A potential distribution induced mapping of free energies for simple fluids

The potential distribution theorem (PDT) is utilized to construct an effective density, the pseudo-density ρ p s e u d o ( z ) , that enables mapping of the free energies of the uniform fluid exactly onto the nonuniform system values. In addition, a similar quantity, the pseudo-chemical potential μ p s e u d o ( z ) , is given as the chemical potential produced by the uniform equation of state upon using the nonuniform density ρ w ( 1 ) ( z ) as input. The PDT connects three quantities: the work W i n s ( z ) for inserting a test particle into the fluid, the chemical potential μ 0 of the bulk fluid, and the nonuniform singlet density ρ w ( 1 ) ( z ) . We perform Metropolis NVT ensemble Monte Carlo (MC) simulations to obtain the insertion work W i n s ( z ) (via Widom’s particle insertion) and the densities ρ w ( 1 ) ( z ) . We illustrate the mapping on two simple fluids adsorbed on a hard wall: the Lennard-Jones and the attractive Yukawa fluids. The pseudo-density is determined via an accurate uniform-fluid equation of state for the Lennard-Jones system, and for the Yukawa fluid via direct MC simulations. We characterize the behavior of the effective density and the pseudo-chemical potential vis-à-vis the cases of enhancement and depletion of the fluid density near the wall. These quantities ( ρ p s e u d o & μ p s e u d o ) are found to exhibit for enhanced adsorption out-of-phase oscillations compared to ρ w ( 1 ) ( z ) and β W i n s ( z ) . For depleted adsorption, we do not observe oscillations and the trends of ρ p s e u d o and μ p s e u d o are in good agreement with those of ρ w ( 1 ) ( z ) and β W i n s . We analyze the differences in behavior in terms of the concavity of the chemical-potential function. We also show the equivalence of the potential distribution theorem to the Euler–Lagrange equation of the density functional theory.