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Rare events out of equilibrium: mobility through the liquid–liquid interface

Transition state theory provides a well established means to compute the rate at which rare events occur; however, this is strictly an equilibrium approach. Here we consider a nonequilibrium problem of this nature in the form of transport through a liquid–liquid interface. When two immiscible liquids are coexisting in equilibrium, there will be a certain amount of mixing between the two phases, resulting in a finite linear mobility across the liquid–liquid interface. We derive an exact relationship between the mobility and the local diffusion in the direction perpendicular to the interface. We compute the mobility using both nonequilibrium molecular dynamics and a variety of linear response type approaches, with accurate agreement being obtained for the best of these. Our analysis makes it clear how the local diffusion is influenced by the inhomogeneities of the interface, even when at a distance from it. This nonlocal character to the mobility has not been appreciated before and results in a strong variation in the local diffusion, which is formally coupled to the variation in the potential of mean force. The nonlocal aspect of the diffusion requires the velocity autocorrelation function to be integrated out to far longer times than is the case for homogeneous liquids, and requires special care with regard to the choice of numerical approach.