Irving and Kirkwood (IK) 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 as point functions in terms of molecular variables. The IK formalism has inspired numerous formulations. Many of the later developments, however, considered it more rigorous or convenient to replace the Dirac delta with a continuous volume-weighted averaging function and subsequently define fluxes as a volume density. Although these volume-averaged (VA) 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 talk, we review the historical developments that led to the unified physical concept of transport fluxes. We then use MD simulations to show that replacing the Dirac delta with a volume weighting function changes the fundamental nature of fluxes as a surface density. As a result, the dynamic balance between the change of the total momentum or energy in a volume element and the fluxes across the enclosing surface  cannot be established. This leads to the failure of VA flux formulas in satisfying the momentum and energy conservation laws or typical transport boundary conditions. We also use two different approaches to derive fluxes for general many-body potentials. The results show that atomistic formulas for fluxes can be fully consistent with the physical definitions of fluxes and conservation laws.

On the physical foundation and consistent formulation of atomic-level fluxes in transport processes

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