Stress Tensor in Fundamental Measure Theory of Hard-Sphere Fluids: A Thermomechanical Approach
We derive the local pressure tensor generated by the original Rosenfeld fundamental measure theory (FMT) for hard-sphere fluids from an arbitrary infinitesimal deformation. The variation of a finite-distance FMT kernel is written via the line integrals of the local deformation gradient along the straight segment connecting its endpoints. The resulting symmetric pressure tensor consists of a local term expressed through the standard FMT weighted densities and a finite-range chord term. For a monodisperse hard-sphere fluid, differential relations between the Rosenfeld measures reduce the tensor to a closed one-body form. We demonstrate that no independent pair density or inhomogeneous Ornstein--Zernike equation is required. In the limit of homogeneous fluid the tensor reproduces the Percus--Yevick compressibility equation of state. In planar confinement the normal stress satisfies the hard-wall contact relation, the mechanical surface tension agrees with the surface excess grand potential, and the disjoining pressure agrees with the derivative of the confined grand potential with respect to slit width. The finite-range chord term extends the deformation method from local field theories to nonlocal weighted-density functionals.