Cluster expansion and the configurational theory of alloys

A rigorous mathematical foundation for the cluster-expansion method is presented. It is shown that the cluster basis developed by Sanchez et al. Physica A 128, 334 (1984) is a multidimensional discrete Fourier transform while the general formalism of Sanchez Phys. Rev. B 48, 14013 (1993) corresponds to a multidimensional discrete wavelet transform. For functions that depend nonlinearly on the concentration, it is shown that the cluster basis corresponding to a multidimensional discrete Fourier transform does not converge, as it is usually assumed, to a finite cluster expansion or to an Ising-type model representation of the energy of formation of alloys. The multidimensional wavelet transform, based on a variable basis cluster expansion, is shown to provide a satisfactory solution to the deficiencies of the discrete Fourier-transform approach. Several examples aimed at illustrating the main findings and conclusions of this work are given.