A theory of molecular velocity correlations in liquids

Abstract

It is proposed that the frequency spectrum (Fourier transform) of the single particle velocity correlation function for a classical liquid is of the form ρ (ω) = (ωc2−ω2)1/2f (ω) + (ωcω−ω2)1/2g (ω), where ωc is identified with the intermolecular collision frequency. The rationale for this construction is that the features of a cutoff frequency and nonanalytic points characteristic of the phonon spectrum of a solid should be reflected in the associated liquid: ωc defines the cutoff frequency and ρ (ω) is nonanalytic at ω=0. As a consequence, a general analytic expression for the correlation function as a superposition of Bessel functions is developed, whose damped oscillatory behavior parallels that deduced from neutron scattering experiments and molecular dynamics computer simulation studies of liquids. The Bessel function basis set for the correlation function implies an expansion of the frequency spectrum in terms of a Gegenbauer polynomial basis set, both of which can provide analytical tools for molecular dynamics investigations. For long times such that ωct≫1 the correlation function decays as t−3/2[A+B sin(ωct−π/4)]. The first term is identified with the shear viscosity from hydrodynamics, and it is argued that the second term should also exist as a residuum lattice contribution not deriveable from hydrodynamics. A specific two parameter model spectrum is considered to establish a connection between the present formulation and the Langevin theory of Brownian motion.