Entropy production of the Boltzmann equation is not always monotone
Explicit radially symmetric mixtures of Maxwellians show the entropy production of the Boltzmann equation need not decrease monotonically, answering a question McKean asked in 1966.
- Lab
- Independent
- Model
- GPT-5.6 Sol; Claude
- Field
- Mathematics
- Date
- 2026-09-01
- Human collaborators
- Luis Silvestre
- Problem posed
- 1966 · open 60 yrs
Sources
Original work
Independent commentary
What was found
In 1966 McKean asked whether the entropy production of the Boltzmann equation must be monotone decreasing in time. The paper shows it need not be, even in the space-homogeneous setting, for the Boltzmann collision operator with a constant angular cross section and kinetic parameter gamma in the interval from 0 to 1. That range recovers the classical case of hard spheres and the simplest case of Maxwell molecules, so the counterexample reaches the two physically standard models rather than an exotic corner. The examples are radially symmetric mixtures of Maxwellians.
Novelty check
The question is named, dated and attributed to McKean in 1966 in the paper's own opening, and the parameter range is chosen precisely so that the two classical models, hard spheres and Maxwell molecules, are covered. The paper separately flags the corresponding question for the Landau equation as still open, which shows the boundary of the claim is being drawn deliberately. No prior counterexample appears.
Caveats and known objections
A preprint days old at entry, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-led on the paper's disclaimer on the use of AI tools, which records a chain of tools and a substantial human rewrite: a first version of the proof was obtained by ChatGPT 5.6 Sol, run in Codex in Ultra mode with access to the source of one of the author's own earlier papers and other notes, and it was then rewritten first using Claude Code and afterwards reinterpreted and restructured by the author.
Also recorded at
vibemathedmckean-entropy-production-conjecture
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-led.
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Cite this entry
whataifound.org. (2026). Entropy production of the Boltzmann equation is not always monotone. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-09-01-mckean-entropy-productionBibTeX
@misc{whataifound-independent-2026-production,
title = {Entropy production of the Boltzmann equation is not always monotone},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-09-01-mckean-entropy-production}
}