On the spectral theory of Rayleigh's piston. II. The exact singular solution for unit mass ratio

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Abstract
For pt. I see abstr. A70752 of 1973. The authors have solved exactly the singular eigenvalue problem governing the relaxation of velocity for a one-dimensional ensemble of test particles in a heat bath of similar objects. This corresponds to the special case of Rayleigh's piston with mass ratio unity. The eigenfunctions prove to be singular schwartzian distributions involving a combination of delta functions and Hadamard pseudofunctions. No discrete eigenvalues occur other than the isolated point lambda =0, and, correspondingly, no regular eigenfunctions other than the equilibrium maxwellian, to which the whole continuum set is orthogonal. Orthogonality and completeness relations are established and the initial-value problem for the relaxation of an initial delta ensemble is considered. Speed relaxation proves relatively simple and a closed form can be derived for the time-dependent distribution function. Velocity relaxation is considerably more complex, but can be specified in terms of standard solutions to a Carleman-type integral equation.

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