Inertial effects on the escape rate of a particle driven by colored noise: an instanton approach

A recent calculation, in the weak-noise limit, of the rate of escape of a particle over a one-dimensional potential barrier is extended by including an inertial term in the Langevin equation. Specifically, we consider a system described by the Langevin equation {Mathematical expression}, where ? is a Gaussian colored noise with mean zero and correlator =(D/t)exp(-|t-t'|/t). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D?0) limit. This yields an escape rate G~exp(-S/D), where the "action"S is the minimum, over paths characterizing escape over the barrier, of a generalized Onsager-Machlup functional, the extremal path being an "instanton" of the theory. The extremal action S is calculated analytically for small m and t for general potentials, and numerical results for S are displayed for various ranges of m and t for the typical case of the quartic potential V(x)=-x/2+x/4.