Sound attenuation and dispersion in a liquid-gas system near the critical point: Renormalization-group analysis

We generalize the usual hydrodynamic equations of motion to include mode-coupling terms and a Landau-Ginzburg free energy. We perform a renormalization-group analysis of these equations which includes a mode-coupling term between the longitudinal current fluctuations and the order-parameter entropy density. Coupling to the pressure fluctuations is also included by a speed-of-sound term. We calculate from the generalized nonlinear Langevin equation the sound attenuation and dispersion using two different techniques. The first approach utilizes a memory-function expansion for the longitudinal-current generalized transport coefficient ${\Gamma }_l(\omega , \xi )$. By keeping only the lowest-order mode-coupling term, we reproduce the early results of Fixman and Kawasaki. The second technique is a calculation of ${\Gamma }_l(\omega , \xi )$ in a $\frac{1}{n}$ expansion which takes into account both the mode-coupling and time-dependent Ginzburg-Landau vertices. This calculation gives improved agreement with experiment for the high-frequency sound attenuation.