6 - Nonlinear fluctuating hydrodynamics

Summary

The fluctuating-hydrodynamics approach discussed earlier takes into account only the transport properties at the level of completely uncorrelated motion of the fluid particles. The corresponding dissipative processes are expressed in terms of bare transport coefficients of the fluid. The strongly correlated motion of the fluid particles which occurs at high density is not take into consideration here. This is reflected through the Markov approximation of the transport coefficients and the short correlation of the corresponding noise representing the fast degrees of freedom in the system. The Markovian equations for the collective modes involving frequency-independent transport coefficients constitute a model for the dynamics of fluids with exponential relaxation of the fluctuations. The corresponding equations of motion for the collective modes are linear. However, exceptions occur in certain situations in which the description of the dynamics cannot be reduced to a set of linearly coupled fluctuating equations with frequency-independent transport coefficients. In this chapter we will consider the nonlinear dynamics of the hydrodynamic modes for studying the strongly correlated motion of the particles in a dense fluid.

Nonlinear Langevin equations

We present in this section the formulation of a set of nonlinear stochastic equations for the dynamics of the many-particle system. We first discuss the physical motivation for extension of the fluctuating-hydrodynamics approach to include nonlinear coupling of the slow modes.