Year: 2026
Type: preprint
Abstract: We prove Villani's conjecture on entropy production for Kac's walk with uniform collision angles. For every $N\geq2$, 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.
Source: arXiv (Cornell University)
Authors: Gao, Fuqing, Wang, Shaochen, Xia, Xianjie
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3158526266, G364413142
OpenAlex ID: W7214019858