  This paper's objective is to improve the existing proof of the derivation of
the Rayleigh--Boltzmann equation from the nonideal Rayleigh gas [6], yielding a
far faster convergence rate. This equation is a linear version of the Boltzmann
equation, describing the behavior of a small fraction of tagged particles
having been perturbed from thermodynamic equilibrium. This linear equation,
derived from the microscopic Newton laws as suggested by the Hilbert's sixth
problem, is much better understood than the quadratic Boltzmann equation, and
even enable results on long time scales for the kinetic description of gas
dynamics.The present paper improves the physically poor convergence rate that
had been previously proved, into a much more satisfactory rate which is more
than exponentially better.
