Soluble Boltzmann equations for internal state and Maxwell models

Abstract

We consider a class of scalar nonlinear Boltzmann equations describing the evolution of a microcanonical ensemble in which sub-systems exchange internal energy ‘randomly’ in binary interactions.

In the continuous variable version these models can equally be interpreted as Boltzmann equations for Maxwell type molecules in arbitrary dimensionality.

We construct general solutions in the form of a Fourier series; the expansion coefficients (Sonine or Meixner moments) satisfy the same recursive system of coupled equations as the ordinary moments, and can be solved sequentially.