On the sign of successive time derivatives of Boltzmann's H function

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Abstract
It has been conjectured in the literature that for the Boltzmann equation: (i) Boltzmann's H has the property of possessing successive (semi-) definite time derivatives with alternating sign, (ii) this property is suitable for selecting H from further possibly existing (Lyapunov) functions with (semi-) definite first time derivative. The authors show at first that, in contradiction to (ii), for Maxwell's model of a discrete velocity gas the corresponding variant of H is not the unique Lyapunov function having the property (i) and suitable to define a 'non-equilibrium entropy' with respect to the (spatially homogeneous) Boltzmann equation of this model. Secondly, in the case of the complete spatially inhomogeneous Boltzmann equation, H is generally not (semi-)definite in contradiction to the above-mentioned conjecture (i).

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