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Analysis and Stochastics of Growth Processes and Interface Models$
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Peter Mörters, Roger Moser, Mathew Penrose, Hartmut Schwetlick, and Johannes Zimmer

Print publication date: 2008

Print ISBN-13: 9780199239252

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199239252.001.0001

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Validity and Non-Validity of Propagation of Chaos

Chapter:
(p.101) 5 Validity and Non-Validity of Propagation of Chaos
Source:
Analysis and Stochastics of Growth Processes and Interface Models
Author(s):

Karsten Matthies

Florian Theil

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199239252.003.0005

In this chapter a novel, rigorous approach to analyse the validity of continuum approximations for deterministic interacting particle systems is discussed. The focus is on the Boltzmann–Grad limit of ballistic annihilation, a topic which has has received considerable attention in the physics literature. In this situation, due to the deterministic nature of the evolution, it is possible that correlations build up and the mean–field approximation by the Boltzmann equation breaks down. A sharp condition on the initial distribution, which ensures the validity of the Boltzmann equation is given, together with an example demonstrating the failure of the mean-field theory if the condition is violated.

Keywords:   continuum approximation, Boltzmann–Grad limit, ballistic annihilation, mean-field theory, Boltzmann equation

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