On the convergence of Kikuchi's natural iteration method

TL;DR

This paper analyzes the convergence of Kikuchi's natural iteration method used in statistical mechanics, providing conditions for convergence based on lattice geometry and connecting it to a broader minimization framework.

Contribution

It establishes a sufficient convergence condition for the natural iteration method and relates it to a general minimization approach for non-convex functionals.

Findings

Convergence depends on cluster entropy coefficients and lattice geometry.

The sufficient condition is satisfied for many common lattice types.

Natural iteration is a special case of a broader minimization method.

Abstract

In this article we investigate on the convergence of the natural iteration method, a numerical procedure widely employed in the statistical mechanics of lattice systems to minimize Kikuchi's cluster variational free energies. We discuss a sufficient condition for the convergence, based on the coefficients of the cluster entropy expansion, depending on the lattice geometry. We also show that such a condition is satisfied for many lattices usually studied in applications. Finally, we consider a recently proposed general method for the minimization of non convex functionals, showing that the natural iteration method turns out as a particular case of that method.