On the role of the energy loss in turnover theories of activated rate processes

Variational theory of activated rate processes for an arbitrary barrier

The thermally activated escape of a Brownian particle over a smooth barrier of arbitrary shape and height is considered as an eigenproblem of the Fokker-Planck equation. For the case of moderate and large friction, the least nonzero eigenvalue of this equation is found via a Rayleigh-quotient-based perturbation method. A comparison with existing variational approaches and from numerical simulations for bistable potentials with parabolic and quartic barriers shows that the proposed expression gives unprecedentedly accurate results at all barrier heights including the limit of vanishingly low barrier.

Self-similar renormalization approach to barrier crossing processes

An algebraic self-similar renormalization method developed recently for summation of divergent field-theoretical series is applied to the thermally activated escape of a Brownian particle over an arbitrarily shaped barrier. Based on the Mel'nikov-Meshkov result for the underdamped Brownian motion and the inverse friction expansion of the underlying Fokker-Planck equation for strong friction, an overall rate formula is constructed. This formula agrees in the weak friction regime with the rate obtained from a diffusion equation in energy variables and, in the limiting case of strong friction with the rate following from a Smoluchowski equation. Its validity is tested for Brownian motion in bistable potentials with parabolic, cusped, and quartic barriers of different heights. The proposed formula is found to give a reasonable description of activated rate processes even though the barrier is quite low. Our comparison also includes results from various different crossover theories. In most of the cases considered the present formula is in considerably better agreement with exact numerical rates than the other interpolation formulas.