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Backward clusters, hierarchy and wild sums for a hard sphere system in a low-density regime

Kazuo Aoki

Department of Mechanical Engineering and Science, Kyoto University, Katsura, 615-8540 Kyoto, Japan

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Mario Pulvirenti

Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185 Roma, Italy

Corresponding author.

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Sergio Simonella

Zentrum Mathematik, TU München, Boltzmannstrasse 3, 85748 Garching, Germany

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, and

Tetsuro Tsuji

Department of Mechanical Engineering and Science, Kyoto University, Katsura, 615-8540 Kyoto, Japan

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https://doi.org/10.1142/S0218202515500256Cited by:8 (Source: Crossref)

Abstract

We study the statistics of backward clusters in a gas of hard spheres at low density. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We derive upper and lower bounds on the average size of clusters by using the theory of the homogeneous Boltzmann equation combined with suitable hierarchical expansions. These representations are known in the easier context of Maxwellian molecules (Wild sums). We test our results with a numerical experiment based on molecular dynamics simulations.

Keywords:

AMSC: 35Q20, 82C40, 82C22

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