The Energy of Some Microscopic Stochastic Lattices

We introduce a notion of energy for some microscopic stochastic lattices. Such lattices are broad generalizations of simple periodic lattices, for which the question of the definition of an energy was examined in a series of previous works [14–18]. Note that slightly more general deterministic geometries were also considered in [6]. These lattices are involved in the modelling of materials whose microscopic structure is a perturbation, in a sense made precise in the article, of the periodic structure of a perfect crystal. The modelling considered here is either a classical modelling, where the sites of the lattice are occupied by ball-like atomic systems that interact by pair potentials, or a quantum modelling where the sites are occupied by nuclei equipped with an electronic structure spread all over the ambient space. The corresponding energies for the infinite stochastic lattices are derived consistently with truncated systems of finite size, by application of a thermodynamic limit process. Subsequent works [7, 8] will be devoted to the macroscopic limits of the energies of such microscopic lattices, thereby extending to a stochastic context the results of [4, 5]. Such convergences in a stochastic setting (in dimension 1) have been studied in [21, 22]. We will also study in [8] some variants and extensions of the stationary setting presented here.