Propagating density fluctuations in hydrodynamics and beyond

Density fluctuations in simple liquids are analysed in the context of three different and widely used formalisms, whose equivalence in the hydrodynamic limit is shown. We, furthermore, address the issue of the dispersion of the propagating modes outside the hydrodynamics, by comparing three different definitions of the generalized sound velocity. The first definition is standard in statistical mechanics. It relates the sound velocity to the imaginary part of the complex conjugate poles of the so-called intermediate scattering function. Other definitions, frequently used in the literature, identify the characteristic frequencies of the inelastic excitations with the maxima of the inelastic features of the dynamic structure factor, or with the maxima of the current function. The behaviour of these three quantities in the hydrodynamic limit is discussed. Deviations from hydrodynamic dispersion law are also considered with particular emphasis given to the analysis of different sound propagation regimes related to different density fluctuations decay channels.