Exact solution of Kramersʹ problem for piecewise parabolic potential profiles

This paper presents a general procedure for deducing exact time characteristics of Brownian diffusion in the high damping limit in piecewise parabolic pontential profiles. Our approach to solve this problem is based on the Laplace transform method for obtaining the desired time characteristics. For some concrete examples of piecewise parabolic profilesʹ exact values of metastable state decay times and bistable system relaxation times are found. These results are valid for any height of the potential barrier and coincide with Kramerʹs results for a high potential barrier. Conditions for the validity of Kramerʹs formula are also derived.