{"title":"直线上硬球的最大标度碰撞和不变度量","authors":"Mark Wilkinson","doi":"10.1007/s10955-024-03310-y","DOIUrl":null,"url":null,"abstract":"<p>For any <span>\\(N\\ge 3\\)</span>, we study invariant measures of the dynamics of <i>N</i> hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold <span>\\(\\mathcal {M}\\)</span> of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all <i>N</i> hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable <i>N</i>-body scattering maps which generate a billiard dynamics on <span>\\(\\mathcal {M}\\)</span> admitting a canonical weighted Hausdorff measure on <span>\\(\\mathcal {M}\\)</span> (that we term the <i>Liouville measure on</i> <span>\\(\\mathcal {M}\\)</span>) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving <i>linear</i> <i>N</i>-body scattering map yields a billiard dynamics which admits the Liouville measure on <span>\\(\\mathcal {M}\\)</span> as an invariant measure.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Maximal Codimension Collisions and Invariant Measures for Hard Spheres on a Line\",\"authors\":\"Mark Wilkinson\",\"doi\":\"10.1007/s10955-024-03310-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For any <span>\\\\(N\\\\ge 3\\\\)</span>, we study invariant measures of the dynamics of <i>N</i> hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold <span>\\\\(\\\\mathcal {M}\\\\)</span> of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all <i>N</i> hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable <i>N</i>-body scattering maps which generate a billiard dynamics on <span>\\\\(\\\\mathcal {M}\\\\)</span> admitting a canonical weighted Hausdorff measure on <span>\\\\(\\\\mathcal {M}\\\\)</span> (that we term the <i>Liouville measure on</i> <span>\\\\(\\\\mathcal {M}\\\\)</span>) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving <i>linear</i> <i>N</i>-body scattering map yields a billiard dynamics which admits the Liouville measure on <span>\\\\(\\\\mathcal {M}\\\\)</span> as an invariant measure.</p>\",\"PeriodicalId\":667,\"journal\":{\"name\":\"Journal of Statistical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Statistical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s10955-024-03310-y\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10955-024-03310-y","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Maximal Codimension Collisions and Invariant Measures for Hard Spheres on a Line
For any \(N\ge 3\), we study invariant measures of the dynamics of N hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold \(\mathcal {M}\) of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all N hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable N-body scattering maps which generate a billiard dynamics on \(\mathcal {M}\) admitting a canonical weighted Hausdorff measure on \(\mathcal {M}\) (that we term the Liouville measure on\(\mathcal {M}\)) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linearN-body scattering map yields a billiard dynamics which admits the Liouville measure on \(\mathcal {M}\) as an invariant measure.
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.