In a 1966 paper (Speed of Approach to Equilibrium for Kac's Caricature of a Maxwellian Gas, Arch. Rational Mech. Anal., Vol. 21), McKean seems to suggest that the successive derivatives of entropy $H (f) = - \int f \log f $ of a probability density $f$ along the heat flow have alternating signs. Is there anything proved today in this respect?
2 Answers
McKean's conjecture was proven already in 1967 by Stewart Harris, see Proof that Successive Derivatives of Boltzmann's H Function for a Discrete Velocity Gas Alternate in Sign.
See also On the sign of successive time derivatives of Boltzmann's H function (1970); Do the higher derivatives of the H-function alternate in sign? (1983); Alternating signs of the higher derivatives of the H- function for a nonlinear model Boltzmann equation (1985).
For scalar random variables, the signs of the first four derivatives have been shown to be alternating. This is in "Higher order derivatives in Costa’s entropy power inequality" by Fan Cheng and Yanlin Geng. The question for higher derivatives seem open to the best of my knowledge.
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$\begingroup$ Do you know the name of the paper? $\endgroup$ Commented Feb 19, 2023 at 13:27
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$\begingroup$ ieeexplore.ieee.org/document/7274074 $\endgroup$– nc78Commented Feb 19, 2023 at 13:46