Abstract: In this talk, we explore large-scale interacting systems that explain macroscopic phenomena through the movement of microscopic particles. These systems are modeled by discrete lattices where microscopic dynamics occur and the transitions between adjacent sites, referred to as “interactions” in our definition. We classify interactions based on conserved quantities, which reflect macroscopic properties, and introduce wedge sums and box products as methods to construct new interactions from existing ones. We also present a discrete harmonic theory for large-scale interacting systems on discrete lattices with finite local state sets. By assuming exchangeable interactions, we define the inverse harmonic period matrix, which we expect to correspond to the diffusion matrix in the hydrodynamic limit. Our result offers an interpretation of the diffusion matrix based on the geometry of the microscopic model.

[Founded by the European Union – Next Generation EU.]