On the cardinality of collisional clusters for hard spheres at low density

Mario Pulvirenti; Sergio Simonella

doi:10.3934/dcds.2021021

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2021, Volume 41, Issue 8: 3903-3914. Doi: 10.3934/dcds.2021021

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On the cardinality of collisional clusters for hard spheres at low density

Mario Pulvirenti^1, and
Sergio Simonella^2,

1.
Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185 Rome – Italy, and, International Research Center M & MOCS, Università dell'Aquila, Palazzo Caetani, 04012 Cisterna di Latina, Italy
2.
UMPA UMR 5669 CNRS, ENS de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France

Received: May 2020

Revised: December 2020

Early access: January 2021

Published: August 2021

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Abstract

We resume the investigation, started in [2], of the statistics of backward clusters in a gas of $ N $ hard spheres of small diameter $ \varepsilon $. 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 obtain an estimate of the average cardinality of clusters with respect to the equilibrium measure, global in time, uniform in $ \varepsilon, N $ for $ \varepsilon^2 N = 1 $ (Boltzmann-Grad regime).

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Mathematics Subject Classification: Primary:35Q20, 82C40, 82C22.

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Figure 1. The trajectory of a backward cluster $ BC(1) $ at time $ t $ (of cardinality $ 3 $) is represented on the left. Its tree structure $ {\Gamma}_3 = (1,1,2) $ is given by the graph on the right

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