Exact transform solution of the one-dimensional special Rayleigh problem

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Abstract
We examine the full-range initial-value problem for velocity relaxation in a one-dimensional ensemble of Rayleigh test-particles in an arbitrary heat-bath at unit mass-ratio (the special Rayleigh problem). A Laplace transformation method is used to obtain an explicit solution in two independent parity components. Of these, the even component, giving the speed relaxation is obtained exactly and agrees with our previous result obtained by the singular eigenfunction method; the odd component, explicit to within a Laplace inverse provides the time-dependent flux of particles with given velocity. By numerical inversion of the latter, we obtain both time-dependent velocity distributions and the velocity autocorrelation function for equilibrium fluctuations. In keeping with the singular character of the transport equation, the velocity ACF proves to be appreciably non-exponential in character, behaving asymptotically as t−5/2e−t independently of the heat-bath distribution. The complex admittance of a system of charged Rayleigh test-particles is also derived and shown to lead to expected behaviour for dissipation to the heat-bath and phase-lag of response to the applied field.

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