7 - Renormalization of the dynamics

Summary

We have discussed the construction of the nonlinear Langevin equations for the slow modes in a number of different systems in the previous chapter. Next, we analyze how the nonlinear coupling of the hydrodynamic modes in these equations of motion affects the liquid dynamics. In particular, we focus here on the case of a compressible liquid in the supercooled region. In this book we will primarily follow an approach in which the effects of the nonlinearities are systematically obtained using graphical methods of quantum field theory. Such diagrammatic methods have conveniently been used for studying the slow dynamics near the critical point (Kawasaki, 1970; Kadanoff and Swift, 1968) or turbulence (Kraichnan 1959a, 1961a; Edwards, 1964). The present approach, which is now standard, was first described by Martin, Siggia, and Rose (1973). The Martin–Siggia–Rose (MSR) field theory, as this technique is named in the literature, is in fact a general scheme applied to compute the statistical dynamics of classical systems.

The field-theoretic method presented here is an alternative to the so-called memoryfunction approach. The latter in fact involves studying the dynamics in terms of non-Markovian linearized Langevin equations (see, for example, eqn. (6.1.1) which are obtained in a formally exact manner with the use of so called Mori–Zwanzig projection operators. This projection-operator scheme is described in Appendix A7.4). The generalized transport coefficients or the so-called memory functions in this case are frequency-dependent and can be expressed in terms of Green–Kubo forms of integrals of time correlation functions.