The pressure tensor across a liquid-vapour interface

Inhomogeneous fluids exhibit physical properties that are neither uniform nor isotropic. The pressure tensor is a case in point, key to the mechanical description of the interfacial region. Kirkwood and Buff and, later, Irving and Kirkwood, obtained a formal treatment based on the analysis of the pressure across a planar surface [J. G. Kirkwood and F. P. Buff, J. Chem. Phys. 17(3), 338 (1949); J. H. Irving and J. G. Kirkwood, J. Chem. Phys. 18, 817 (1950)]. We propose a generalisation of Irving and Kirkwood's argument to fluctuating, non-planar surfaces and obtain an expression for the pressure tensor that is not smeared by thermal fluctuations at the molecular scale and corresponding capillary waves [F. P. Buff et al., Phys. Rev. Lett. 15, 621-623 (1965)]. We observe the emergence of surface tension, defined as an excess tangential stress, acting exactly across the dividing surface at the sharpest molecular resolution. The new statistical mechanical expressions extend current treatments to fluctuating inhomogeneous systems far from equilibrium.