Cluster expansion and the configurational theory of alloys

TLDR: In this paper, a rigorous mathematical foundation for the cluster-expansion method is presented and it is shown that the cluster basis developed by Sanchez et al. is a multidimensional discrete Fourier transform while the general formalism of Sanchez [Phys Rev B 48, 14013 (1993) corresponds to a multi-dimensional discrete wavelet transform.

Abstract: 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