Persistent homology and many-body atomic structure for medium-range order in the glass

Abstract

Abstract. Characterization of medium-range order in amorphous materials and its relation to short-range order is discussed. A new topological approach is presented here to extract a hierarchical structure of amorphous materials, which is robust against small perturbations and allows us to distinguish it from periodic or random configurations. The method is called the persistence diagram (PD) and it introduces scales into manybody atomic structures in order to characterize the size and shape. We first illustrate how perfect crystalline and random structures are represented in the PDs. Then, the medium-range order in the amorphous silica is characterized by using the PD. The PD approach reduces the size of the data tremendously to much smaller geometrical summaries and has a huge potential to be applied to broader areas including complex molecular liquid, granular materials, and metallic glasses.

Persistent Homology and Materials Informatics

“…A simple computation yields that Ker 1 = k( [1,4] To finish, remark that 2]) and that 0 is a zero map. Therefore, [5] and H 0 (X) ≅ k 2 which indicates the presence of two connected components.…”

“…The interval spans from the second to the fourth so we denote it I [2,4]. All maps between the nonzero vector spaces are identity maps.…”

“…The lowest nonzero entry corresponds to line [2] so [3] kills the feature created by [2]. In the same way, [4] creates a new connected component killed by [5].…”

“…Therefore, we need to be careful about the dependence of the system sizes. The scaling properties of PDs with respect to the system size are computationally studied in [4]. Recently, the existence and uniqueness of limiting persistence diagram is mathematically solved in [15].…”

“…The purpose of this paper is to provide a self-contained introduction to persistent homology and survey several applications to materials science [2][3][4][5]. We only assume knowledge of elementary linear algebra and show several examples to help the readers' understanding.…”