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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. On the Dynamics of Large Particle Systems in the Mean Field Limit

This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics—such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics—can be rigorously derived from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these derivations, such as Dobrushin’s stability estimate in the Monge-Kantorovich distance for the empirical measures built on the solution of the N-particle motion equations in classical mechanics, or the theory of BBGKY hierarchies in the case of classical as well as quantum problems. We explain in detail how these different approaches are related; in particular we insist on the notion of chaotic sequences and on the propagation of chaos in the BBGKY hierarchy as the number of particles tends to infinity.
François Golse

### Chapter 2. Continuum Limits of Discrete Models via $$\varGamma$$ Γ -Convergence

These lecture notes are a step-by-step guide to $$\varGamma$$-convergence and its applications to the upscaling of discrete systems. In many cases of interest—atoms, defects in metals, crowds—one has to face the challenging problem of deriving a macroscopic model to describe the collective behaviour of the interacting particles. $$\varGamma$$-convergence is a mathematically rigorous approach to the upscaling, and has been successfully applied to several problems in material science. The focus of this contribution is on one-dimensional chains of particles, and the starting point is the interaction energy of the system. The nature of the interaction depends on the application and we will consider the case of convex and non convex potentials and both short and long-range interactions.
Lucia Scardia

### Chapter 3. On Evolutionary $$\varGamma$$ Γ -Convergence for Gradient Systems

In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional $${\mathscr {E}}_\varepsilon$$ and the dissipation potential $${\mathscr {R}}_\varepsilon$$ or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions $$u_\varepsilon$$. We investigate the question under which conditions the limits u of (subsequences of) the solutions $$u_\varepsilon$$ are solutions of the gradient system generated by the $$\varGamma$$-limits $${\mathscr {E}}_0$$ and $${\mathscr {R}}_0$$. Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where $${\mathscr {R}}_\varepsilon$$ is quadratic, and rate-independent systems as well as the passage from classical gradient to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.
Alexander Mielke

### Chapter 4. Homoclinic Points of Principal Algebraic Actions

The 1999 paper by D. Lind and K. Schmidt on homoclinic points of a special class of dynamical systems—the so called algebraic $${{\mathrm{\mathbb {Z}^d}}}$$-actions—attracted a lot of interest to the study of homoclinic points. In the present paper we review the developments over the past 15 years. Major progress has been made in questions of existence of homoclinic points for $${{\mathrm{\mathbb {Z}^d}}}$$-actions. More recently, first results were obtained for actions of non-abelian discrete groups. Summable homoclinic points were successfully used in the computation of entropy and in the study of probabilistic properties of dynamical systems, e.g., Central Limit Theorems. Moreover, homoclinic points allow one to construct coding maps which link certain particle systems to algebraic dynamical systems.
Martin Göll, Evgeny Verbitskiy

### Backmatter

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