Cluster expansion methods from physical concepts

TL;DR

This paper reconstructs the cluster expansion formalism in materials science using an axiomatic approach, emphasizing intrinsic definitions and Hilbert space geometry to improve clarity and computational robustness.

Contribution

It introduces an axiomatic, intrinsic framework for cluster expansion, avoiding conventional cluster functions and addressing underdetermination issues with a geometry-based fitting method.

Findings

Intrinsic cluster components defined via Moebius inversion

Model fitting grounded in Hilbert space geometry

Avoids underdetermination in model construction

Abstract

The cluster expansion formalism used in materials science is reconstructed on an axiomatic basis with the aims of clarifying underlying concepts and improving computational procedures, and without using conventional cluster functions. Instead, cluster components of configuration functions are defined in an intrinsic manner, which can be viewed as Moebius inversion of conditional expectation. The associated method for fitting a model to a configurational sample is grounded entirely in Hilbert space geometry. By constructing models directly from the given data, we avoid an underdetermination problem to which the conventional approach is subject. Tensor observables are treated on an equal footing with scalar observables.