Mean first passage times across a potential barrier in the lumped state approximation

The lumped state approximation ~LSA! is a method for handling boundary conditions for diffusion on an interval which simplifies the description of transitions into and out of the interval. It was originally motivated by the problem of proton conduction through the ion channel gramicidin. This paper discusses the mean first passage time of a diffuser crossing a potential barrier in the lumped state approximation. The LSA mean first passage time is shown to be identical to a different quantity, the interior mean first passage time, clarifying the nature of the approximation. We also discuss a variant of the LSA in which dependence on an applied electrical potential is made explicit; an optimal value for an effective electrical distance is found. A detailed comparison is made of the LSA mean first passage time with several other formulations of the mean time to cross a barrier. © 2001 American Institute of Physics. @DOI: 10.1063/1.1330215#