Fluctuation-dissipation framework for size-dependent surface tension

The size-dependent liquid-vapor surface tension controls phase change, wetting, and transport at nanoscales, yet its first curvature correction, the Tolman length, remains difficult to determine. We develop a thermodynamic and statistical-mechanical framework that relates this correction to bulk response properties of a one-component liquid near liquid-vapor coexistence. For curved interfaces, the analysis considers two local formulations of the same capillary-chemical balance, in excess pressures and in relative density deviations. For weakly compressible liquids in the regime emphasized here, the adopted asymmetric density-based formulation is the practically relevant one, with finite-curvature effects entering through vapor supersaturation under capillary equilibrium. At coexistence, the planar-limit value of the same Tolman length reduces to a combination of the liquid isothermal compressibility and its pressure derivative, which can be recast as a bulk fluctuation-response observable of the homogeneous liquid in the isothermal-isobaric ensemble. In this representation, the planar-limit coefficient is determined by second and third central moments of the volume distribution, equivalently by the pressure response of the relative fluctuation width. For water, homogeneous (N, P, T) simulations of the extended simple point charge and TIP4P/2005 water models sample the bulk liquid, not an explicit liquid-vapor interface, and yield estimates near -0.7 Å at 300 K. An independent evaluation based on the IAPWS-IF97 industrial formulation gives -0.713 ± 0.004 Å at the same coexistence state and predicts a weakly nonmonotonic temperature dependence along coexistence. Beyond water, the framework applies to other one-component liquids in regimes where an accurate thermal equation of state or sufficiently converged bulk volume statistics is available.