Hard-sphere fluids with chemical self-potentials

The existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container Λ- ^R3 and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at r{small element of}Λ is the sum of the matter reservoir's contribution and a self-contribution -(V*ρ)(r), where ρ is the fluid density function and V a non-negative linear combination of the Newton kernel ^VN({pipe}r{pipe})=- ^{{pipe}r{pipe}-1}, the Yukawa kernel ^VY({pipe}r{pipe})=- ^{{pipe}r{pipe}-1e-κ{p ipe}r{pipe}}, and a van der Waals kernel ^VW({pipe}r{pipe})=- ^{(1+κ{script}2{p ipe}r{pipe}2)-3}. The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic partial differential equations (PDEs) of second order with a nonlinear, nonlocal boundary condition. We prove the existence of a grand canonical phase transition and of a petit canonical phase transition which is embedded in the former. The proofs suggest that, except for boundary layers, the grand canonical transition is of the type "all gas↔all liquid" while the petit canonical one is of the type "all vapor↔liquid drop with vapor atmosphere." The latter proof, in particular, suggests the existence of solutions with interface structure which compromise between the all-liquid and all-gas density solutions.