Using Path-Integral Methods to Calculate Noise-Induced Escape Rates in Bistable Systems: The Case of Quasi-Monochromatic Noise

Abstract

We review the basic steps leading from the definition of a stochastic process as a set of Langevin equations to the calculation of escape rates from path-integral representations for probability distribution functions. While the construction of the path-integral itself and the use of the method of steepest descents in the weak-noise limit can be formally carried out for a system described by rather general Langevin equations with complicated colored noise, the analysis of the resulting extremal equations is not, in general, so straightforward. However, we show that, even when the noise is colored, these may be put into a Hamiltonian form, which leads to improved numerical treatments and better insights. We concentrate on discussing the solution of Hamilton’s equations over an infinite time interval, in order to determine the leading order contribution to the mean escape time for a bistable potential. The paths may be oscillatory and inherently unstable, in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy. All of this is illustrated on the simple example consisting of an overdamped particle in a bistable potential acted upon by quasi-monochromatic noise. We first show how an approximate solution of the extremal equations leads to the conclusion that the bandwidth parameter has a certain critical value above which particle escape is by white-noise-like outbursts, but below which escape is by oscillatory type behavior. We then discuss how a numerical investigation of Hamilton’s equations for this system verifies this result and also indicates how this change in the nature of the optimal path may be understood in terms of singularities in the configuration space of the corresponding dynamical system.