Oscillating and absolute Maxwellians: Exact solutions for (d>1)-dimensional Boltzmann equations

Abstract

A study of (d>1)-dimensional nonlinear Boltzmann equations is made for which both momentum and energy conservations hold. Maxwell particles in the presence of outside forces are assumed. For either linear spatial force or linear velocity force plus source term, a class of exact solutions for the homogeneous and inhomogeneous distributions is determined. In particular, a look is taken at oscillating external forces with a varying parameter and distributions relaxing towards oscillating Maxwellians are found. On the other hand, exact inhomogeneous distributions relaxing towards absolute Maxwellians are obtained. The exact solutions have asymptotic regimes that are solutions of the linear part of the Boltzmann equations and the relaxations towards these regimes are studied. Furthermore when equilibrium absolute Maxwellian states exist, whether the relaxation towards these states are from above or from below is investigated and the possibility of finding an overpopulation of high velocity particles at intermediate times is looked at.