Surmounting oscillating barriers: Path-integral approach for weak noise

Abstract

We consider the thermally activated escape of an overdamped Brownian particle over a potential barrier in the presence of periodic driving. A time-dependent path-integral formalism is developed which allows us to derive asymptotically exact weak-noise expressions for both the instantaneous and the time-averaged escape rate. Our results comprise a conceptually different, systematic treatment of the rate prefactor multiplying the exponentially leading Arrhenius factor. Moreover, an estimate for the deviations at finite noise strengths is provided and a supersymmetry-type property of the time-averaged escape rate is verified. For piecewise parabolic potentials, the rate expression can be evaluated in closed analytical form, while in more general cases, as exemplified by a cubic potential, an action-integral remains to be minimized numerically. Our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies.

Spectral Gap of the Erlang A Model in the Halfin-Whitt Regime

“…The results in (21) and (22) are also consistent with Figure 1, which suggests that r(0, η) increases as a concave function of η. The queueing counterpart is such that the load is one, and hence the reneging is necessary to alleviate the system.…”

“…As discussed in Section 5 the transition to oscillatory behavior occurs when R ≈ γ 2 /4 and then we can approximate D R−1 (γ) by Airy functions, with the leading term given in (87). Thus with R = γ 2 /4 − (γ/2) 2/3 δ and γ → ∞ the minimal root corresponds to the maximal root of Ai(δ) = 0, which occurs at δ = a 0 , leading to (22).…”

Chemical reactions induced by oscillating external fields in weak thermal environments

“…This is similar to the POs used by Lehmann et al [63][64][65] for the case of thermal activation with additive periodic driving. Note that this TS trajectory Γ ‡ is an exact solution to the equations of motion.…”

“…For thermally induced reactions, Lehmann, Reimann, and Hänggi [63][64][65] have shown that in the overdamped (large-γ) regime, when a chemical reaction is forced by a periodic field the reaction rate is determined in part by the geometry of periodic trajectories in the purely deterministic phase space. This work was later extended to cases with different scaling behaviors between the strength of thermal activation and the strength of the external field.…”

Activated escape over oscillating barriers: The case of many dimensions

“…The subject of our present paper is one of the simplest and experimentally most natural such non-equilibrium descendants of Kramers' original escape problem [17], namely the thermally activated escape of a Brownian particle over a potential barrier in the presence of periodic driving which modulates both the corresponding potential well region and the activation barrier. While most previous attempts have been restricted to weak [25,26,27], slow [28,29], or fast [25,28,30] driving, we have addressed in recent analytical explorations [31,32] by means of a path-integral technique the most challenging intermediate regime of moderately strong and moderately fast driving for a one-dimensional, overdamped escape problem. Closely related to our recent works are the subsequent appealing efforts in Ref.…”