  A popular approach to modeling bimolecular reactions between diffusing
molecules is through the use of reactive boundary conditions. One common model
is the Smoluchowski partial absorption condition, which uses a Robin boundary
condition in the separation coordinate between two possible reactants. This
boundary condition can be interpreted as an idealization of a reactive
interaction potential model, in which a potential barrier must be surmounted
before reactions can occur. In this work we show how the reactive boundary
condition arises as the limit of an interaction potential encoding a steep
barrier within a shrinking region in the particle separation, where molecules
react instantly upon reaching the peak of the barrier. The limiting boundary
condition is derived by the method of matched asymptotic expansions, and shown
to depend critically on the relative rate of increase of the barrier height as
the width of the potential is decreased. Limiting boundary conditions for the
same interaction potential in both the overdamped Fokker-Planck equation
(Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are
investigated. It is shown that different scalings are required in the two
models to recover reactive boundary conditions that are consistent in the high
friction limit where the Kramers equation solution converges to the solution of
the Fokker-Planck equation.
