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.
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Ilyin, O. Exact Stationary and Non-stationary Solutions to Inelastic Maxwell Model with Infinite Energy. J Stat Phys 165, 755–764 (2016). https://doi.org/10.1007/s10955-016-1643-4
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DOI: https://doi.org/10.1007/s10955-016-1643-4