Barrier-crossing processes in nature are often non-Markovian and typically occur over an asymmetric double-well free-energy landscape. However, most theories and numerical studies on barrier-crossing rates assume symmetric free-energy profiles. Here, we use a one-dimensional generalized Langevin equation (GLE) to investigate non-Markovian reaction kinetics in asymmetric double-well potentials. We derive a general formula, confirmed by extensive simulations, that accurately predicts mean first-passage times from well to barrier top in an asymmetric double-well potential with arbitrary memory time and reaction coordinate mass. We extend our formalism to non-equilibrium non-Markovian systems, confirming its broad applicability to equilibrium and non-equilibrium systems in biology, chemistry, and physics.
