A stochastic molecular model was presented for the diffusive motion of interacting particles in an external field of force, and effective partial differential equations and boundary conditions were derived that described a stationary non-equilibrium system. The interactions could include electrostatic, Lennard-Jones and other pair-wise forces. The analysis yielded a new type of Poisson–Nernst–Planck equation that involved conditional and unconditional charge densities and potentials. The conditional charge densities were non-equilibrium analogues of the widely studied pair correlation functions of equilibrium statistical physics. The proposed theory was an extension of the equilibrium statistical mechanics, of simple fluids, to stationary non-equilibrium problems. The proposed system of equations differed from the standard Poisson–Nernst–Planck system in 2 ways. Firstly, the force term depended upon conditional densities and therefore upon the finite size of ions. Secondly, it contained the dielectric boundary force on a discrete ion near to dielectric interfaces. It had been shown that both terms were important for diffusion through confined geometries; in the context of ion channels.

Ionic Diffusion through Confined Geometries - from Langevin Equations to Partial Differential Equations. B.Nadler, Z.Schuss, A.Singer, R.S.Eisenberg: Journal of Physics - Condensed Matter, 2004, 16[22], S2153-65