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. The path-integral formulation of quantum mechanics introduced by Feynman in 1948 became extremely popular as a tool in quantum field theory during the 1970's, to such an extent that most textbooks published on this subject since then use this approach to quantization. There were several reasons for this, but one of the most important was the power of the path-integral method in the study of non-perturbative phenomena. In quantum mechanics the most natural quantity to express in path-integral form is G(x, t| x 0 , t 0)-the Green function associated with the Schrödinger equation which describes the time evolution of the system. Since the Fokker-Planck equation has a very similar structure to the Schrödinger equation, exactly the same set of mathematical procedures might be expected to give an expression for P (x, t| x 0 , t 0), the conditional probability density. That this was so was realized many years ago, but it has only been in the last few years that the method has been used as a tool to do calculations and not just as a formal device. It has been particularly useful in the limit of small noise strength (the equivalent of taking the ¯ h → 0 or classical limit) where steepest descent methods have enabled new results to be obtained in a quick and controlled fashion. The use of this approach on the barrier crossing problem gives a trajectory in configuration space x c (t) (the "c" denotes classical-keeping the analogy with quantum mechanics). These so-called