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

YP Chen and A Diaz, PHYSICAL REVIEW E, 98, 052113 (2018).

DOI: 10.1103/PhysRevE.98.052113

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.

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