Multicomponent formalism in the mean-field approximation: a geometric interpretation of Chebychev polynomials

Abstract A simple geometrical interpretation is given for the Chebychev polynomials used in the definition of the correlation functions for multicomponent Ising models. This interpretation is based on the fact that each chemical species is associated with one basis unit vector of the canonical reference frame of a M -dimensional space where M is the number of chemical components of the system. Under this scheme the Flinn operator attached to each lattice site is written simply as a scalar product from which multisite Flinn operators are easily derived. The method is exemplified in the framework of the cluster variation method approximation for the practical case of ternary systems.