PreprintCC BY 4.0arXiv
The entropy production is not always monotone for the Kac equation
Abstract
In 1966, McKean conjectured that the entropy production of Kac's caricature of a Maxwellian gas is nonincreasing in time \cite{McKean66}. We disprove this conjecture for the standard uniform-angle Kac equation. Our counterexamples are smooth, strictly positive, even mixtures of two Gaussians at different temperatures. We also show that every entropy derivative beyond the first violates its expected sign at some time along the same solution. The initial densities can be chosen with unit mass and second moment. mass and second moment.
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BibTeX
@misc{wang2026entropy,
title = {{The entropy production is not always monotone for the Kac equation}},
author = {Weinan Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06290},
eprint = {2609.06290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06290},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The entropy production is not always monotone for the Kac equation
AU - Weinan Wang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06290
UR - https://arxiv.org/abs/2609.06290
AB - In 1966, McKean conjectured that the entropy production of Kac's caricature of a Maxwellian gas is nonincreasing in time \cite{McKean66}. We disprove this conjecture for the standard uniform-angle Kac equation. Our counterexamples are smooth, strictly positive, even mixtures of two Gaussians at different temperatures. We also show that every entropy derivative beyond the first violates its expected sign at some time along the same solution. The initial densities can be chosen with unit mass and second moment. mass and second moment.
ER -