendency towards defect clustering at temperatures above the critical one is observed. A comparative analysis of the accuracy and computational efficiency of both approaches is carried out.