Long-time correlations in a binary mixture: analysis of the nonlinearities of fluctuating-hydrodynamic equations

Abstract

Abstract The equations of fluctuating nonlinear hydrodynamics (FNH) following the proper conservation laws are considered for a binary mixture. We focus on the density (ρ) correlations’ renormalization due to the FNH equations’ nonlinearities. The consequence of density nonlinearities treated in simplest approximations gives rise to the well-studied form of the mode coupling theory (MCT) for a two-component system. The MCT predicts a sharp ergodicity–nonergodicity transition similar to the one-component fluid in this idealized form. In the first part of the present paper, we compare the predictions of the idealized MCT model with the computer simulation results for a hard sphere mixture. We show that there is clear disagreement in long-time dynamic behaviour. Next, we consider the full set of nonlinearities in the FNH equations using a Martin–Siggia–Rose field theory. From the time reversal properties of the correlation and response function of the associated field theory, a set of fluctuation–dissipation relations (FDR) are obtained. These FDRs impose constraints on the long-time behaviour of the correlation functions. Our non-perturbative analysis considers the viability of freezing the time correlations for the two-component fluid over the longest time scale. Due to the FDR constraints arising from the 1 / ρ nonlinearity in the FNH equations, a sharp ergodicity–nonergodicity transition for the binary mixture is not supported. If the 1 / ρ are replaced as 1 / ρ 0 in terms of the average density ρ 0, ad hoc, while the density-nonlinearities in the pressure term of the corresponding FNH equations are kept, the ideal transition model of the simplified MCT is recovered.