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

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• 作者：M. Pulvirenti ; S. Simonella
• 刊名：Inventiones mathematicae
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：207
• 期：3
• 页码：1135-1237
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1297
• 卷排序：207

文摘

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 \(k < \varepsilon ^{-\alpha }\), such an error flows to zero with the average density \(\varepsilon \), for short times, as \(\varepsilon ^{\gamma k}\), for some positive \(\alpha ,\gamma \in (0,1)\). 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 \(\varepsilon \). The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.