Generalized hydrodynamics for simple fluids

Abstract

Abstract Linear response theory is used to derive generalized nonlocal macroscopic equations which descibe the free relaxation of a system initially constrained to be in a state not far from its unperturbed equilibrium state. When the complete set of macroscopic variables is the set of hydrodynamic variables, it is then demonstrated that the matrix of coefficients of these generalized equations can rigorously be used to relate the entire matrix of time-dependent conserved-variable correlation functions to the matrix of their equal-time correlation functions. By restricting the regime of interest to be the hydrodynamic regime, where the conserved variables vary much more slowly than the remaining dynamical variables of the system, the conservation laws are used to develop a procedure for obtaining the terms in the matrix of coefficients as an expansion in powers of the wavevector, k , with explicit molecular expressions worked out for the terms through order k4. The expressions are shown to involve limits of generalized current-density correlation functions containing no conserved parts which are therefore well behaved for small frequencies. The forms of the macroscopic equations and the correlation functios implied by this order approximation to the matrix of coefficients are discussed in detail. Reasons are given why a double generalization, which considers both the time dependence and k dependence of the terms in the matrix, is necessary beyond order k2 in simple fluids.