Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity

by Adrian Muntean, Jens D. M. Rademacher, Antonios Zagaris

Series: Lecture Notes in Applied Mathematics and Mechanics (No. 3)

This book is the offspring of a summer school school “Macroscopic andlarge scale

phenomena: coarse graining, mean field limits and ergodicity”, which washeld in 2012 at the University of Twente, the Netherlands. The focus lies onmathematically rigorous methods for multiscale problems of physical origins.

Each of the four book chapters is based on a set of lectures deliveredat the school, yet all authors have expanded and refined their contributions. 

Francois Golsedelivers a chapter on the dynamics of large particle systems in the mean fieldlimit and surveys the most significant tools and methods to establish suchlimits with mathematical rigor. Golse discusses in depth a variety of examples,including Vlasov--Poisson and Vlasov--Maxwell systems.

Lucia Scardia focuseson the rigorous derivation of macroscopic models using $\Gamma$-convergence, amore recent variational method, which has proved very powerful for problems inmaterial science. Scardia illustrates this by various basic examples and a moreadvanced case study from dislocation theory.

Alexander Mielke'scontribution focuses on the multiscale modeling and rigorous analysis ofgeneralized gradient systems through the new concept of evolutionary$\Gamma$-convergence. Numerous evocative examples are given, e.g., relating toperiodic homogenization and the passage from viscous to dry friction.

Martin Göll and EvgenyVerbitskiy conclude this volume, taking a dynamical systems and ergodic theoryviewpoint. They review recent developments in the study of homoclinic pointsfor certain discrete dynamical systems, relating to particle systems viaergodic properties of lattices configurations.