Exact Generalization of Local Thermodynamics of Inhomogeneous Fluids to   Arbitrary Spatial Scales
  In order to provide a formally correct thermodynamical description of inhomogeneous fluids valid on all length scales down to the classical limit we postulate that all extensive quantities have locally extensive analogues. We derive local thermodynamic relations for the canonical and isothermal- isobaric ensembles without limitations with respect to the nature of intermolecular interactions. The inhomogeneity of the system is shown to result in two distinct entropic contributions. The intrinsic contribution is due the specific distribution of local environments experienced by a single molecule while the extrinsic one originates exclusively from the local variation of the density from its bulk value. 
Alja\v{z} Godec, Janko Jamnik, Franci Merzel
cond-mat.stat-mech