On the spectral theory of Rayleigh's piston. III. Exact solution of the absorbing barrier problem ( =1)

Abstract

For pt.II see ibid., vol.7, p.1070 (1974). The authors derive an exact solution for the evolution of a one-dimensional ensemble of Rayleigh test particles at unit mass ratio ( gamma =1) in the presence of a speed absorbing barrier. The result, which entails an Arrhenius-law dependence of mean first-passage time on barrier height, appears to be a hitherto unique example of a singular passage-time problem exactly soluble through a continuous spectrum of eigenvalues and their associated eigen distributions.