Mathematical methods for scale-bridging: from interacting particle systems to differential equations

Many phenomena in the natural and social sciences can be modeled on a small (or microscopic) scale by many “particles” that change their state according to a random input (noise) and the state of the others, think of molecules in solution or agents in a market. Due tot he high dimensionality, these models are difficult to analyse computationally. On a larger scale however, the behaviour of such systems can often be described by differential equations which are numerically much more tractable. This leads to the problem of scale bridging, i.e. how to connect rigorously these different descriptions at different scales by proving limit theorems. The topic of scale-bridging is a long-standing challenge for mathematics. In the explanation of his 6th problem, Hilbert set the task “of developing mathematically the limiting process… which lead from the atomistic view to the laws of motion of continua” Scale-bridging lies at the intersection of several mathematical disciplines.

The focus of the project is on applying two new mathematical developments for the purpose of scale bridging: First, the theory of gradient flows and Wasserstein metrics developed by Otto and co-authors and for the first time applied to the analysis of interacting particle systems by the 1st supervisor and co-authors (ADPZ 2013), and second the recent progress in the theory of stochastic homogenization by Armstrong, Cardaliaguet, Souganidis and others.
Typical examples will include interacting diffusions, zero-range processes and variants of the Ising model.