On the Non-uniqueness of Physical Scattering for Hard Non-spherical Particles

  • Mark WilkinsonEmail author


We prove the existence of uncountably-many physical scattering maps for non-spherical hard particles which, when used to construct global-in-time weak solutions of Newton’s equations of motion, conserve the total linear momentum, angular momentum and kinetic energy of the particle system for all time. We prove this result by first exhibiting the non-uniqueness of a classical solution to a constrained Monge–Ampère equation posed on Euclidean space, and then applying the deep existence theory of Ballard for hard particle dynamics. In the final section of the article, we briefly discuss the relevance of our observations to the statistical mechanics of hard particle systems.



I would like to extend my sincere thanks and gratitude to Gilles Francfort and to Patrick Ballard for several extended conversations and encouragement during the preparation of this manuscript. Finally, I would like to thank two anonymous referees whose comments helped to improve an earlier draft of this work. The author is supported by the EPSRC Standard Grant EP/P011543/1.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science (MACS) and the Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburghScotland

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