A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension d≥ 4
Abstract
We prove a periodic long-time Boltzmann--Grad limit for hard spheres in every fixed spatial dimension d≥4. Under the assumptions of the main theorem, the rescaled s-particle correlations converge in L1 to the corresponding tensor-product Boltzmann profile at rate ε1/(400d), uniformly for 1≤ s≤|logε| and 0≤ t≤ tfin. In particular, when the activity and weighted solution bounds are fixed, tfin=O(log|logε|). 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≥1, 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 ε(d−1)(k−1)ε*−k(d−1); k=2 is the first gain-producing case. A complementary full-chord estimate provides a coarse fallback.
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BibTeX
@misc{hua2026periodic,
title = {{A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension $d\ge 4$}},
author = {Zhixu Hua},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29566},
eprint = {2609.29566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29566},
}
RIS
TY - UNPB
TI - A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension d≥ 4
AU - Zhixu Hua
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.29566
UR - https://arxiv.org/abs/2609.29566
AB - We prove a periodic long-time Boltzmann--Grad limit for hard spheres in every fixed spatial dimension d≥4. 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 ε^1/(400d), uniformly for 1≤ s≤|logε| and 0≤ t≤ t_fin. In particular, when the activity and weighted solution bounds are fixed, t_fin=O(log|logε|). 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≥1, 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 ε^(d−1)(k−1)ε_*^−k(d−1); k=2 is the first gain-producing case. A complementary full-chord estimate provides a coarse fallback.
ER -