A microscopic perspective on the physical foundations of continuum mechanics--II. A projection operator approach to the separation of reversible and irreversible contributions to macroscopic behaviour

摘要

Reproducible macroscopic behaviour (at some pair of length-time scales), for a confined material system with passive environment, is analysed in terms of a projection operator P on the space of functions f of microscopic state. When f represents a local (macroscopic) spatial average, assumptions of local equilibrium and dynamic ergodicity establish Pf as the corresponding reproducible space-time average. Time evolution of the macroscopic probability density, induced by the microscopic probability density associated with partial initial information, is shown to equal the sum of explicit time-reversible and time-irreversible contributions via an analysis involving δ-function formalism and operators defined in terms of P, its complementary projection, and the Liouville operator.