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 (Vsolute≈Vwater<