This book is the offspring of a summer school school “Macroscopic and
large scale
phenomena: coarse graining, mean field limits and ergodicity”, which was
held in 2012 at the University of Twente, the Netherlands. The focus lies on
mathematically rigorous methods for multiscale problems of physical origins.
Each of the four book chapters is based on a set of lectures delivered
at the school, yet all authors have expanded and refined their contributions.
Francois Golse
delivers a chapter on the dynamics of large particle systems in the mean field
limit and surveys the most significant tools and methods to establish such
limits with mathematical rigor. Golse discusses in depth a variety of examples,
including Vlasov--Poisson and Vlasov--Maxwell systems.
Lucia Scardia focuses
on the rigorous derivation of macroscopic models using $Gamma$-convergence, a
more recent variational method, which has proved very powerful for problems in
material science. Scardia illustrates this by various basic examples and a more
advanced case study from dislocation theory.
Alexander Mielke's
contribution focuses on the multiscale modeling and rigorous analysis of
generalized gradient systems through the new concept of evolutionary
$Gamma$-convergence. Numerous evocative examples are given, e.g., relating to
periodic homogenization and the passage from viscous to dry friction.
Martin Göll and Evgeny
Verbitskiy conclude this volume, taking a dynamical systems and ergodic theory
viewpoint. They review recent developments in the study of homoclinic points
for certain discrete dynamical systems, relating to particle systems via
ergodic properties of lattices configurations.
