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Published in:
On the Dynamics of Large Particle Systems in the Mean Field Limit Read chapter Read first chapter

2016 | OriginalPaper | Chapter

2. Continuum Limits of Discrete Models via \(\varGamma \)-Convergence

Author : Lucia Scardia

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

Publisher: Springer International Publishing

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Abstract

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.

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previous chapter On the Dynamics of Large Particle Systems in the Mean Field Limit
next chapter On Evolutionary -Convergence for Gradient Systems
Footnotes
1
This bound can be guaranteed by some additional term in the discrete energy, or by Dirichlet boundary conditions for the admissible displacements, e.g. \(u_0^n=0=u_n^n\) for every n.
 
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Metadata
Title
Continuum Limits of Discrete Models via -Convergence
Author
Lucia Scardia
Copyright Year
2016
Book
Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity

Print ISBN: 978-3-319-26882-8

Electronic ISBN: 978-3-319-26883-5

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-26883-5_2

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