The Kramers Problem in the Turnover Regime: The Role of the Stochastic Separatrix

Publisher Website
Google Scholar
Abstract
We consider the problem of activated escape of a Brownian particle from a potential well. We find the stochastic separatrix S (the locus of starting points of the phase space trajectories which have equal probabilities of ending up inside or outside the well) for (i) the extremely anisotropic overdamped motion of a two‐dimensional Brownian particle in a bistable potential, and (ii) the damped and underdamped motion of a one‐dimensional Brownian particle in a single metastable state. The significance of S is that (1) it defines the reactant and product wells in a natural though not necessarily intuitive way, and (2) it reduces the calculation of the escape rate to the solution of the stationary Fokker‐Planck equation inside S, with absorbing boundary conditions on S. Finally, employing this approach we derive an expression for the Kramers escape rate which bridges uniformly between Kramers weak damping regime and transition state theory.