Thermal activation in extremely underdamped Josephson-junction circuits

Noise-activated escape of a particle out of a metastable well was treated by Kramers, who derived results covering the three cases of heavy damping, moderate damping, and extreme underdamping. In the case of extreme underdamping, the escape occurs through diffusive motion along the energy coordinate. We extend Kramers's treatment of this extremely underdamped case to cover a wider range of damping constants, taking into account that at energies just above the barrier peak and within the initial well, the distribution is controlled both by uphill diffusion in energy, and by the flow out of the well. This new result is compared with computer simulations of escape events for the case of Josephson junctions under constant current bias. In addition to a conventional simulation which initially puts the particles at the bottom of the well, a method is developed which requires simulation only in an energy range close to the peak of the barrier.