Physical foundation and consistent formulation of atomic-level fluxes in transport processes

Irving and Kirkwood [J. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics, J. Chem. Phys. 48, 817 (1950)] derived the transport equations from the principles of classical statistical mechanics using the Dirac delta to define local densities. Thereby, formulas for fluxes were obtained in terms of molecular variables. The Irving and Kirkwood formalism has inspired numerous formulations. Many of the later developments, however, considered it more rigorous to replace the Dirac delta with a continuous volume-weighted averaging function and subsequently defined fluxes as a volume density. Although these volume-averaged flux formulas have dominated the literature for decades and are widely implemented in popular molecular dynamics (MD) software, they are a departure from the well-established physical concept of fluxes. In this paper, we review the historical developments that led to the unified physical concept of fluxes for transport phenomena. We then use MD simulations to show that these popular flux formulas conserve neither momentum nor energy, nor do they produce fluxes that are consistent with their physical definitions. We also use two different approaches to derive fluxes for general many-body potentials. The results of the formulation show that atomistic formulas for fluxes can be fully consistent with the physical definitions of fluxes and conservation laws.