Kramers' escape problem serves as a paradigm for understanding transitions and various types of reactions, such as chemical and nuclear reactions, in systems driven by thermal fluctuations and damping, particularly within the realm of classical variational dynamics. Here, we generalize Kramers' problem to processes with non-normal dynamics characterized by asymmetry, hierarchy, and stochastic fluctuations, where transient amplification and stochastic perturbations play a critical role. The obtained generalized escape rates are structurally similar to those for variational systems, but with a renormalized temperature proportional to the square of the condition number $\kappa$ which measures non-normality. Because $\kappa$ can take large values (e.g., $10$ or more) in many systems, the resulting acceleration of transition rates can be enormous, given the exponential dependence on the inverse temperature in Kramers' formula. As an illustration, our framework offers a potential resolution to the long-standing question in epigenetics of how DNA methylation can proceed with such remarkable speed. By uncovering the role of strong non-normal dynamics, we derive a robust, universal expression for methylation timing that predicts fast, switch-like transitions, on the order of minutes, for a wide range of noise amplitudes and methylation levels, and in agreement with experimental observations. More broadly, our findings indicate that non-normal accelerated escape rates are likely to be a key mechanism underlying rapid transitions in diverse systems, from biological metabolism and ecosystem shifts to climate dynamics and socio-economic processes.
Comment: main text of 7 pages and 16 pages of supplementary calculations; new revised version including an improved derivation + a full derivation of DNA methylation rates in epigenics