Exact Stationary and Non-stationary Solutions to Inelastic Maxwell Model with Infinite Energy

Abstract

The one-dimensional inelastic Boltzmann equation with a constant collision rate (the Maxwell model) is considered. It is shown that for special values of restitution parameter there exists a stationary solution with the characteristic function in the form \(e^{-P(\log (z))z},\) where P is a periodic function. The corresponding distribution function belongs to a one special class of stochastic processes termed as a generalized stable in the probability theory. The Fourier transform of the non-stationary equation has the solution \(\bigl (1+P(\log (z))z\bigr )e^{-Q(\log (z))z}\). It is proved that this solution is a characteristic function if periodic functions P, Q satisfy some not very restrictive conditions. The stationary and non-stationary solutions correspond to a gas with infinite temperature.