On Boltzmann’s H-Theorem

It is a classical result for solutions of the Boltzmann equation that Boltzmann’s H-function decreases monotonically with time, but ceases to decrease if the distribution function becomes locally Maxwellian. If, after a time, the system were to become precisely locally Maxwellian it would cease to be dissipative and the H-function (which then reduces to the negative thermodynamic entropy) would henceforth remain constant at a value higher than that appropriate to thermodynamic equilibrium. In order to demonstrate an approach to equilibrium, it is necessary to show not that the distribution approaches a Maxwellian, but that it does not approach a local Maxwellian too closely or too soon. In this paper we prove, for a general form of linear Boltzmann equation, that under appropriate boundary conditions all solutions approach equilibrium. The proof uses recent strong estimates of solutions of the linear Boltzmann equation.