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Villani's conjecture for Kac's walk
Abstract
We prove Villani's conjecture on entropy production for Kac's walk with uniform collision angles. For every N≥2, with total collision rate N, the optimal entropy production constant is 2/(N−1). We show that the known lower bound is sharp by constructing two families of smooth strictly positive probability densities, invariant under coordinate permutations and sign changes. The first consists of normalized products of inverse powers; the second is obtained by conditioning products of Gaussian mixtures on the energy sphere. With N fixed, we compute the leading terms of entropy and entropy production. Successive parameter limits yield two independent proofs of sharpness.
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BibTeX
@misc{gao2026villanis,
title = {{Villani's conjecture for Kac's walk}},
author = {Fuqing Gao and Shaochen Wang and Xianjie Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22675},
eprint = {2609.22675},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22675},
}
RIS
TY - UNPB
TI - Villani's conjecture for Kac's walk
AU - Fuqing Gao
AU - Shaochen Wang
AU - Xianjie Xia
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22675
UR - https://arxiv.org/abs/2609.22675
AB - We prove Villani's conjecture on entropy production for Kac's walk with uniform collision angles. For every N≥2, with total collision rate N, the optimal entropy production constant is 2/(N−1). We show that the known lower bound is sharp by constructing two families of smooth strictly positive probability densities, invariant under coordinate permutations and sign changes. The first consists of normalized products of inverse powers; the second is obtained by conditioning products of Gaussian mixtures on the energy sphere. With N fixed, we compute the leading terms of entropy and entropy production. Successive parameter limits yield two independent proofs of sharpness.
ER -