The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error

• 成果类型: Article
• 署名作者: Pulvirenti, M.; Simonella, S.
• 署名单位: Sapienza University Rome; University of L'Aquila; Technical University of Munich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0682-4
• 发表日期: 2017
• 页码: 1135-1237
• 关键词: global-solutions; rare-gas; equation; validity; DYNAMICS

摘要

We present a quantitative analysis of the Boltzmann-Grad (low-density) limit of a hard sphere system. We introduce and study a set of functions, the correlation errors, measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order k measures the event where k particles are connected by a chain of interactions preventing the factorization. We show that, provided , such an error flows to zero with the average density , for short times, as , for some positive . This provides an information on the size of chaos, namely j different particles behave as dictated by the Boltzmann equation even when j diverges as a negative power of . The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.