The Rayleigh model: singular transport theory in one dimension

We present a comprehensive account of the special ‘Rayleigh piston’ model for the spatial and velocity relaxation of an ensemble of labelled test-particles in a one-dimensional heat-bath of particles with identical mass. This model, originally formulated by Rayleigh in 1891 but since largely neglected, is in effect a prototype for all later models in singular particle transport theory and serves to illustrate the mathematical problems associated with the occurrence of singular eigenfunctions and continuous spectra of a scattering operator. Although other idealized scattering models are known, the Rayleigh model remains a unique example of an exactly soluble singular system which, in including conservation laws and time-reversal symmetry in scattering, retains a degree of mechanical realism.

Non-linear Brownian movement of a generalized Rayleigh model. II. Extension of the model to include ‘sticky’ collisions

This paper is a sequel to an earlier one by Alkemade, van Kampen & MacDonald (1963) analyzing the statistical behaviour of a non-linear generalized Rayleigh piston model. In the previous paper, the collisions of the piston with surrounding gas molecules were ideally classical and instantaneous. Under these circumstances the piston may always be said to be loosely coupled to its environment. In the present paper another type of collision is introduced which provides a degree of strong coupling between piston and environment, but which still permits us to analyze the behaviour starting from a master equation and using a Markovian standpoint. The expansion of the master equation and the analysis for both models are carried to one higher order of non-linearity than was done in the previous paper; in particular this enables us to discuss the calculation of the auto-correlation function (and thence the power spectrum) beyond the purely linear approximation. More generally, interesting differences between the behaviour of the two models appear as soon as we proceed beyond the linear approximation. This 'sticky collision', or 'sticky piston', model may also be of some value as a model for a liquid.