Gradient systems and the derivation of effective kinetic relations via EDP convergence

Many ordinary and partial differential equations can be written as a gradient flow, which means that there is an energy functional that drives the evolution of the the solutions by flowing down in the energy landscape. The gradient is defined in terms of a dissipation structure, which in the simplest case is a Riemannian metric. We discuss classical and nontrivial examples in reaction-diffusion systems or friction mechanics. We will emphasize that having a gradient structure for a given differential equation means that we add additional physical information.

Considering a family of gradient systems depending on a small parameter, it is natural to ask for the limiting (also called effective) gradient system if the parameter tends to 0. This can be achieved on the basis of De Giorgi's Energy-Dissipation Principle (EDP). We discuss the new notion of "EDP convergence" and show by examples that this theory is flexible enough to allow for situations where starting from a linear kinetic relation (or quadratic dissipation potentials) we arrive at physically relevant, nonlinear effective kinetic relations. The connections between macroscopic kinetic kinetic relations and microscopic non-equilibrium steady states will be discussed.