Proof that Successive Derivatives of Boltzmann's <i>H</i> Function for a Discrete Velocity Gas Alternate in Sign

Abstract

It has been conjectured by McKean that the particular property of Boltzmann's H function which singles it out from a wide class of functionals of the Boltzmann solution may be that its successive derivatives alternate in sign. We consider here the proof of this alternating property for a discrete velocity gas. For the linearized-model Boltzmann equation, the proof is trivial. For the full (i.e., nonlinear) model Boltzmann equation, the proof is shown to be equivalent to demonstrating the positivity of a particular polynomial. The proof of this property is then demonstrated. It is also shown that H(n), like H(1), is zero only for the equilibrium distribution.