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A popular approach to modeling bimolecular reactions between diffusingmolecules is through the use of reactive boundary conditions. One common modelis the Smoluchowski partial absorption condition, which uses a Robin boundarycondition in the separation coordinate between two possible reactants. Thisboundary condition can be interpreted as an idealization of a reactiveinteraction potential model, in which a potential barrier must be surmountedbefore reactions can occur. In this work we show how the reactive boundarycondition arises as the limit of an interaction potential encoding a steepbarrier within a shrinking region in the particle separation, where moleculesreact instantly upon reaching the peak of the barrier. The limiting boundarycondition is derived by the method of matched asymptotic expansions, and shownto depend critically on the relative rate of increase of the barrier height asthe width of the potential is decreased. Limiting boundary conditions for thesame interaction potential in both the overdamped Fokker-Planck equation(Brownian Dynamics), and the Kramers equation (Langevin Dynamics) areinvestigated. It is shown that different scalings are required in the twomodels to recover reactive boundary conditions that are consistent in the highfriction limit where the Kramers equation solution converges to the solution ofthe Fokker-Planck equation.

Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics