Unstable decay and state selection.
The decay of unstable states when several metastable states are available for occupation is investigated using path-integral techniques. Specifically, a method is described that enables the probabilities with which the metastable states are occupied to be calculated by finding optimal paths, and fluctuations about them, in the weak-noise limit. The method is illustrated on a system described by two coupled Langevin equations, which are found in the study of instabilities in fluid dynamics and superconductivity. The problem involves a subtle interplay between nonlinearities and noise, and a naive approximation scheme that does not take this into account is shown to be unsatisfactory. The use of optimal paths is briefly reviewed and then applied to finding the conditional probability of ending up in one of the metastable states, having begun in the unstable state. There are several aspects of the calculation that distinguish it from most others involving optimal paths: (i) the paths do not begin and end on an attractor, and moreover, the final point is to a large extent arbitrary, (ii) the interplay between the fluctuations and the leading-order contribution are at the heart of the method, and (iii) the final result involves quantities that are not exponentially small in the noise strength. This final result, which gives the probability of a particular state being selected in terms of the parameters of the dynamics, is remarkably simple and agrees well with the results of numerical simulations. The method should be applicable to similar problems in a number of other areas, such as state selection in lasers, activationless chemical reactions, and population dynamics in fluctuating environments.