Abstract

The nature is dominated by systems composed of many individuals, belonging to various species, with a collective behavior. Instead of calculating the trajectories of all individuals, it is computationally much simpler to describe the dynamics of the individuals on a macroscopic level by averaged quantities such as population densities. This leads to systems of highly nonlinear partial differential equations with cross diffusion, which may reveal surprising effects such as uphill diffusion and diffusion-induced instabilities. In this talk, we detail some approaches on the derivation of cross-diffusion equations from more basic models. The relation between principles from nonequilibrium thermodynamics and the entropy structure of cross-diffusion systems is detailed. The entropy structure can be found in nonstandard models like van-der-Waals fluids, spin diffusion in semiconductors,and exotic financial derivatives.