Kac 方程的熵产生并非总是单调的
The entropy production is not always monotone for the Kac equation
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中文总结 AI 辅助
本文否证了 McKean 关于 Kac 方程熵产生单调性的猜想，通过构造光滑正偶的高斯混合反例，并证明高阶熵导数符号也会违背预期。
中文摘要 AI 辅助
1966年，McKean 猜想 Kac 的麦克斯韦气体简化模型（Kac caricature）的熵产生随时间非增 \cite{McKean66}。我们针对标准均匀角 Kac 方程否证了这一猜想。我们的反例是光滑、严格正、偶的，由两个不同温度的高斯分布混合而成。我们还证明，沿着同一解，除一阶外的每个熵导数在某一时刻都会违背其预期符号。初始密度可选取为单位质量和二阶矩。
英文摘要
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.
