Asymptotic Properties of Autocorrelation Functions and the Enskog Expansion in Three-Dimensional Simple Classical Fluids

Abstract

The ${t}^{\frac{\ensuremath{-}3}{2}}$ long-time behavior of the Green's-Kubo autocorrelation functions is the first term in an infinite series of general order ${t}^{\frac{1}{{2}^{n}\ensuremath{-}2}}$, $n$ integer \ensuremath{\ge} 1. The coefficients of these series for the shear and bulk viscosity and for the heat conductivity are given in terms of linear recurrence relation. Similarly, after the usual Navier-Stokes order (in ${k}^{2}$), there exists an infinite expansion for the frequencies of the hydrodynamical modes with terms of general order ${k}^{3\ensuremath{-}\frac{1}{{2}^{n}}}$. The mean square displacement of a particle in a fluid is given, for large times, by an infinite series, the first term being the well-known Einstein displacement $6Dt$, and the following ones proportional to ${t}^{\frac{1}{{2}^{n}}}$. As an application of this expansion of the hydrodynamical theory beyond the Navier-Stokes order, the pressure pattern in a weak shock wave is computed.