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.
Nonlinear equation of correlation function of galaxies in an expanding universe and the solution in linear approximation
