Barrier crossing induced by very slow external noise

TL;DR

This paper investigates how very slow external noise influences particle escape over a potential barrier, deriving a new probability equation that accounts for non-Gaussian noise and predicts escape rates with a Kramers-like turnover behavior.

Contribution

It introduces a novel equation of motion for the probability distribution under slow, arbitrary noise, extending the classical theory to include third order noise effects and non-Gaussian fluctuations.

Findings

Derived a third order noise-inclusive probability equation.

Calculated escape rates showing Kramers-like turnover behavior.

Demonstrated boundary conditions allow steady states despite the system being open.

Abstract

We consider the motion of a particle in a force field subjected to adiabatic, fluctuations of external origin. We do not put the restriction on the type of stochastic process that the noise is Gaussian. Based on a method developed earlier by us [ J. Phys. A {\bf 31} (1998) 3937, 7301] we have derived the equation of motion for probability distribution function for the particle on a coarse-grained timescale $\Delta t$ assuming that it satisfies the separation of timescales; $|\mu |^{-1} \ll \Delta t \ll \tau_c$, where $\tau_c$ is the correlation time of fluctuations. $|\mu |^{-1}$ refers to the inverse of the damping rate (or, the largest of the eigenvalues of the unperturbed system) and sets the shortest timescale in the dynamics in contrast to the conventional theory of fast fluctuations. The equation includes a third order noise term. We solve the equation for a Kramers' type potential and show that although the system is thermodynamically open, appropriate boundary conditions allow the distinct steady states. Based on the exact solution of the third order equation for the linearized potential and the condition for attainment of the steady states we calculate the adiabatic noise-induced rate of escape of a particle confined in a well. A typical variation of the escape rate as a function of dissipation which is reminiscent of Kramers' turn-over problem, has been demonstrated.