11 research outputs found
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