Generalization of Hydrodynamics to Include Single Particle Modes and Fluctuations

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Two exact hierarchies of hydrodynamic equations which follow from Newton's equations for systems of identical particles interacting with a spherically symmetric potential are presented. The first hierarchy, essentially given by Born and Green in 1947, is shown to be capable of predicting hydrodynamic fluctuations and therefore has important applications near the critical point. Explicit microscopic forms of the higher level stress tensors and heat fluxes are given. Detailed macroscopic approximations for these quantities are obtained using the isotropic symmetry of the fluid, reduction conditions which relate these quantities to the ordinary stress tensor and heat flux, and the known equilibrium limit of the hierarchy (the Yvon-Born-Green equation). The second hierarchy is a generalization of Born and Green's hierarchy to include time correlations. At equilibrium it yields an equation for the Van Hove correlation function, G(R, t), which contains both single-particle and collective modes and is the analog of the Yvon-Born-Green equation. From it and the self-part of the Van Hove correlation function, Gs(R, t), the distinct part of the Van Hove correlation function, Gd(R, t), may be obtained by numerical integration.