Delone Sets and Material Science: a Program

These notes are proposing a program liable to provide physicists working in material science, especially metallic liquids and glasses, the mathematical tools they need to build an atomic scale theory of Continuous Mechanics including plasticity, fluidity and, hopefully, fractures. Using the long list of datas and numerical simulations accumulated during the last forty years, physicists have identified a new class of degrees of freedom, besides the elastic ones, which will be called

anankeons

here [7]. They are dominant in the liquid phase and they explain the properties related to plastic deformations of the solid phase. It is advocated that Delone sets provide a natural frame within which such a theory can be expressed. The use of Voronoi tiling and its dual construction, called the Delaunay triangulation, gives a discretization of the data. The concept of Pachner move or Delaunay flips permits to describe very precisely what the anankeons are. A partition of the configuration space into contiguity domains leads to a graph on which a Markov process can be built to describe the anakeon dynamics. At last, a speculative Section is giving an attempt to describe the Continuous Mechanics of a condensed material in terms of a Noncommutative Geometry of the configuration space.