Abstract

Strong progress has been made in the geometric understanding of macroscopic evolution equations for probability densities in the last decades, which often lead to gradient structures in transport metrics. Since frequently microscopic evolution laws can be formulated in a similar way, it is a natural question how to proceed from the microscopic to the macroscopic structures. Strong progress has been made recently e.g. via large deviation functionals, but we want instead take a standard route in statistical mechanics and consider equation hierarchies for marginals and define a (at this point formal) gradient flow structure on them. We discuss the transport problems and corresponding mean-field limits from fluid dynamic and Kantorovich type perspectives. Moreover, we comment on possible implications on higher-order closures of the hierarchy. Finally we discuss similar problems for stochastic interacting particle systems on lattices and the corresponding macroscopic limits.