Refinement of the Correlation Effects of Interacting Particles in the Ising Model

A numerical technique has been developed on the basis of the cluster variation method (CVM) for calculating the spatial distribution of particles in the Ising lattice model. The lattice is approximated by a basic cluster, which makes it possible to obtain an exact solution to the problem by increasing the cluster size. Universal expressions for the microscopic distribution of particles in clusters of any size were derived by expanding the cluster probabilities in terms of the correlation factors determined on smaller clusters inside the basic cluster. For the sites of the basic cluster, new variables were introduced in order to preserve the structure of the relationships of probabilistic correlators with smaller clusters at any size of the basic cluster. It was shown that the correlation factor symmetrically reflects multisite spin correlations. It was found that expansion in terms of correlation factors allows one to avoid the difficulties in calculations in the CVM caused by the need to use the method of indefinite Lagrange multipliers for cluster probabilities. The difference between the new spin variables and the conventional variables in the CVM was discussed. The efficiency of the technique was demonstrated using a square planar lattice as an example, for which the exact Onsager solution is known. For clusters of 16 sites, the difference between the critical temperature and the exact value was found to be ~2%.