Unstable decay and state selection II

The decay of unstable states when several metastable states are available for occupation is investigated using path-integral techniques. Specifically, a method is described which allows 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 non-linearities and noise, and a naive approximation scheme which 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 which 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 which 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.Comment: 28 pages, 6 figures. Accepted for publication in Phys. Rev.

Functional determinants for general Sturm-Liouville problems

Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm-Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses the properties of generalised zeta-functions. The general form of the final results are the same for the entire range of problems considered.Comment: 28 pages, LaTe

Multiply-connected Bose-Einstein condensed alkali gases: Current-carrying states and their decay

The ability to support metastable current-carrying states in multiply-connected settings is one of the prime signatures of superfluidity. Such states are investigated theoretically for the case of trapped Bose condensed alkali gases, particularly with regard to the rate at which they decay via thermal fluctuations. The lifetimes of metastable currents can be either longer or shorter than experimental time-scales. A scheme for the experimental detection of metastable states is sketched.Comment: 4 pages, including 1 figure (REVTEX

Regularization of functional determinants using boundary perturbations

The formalism which has been developed to give general expressions for the determinants of differential operators is extended to the physically interesting situation where these operators have a zero mode which has been extracted. In the approach adopted here, this mode is removed by a novel regularisation procedure, which allows remarkably simple expressions for these determinants to be derived