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

Abstract

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\(m\ddot x + \dot x + V\prime (x) = \xi \), whereξ is a Gaussian colored noise with mean zero and correlator 〈ξ(t)ξ(t')〉=(D/τ)exp(−|t−t'|/τ). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D→0) limit. This yields an escape rateΓ∼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 actionS is calculated analytically for smallm andτ for general potentials, and numerical results forS are displayed for various ranges ofm andτ for the typical case of the quartic potentialV(x)=−x2/2+x4/4.