Correlation Functions for Dilute Systems

Space- and time-dependent correlation functions are considered for particles undergoing Brownian motion and for a low-density gas described by the Boltzmann equation. A Fourier transform is performed with respect to the space coordinates, and a Laplace transform with respect to time, the transform variables being k and −z. It is shown that the transformed correlation functions contain a hydrodynamic pole at z = p(k), where p(k) is a dispersion relation obtained from the hydrodynamic equations. In addition, there are singularities corresponding to microscopic relaxation. For the case of Brownian motion, these are a sequence of poles accumulating at infinity. For the Boltzmann equation, there is, for k → 0, a branch cut along part of the positive real axis, while for finite k the branch cut opens out into a singular region within which the correlation functions are undefined.