A numerical study of the general Rayleigh's piston model

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Abstract
The authors describe a detailed numerical investigation of the spectrum ( lambda k, lambda ) of the Rayleigh singular integral operator Agamma governing the one-dimensional test-particle gas at mass ratio gamma (Rayleigh's piston). Results confirm that the discrete spectrum lambda k( gamma ) is empty apart from the equilibrium eigenvalue, lambda 0=0, in the range gamma >0.28..., but acquires new points in a regular manner as gamma decreases. Thus the discretum interval lambda k in (0,1) is gradually filled to ever-increasing density as the Brownian motion regime gamma <<1 is reached. Spectra and eigenfunctions, together with derived results for the velocity autocorrelation function and complex admittance for charged test particles, are used to illustrate the effectiveness of the Rayleigh-Fokker-Planck approximation based on the artificial assumption gamma to 0.