A Fokker–Planck equation for a piecewise entropy functional defined in different space domains. An application to solute partitioning at the membrane–water interface

A nonlinear Fokker–Planck equation is proposed for a system subject to different statistics (in the present study, the Gibbs–Boltzmann and Fermi–Dirac statistics) defined in different contiguous regions of space. We solved the time-dependent mono-dimensional equation numerically, and solved the time-independent mono-dimensional equation analytically under the effect of a generic external potential equation. These zones are connected by a sharp but continuous transition region. Accurate numerical procedures ensure the convergence of the Fokker–Planck equation in the transition layer. We applied our general procedure to investigate both the stationary and the time-dependent kinetics of solute partitioning between aqueous and membrane phases. Because of the relative volumes of solute, water, and lipid ( V s o l u t e ≈ V w a t e r < < V l i p i d ), the mixing entropy functional in water contains both solute and solvent contributions, while within the membrane only the solute entropy plays a significant role. Also, the potential term differs in space due to the solute interactions with different environments. Lastly, we added the effect of an electrostatic potential (occurring in all membrane systems) localized at the water interface which may deplete or increase the interfacial solute concentration. Rather surprisingly we found a strong coupling between the surface potential and the imposed asymmetric statistics. The effects are relevant when entropic and potential contributions are comparable; otherwise, the standard Boltzmannian behavior is recovered.