Diffractive point sets with entropy

TL;DR

This paper introduces and analyzes diffractive point sets with entropy, generalizing lattice gases, and demonstrates how stochastic site occupation affects their diffraction properties, including diffuse backgrounds.

Contribution

It defines entropic model sets and explores how stochastic site occupation influences their diffraction, extending the understanding of diffractive point sets with entropy.

Findings

Stochastic site occupation leads to well-defined diffraction with diffuse background.

Both i.i.d. and site-dependent random variables are considered.

Examples illustrate the impact of randomness on diffraction patterns.

Abstract

After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in well-defined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown.