Diffractive Point Sets with Entropy

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. Introduction Diffraction is one of the most important ways of identifying long-range order in mathematical and physical structures. In this paper, we look at the effects on diffraction that occur in certain periodic and quasiperiodic point sets when the occupation of the point sites is taken stochastically rather than deterministically, with in..