Equilibrium Time Correlation Functions in the Low-Density Limit

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).