a Renormalization Group Calculation on Large-Scale Long-Time Properties of Fluctuations in Compressible Fluids.

Dynamic renormalization group methods are applied to investigate the large-scale, long-time properties of the spontaneous fluctuations which occur in compressible fluids near equilibrium. For simplicity, temperature fluctuations are omitted. The investigation is based on a non-linear version of fluctuating hydrodynamics driven by stochastic forces, for which we give a new derivation. We impose a fluctuation-dissipation theorem, and show that it is stable under shell-integration. The model is Galilean invariant, and we derive a Ward identity which leads to the non-renormalization of the most important vertex. We derive and solve the renormalization flow equations for the longitudinal and transverse viscosities, and derive asymptotic results for the correlation and memory functions of this system. In three dimensions, all renormalizations are finite, but we derive non-trivial, non-analytic long time tail corrections to the conventional hydrodynamic results. We do not, however, find the higher order Pomeau corrections to the sound dispersion relation, and we argue that these are spurious. In and below two dimensions we show that the transport coefficients diverge as a result of mode interactions, and that their ratio approaches a universal value. We also derive analytic expressions for the correlation functions in the asymptotic region of small frequency and wave-number. Finally, we investigate the limit when the sound speed becomes large. We find that in this limit, our theory leads to the purely transverse FNS theory for incompressible fluids, decoupled from a linear ghost theory for the longitudinal modes.