发表机构
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在任意固定维度 $d\ge 4$ 中证明硬球系统的周期长时间 Boltzmann--Grad 极限，通过双着陆根包和周期首次失效构造，得到速率 $\varepsilon^{1/(400d)}$。
AI 中文摘要
我们证明了在每一个固定空间维度 $d\ge 4$ 中，硬球系统满足周期长时间 Boltzmann--Grad 极限。在二维和三维中使用的分量式长键估计无法给出所需的任意维数幂次。我们的替代方案将两个相连的时间子层保持为单个正核包，并选择前两个较低碰撞原子作为着陆根。对于不相交的着陆边，两个入射框架直接组合。对于重叠的边，在第二次着陆时首先出现的粒子线提供了一条无碰撞弦；消除该弦产生一个非负的 Jacobi 指标形式，而剩余的单速框架在带根的上层碰撞词下无法聚焦。由此得到的逐胞估计对任意非负联合检验函数均成立，并给出相对因子 $\varepsilon^{2(d-1)}\varepsilon_*^{-2d}$，其中唯一的全分量 $\varepsilon^{-(d-1)}$ 归一化只计数一次。一个有界度的有限纤维余面积论证控制了所有全局恢复分支。我们还用具有永久入射标签的周期首次失效构造替换了欧几里得无双重重叠推断。将这两个几何输入插入长时间累积量展开中，对于 $0\le t\le t_{\rm fin}$ 和 $1\le s\le |\log\varepsilon|$ 一致地得到速率 $\varepsilon^{1/(400d)}$。该证明还分离出任意维度中的固定词多着陆原理；一个可选的混合速度-时间外部条件给出更锐利的局部损失，但未在主定理中使用。
英文摘要
We prove a periodic long-time Boltzmann--Grad limit for hard spheres in every fixed spatial dimension $d\ge4$. Under the assumptions of the main theorem, the rescaled $s$-particle correlations converge in $L^1$ to the corresponding tensor-product Boltzmann profile at rate $\varepsilon^{1/(400d)}$, uniformly for $1\le s\le|\log\varepsilon|$ and $0\le t\le t_{\rm fin}$. In particular, when the activity and weighted solution bounds are fixed, $t_{\rm fin}=O(\log|\log\varepsilon|)$. The componentwise long-bond estimate used in dimensions two and three is insufficient in higher dimension. We replace it by a joint estimate for two connected time sublayers, selecting the first two lower collisions as landing roots. Disjoint landing edges yield a direct tangential frame. When the edges overlap, the newly appearing particle line contains a collision-free chord whose radial relative speed produces a rank-$(d-1)$ positive Schur factor. A positive Jacobi-network argument prevents focusing, while bounded-degree finite-fibre coarea globalizes the estimate for arbitrary nonnegative joint kernels. A first-failure construction repairs periodic double overlaps, and the resulting packet bound closes the exceptional top-layer contribution. A paid epoch restart for sealed complete joint kernels, using the fixed-word multi-landing operator for every fixed integer $k\ge1$, yields the stated time range. Within the regular connected full two-sublayer packet class, every packet with at least $2k-1$ physical lines admits a canonical $k$-birth flag and the operator factor $\varepsilon^{(d-1)(k-1)}\varepsilon_*^{-k(d-1)}$; $k=2$ is the first gain-producing case. A complementary full-chord estimate provides a coarse fallback.
Comments230 pages, 7 figures. Includes Online Resource 1 with supplementary proofs and source-interface material
