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

In this paper, the influence of point defects such as vacancies on the thermodynamic properties of the two-dimensional Ising model is investigated. The main attention is paid to comparing two approaches to describing the system: the traditional analysis of 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 these quantities upon the introduction of defects are calculated. It is established that the interaction between defects is attractive with a effective radius of d ~ 3. It is shown that in the phase transition region, the free energy behaves as a continuous monotonically decreasing function of temperature, while the internal energy exhibits a singularity caused by the behavior of 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.