Hydrodynamic Forms of Correlation Functions and Generalised Hydrodynamics

Abstract

The simple models evaluated in the preceding chapter drew attention to some of the basic features of time-dependent correlations in fluids. These are: (a) the meaning of their Fourier transforms as frequency spectra of dissipation by suitable external probes; (b) their relation to macroscopic fluid dynamics (i.e. hydrodynamics) and to transport coefficients in the limit k→0 and ω→0; and (c) their short-time behaviour as expressible through moments of the frequency spectrum. The most convenient way to develop a theory of time-dependent correlations whose structure incorporates these properties, and thus allows us to draw conclusions on the dynamics of liquids, is through the response function and memory function approach (Mori, 1965a; see also Kubo, 1966). It will be instructive to introduce this formalism through a study of the simplest time-dependent correlation function, which is the velocity autocorrelation function for a particle in a classical fluid, before proceeding to the discussion of S(k, ω).