Abstract
We consider the motion of a finite though large number N of hard spheres in the whole space \({\mathbb R}^n\). Particles move freely until they experience elastic collisions. We use our recent theory of Compensated Integrability in order to estimate how much the particles are deviated by collisions. Our result, which is expressed in terms of hodographs, tells us that only \(O(N^2)\) collisions are significant.
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Notes
Bear in mind that \({\mathrm{Div}}\,S\) involves time and space derivatives.
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Acknowledgements
I am indebted to Laure Saint-Raymond and Reinhard Illner for valuable discussions and their help in gathering the relevant literature. Étienne Ghys remarked that my results can be rephrased in terms of hodographs.
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Communicated by C. Mouhot.
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Serre, D. Hard Spheres Dynamics: Weak Vs Strong Collisions. Arch Rational Mech Anal 240, 243–264 (2021). https://doi.org/10.1007/s00205-021-01610-1
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DOI: https://doi.org/10.1007/s00205-021-01610-1