A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension $d\ge 4$

Abstract

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.