A critical analysis of the current situation with the description of non-equilibrium transport processes allows us to conclude that in order to describe non-equilibrium processes it is necessary to go to a deeper level of description in comparison with the average, macroscopic one. Such a description based on the so called “first principles” is provided by non-equilibrium statistical mechanics. The goal of statistical mechanics is to create a consistent and efficient formalism for describing the macroscopic behavior of many-particle systems based on microscopic theory. Non-equilibrium statistical mechanics provides approaches and tools for describing irreversible processes in real systems within the framework of a unified theoretical method allowing one to calculate, albeit approximately, the transport coefficients that characterize the system relaxation to equilibrium.
The most constructive result in non-equilibrium statistical mechanics is the proof of the fact that the equations describing the behavior of a non-equilibrium thermodynamic system in terms of an incomplete set of variables can no longer be purely differential that is, local in space and time. A fruitful statistical-mechanical method for describing non-equilibrium processes of mass, momentum and energy transport was proposed by D.N. Zubarev [1-3]. Here we briefly consider the method and the main results obtained in its framework. The space-time integral relationships between the conjugate fluxes and macroscopic gradients of momentum and energy densities obtained on the basis of the non-equilibrium statistical operator method contain relaxation transport kernels which generalize the transport coefficients to non-equilibrium conditions.
Integro-differential equations of nonlocal hydrodynamics generalize the equations of classical hydrodynamics to non-equilibrium processes beyond the validity of continuum mechanics. A characteristic feature of the new description is the preservation of integral information about the system in the generalized hydrodynamic equations when describing its local properties. The effects of the spatiotemporal nonlocality are the price that one has to pay for the inevitable incompleteness of the macroscopic description of non-equilibrium processes in open systems. The nonlocal transport equations with memory will serve as the basis for the new approach proposed in Chapter 5 of this book which will be able to describe highly non-equilibrium processes.