5 - Dynamics of collective modes

Summary

Theoretical developments on the dynamics of a dense liquid using a statistical-mechanical approach primarily involve a small set of slow collective densities termed hydrodynamic modes. The time scales of relaxation of these modes are much longer than those for the microscopic modes of the system. The basic approach adopted here is the analysis of the time correlation functions (introduced earlier in Chapter 1) of the slow modes. In the present chapter and the next two chapters we discuss microscopic methods for calculating the correlation functions involving the fluctuation or hydrodynamic approach. We focus primarily on the simplest type of correlation functions involving fluctuations at two different spatial and time coordinates. Owing to time translation invariance, equilibrium two-point correlation functions of hydrodynamic modes at the same time over different spatial points are time-independent and provide us with information on the thermodynamic behavior of the system. On the other hand, the dynamic behavior of the system is linked to the correlation of physically observable quantities at two different times. The time correlation function of density fluctuations is particularly important for our discussion of the slow dynamics in a liquid. In the simplest of the theoretical models, the decay of the correlation with time is exponential. We discuss here how such exponential relaxation behavior can be understood using linear dynamics of the fluctuations. The formalism developed in the later parts of this chapter allows in a natural way the extension of the macroscopic hydrodynamics to intermediate length and time scales, and is referred to as generalized hydrodynamics.