Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations

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Abstract
Stochastic trajectories are described that underly classical diffusion between known concentrations. The description of those experimental boundary conditions requires a phase space using the full Langevin equation, with displacement and velocity as state variables, even if friction entirely dominates the dynamics of diffusion, because the incoming and outgoing trajectories have to be told apart. The conditional flux, probabilities, mean first-passage times, and contents (of the reaction region) of the four types of trajectories—the trans trajectories LR and RL and the cis trajectories LL and RR—are expressed in terms of solutions of the Fokker–Planck equation in phase space and are explicitly calculated in the Smoluchowski limit of high friction. With these results, diffusion in a region between fixed concentrations can be described exactly as a chemical reaction for any potential function in the region, made of any combination of high or low barriers or wells.