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

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  • 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).

    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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