Abstract
We study the activated-barrier-crossing (ABC) problem using the Hamiltonian approach with general memory friction kernels and for a parabolic barrier joined to an infinite wall. We solve the problem using the Grote-Hynes (GH) theory and the more recent Pollak-Grabert-H\"anggi (PGH) approach. We show that the singular behavior of the rate for large memory correlation times is an example of critical phenomena. We determine all the relevant critical exponents in different regimes and explicitly show that the rate has a scaling behavior. We verify that the universality of exponents and amplitudes is applicable in both the GH and the PGH solutions. Studying the ABC problem with techniques borrowed from critical phenomena reveals its rich mathematical structure and points out the ways in which one may discover the critical behavior of this problem experimentally.
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