Self-referential order

We introduce the concept of self-referential order which provides a way to quantify structural organization in non-crystalline materials. The key idea consists in the observation that, in a disordered system, where there is no ideal, reference, template structure, each sub-portion of the whole structure can be taken as reference for the rest and the system can be described in terms of its parts in a self-referential way. Some parts carry larger information about the rest of the structure and they are identified as motifs. We discuss how this method can efficiently reduce the amount of information required to describe a complex disordered structure by encoding it in a set of motifs and matching rules. We propose an information-theoretic approach to define a self-referential-order-parameter and we show that, by means of entropic measures, such a parameter can be quantified explicitly. A proof of concept application to equal disk packing is presented and discussed.