title: Optimal paths and the calculation of state selection probabilities.
abstract: The addition of noise to a dynamical system means that initial states near points of instability may no longer decay to a unique stable state. A common example of this behavior occurs in a dynamical system with two degrees of freedom and with two or more stable states. If the initial state of the system is near the separatrices bounding the basins of attraction of these stable states, then the addition of noise to the system means that there is a nonzero probability that the stable state selected is in a different basin of attraction to that of the initial state. We discuss a method of calculating these state-selection probabilities based on a path-integral representation of the stochastic dynamics. The relationship of this approach to a method based on the solution of the backward Fokker-Planck equation is particularly stressed, since this was used in previous studies of problems of this type. However, while the method based on the backward Fokker-Planck equation is a powerful one for systems with one degree of freedom, in systems with more degrees of freedom it is much less useful. Since the standard method of solution in this case involves a series of mappings onto a deterministic dynamics which is simply the classical dynamics associated with the path-integral formulation, we argue that for systems with more than one degree of freedom, the path-integral method is a very natural way of calculating state-selection probabilities. We illustrate this on a simple example taken from population biology, and find that the state-selection probabilities are in excellent agreement with Monte Carlo simulations.
authors.affiliation: Department of Theoretical Physics, University of Manchester, Manchester M13 9PL, United Kingdom.
authors.forename: Alan J
authors.initials: AJ
authors.lastname: McKane
authors.forename: Martin B
authors.initials: MB
authors.lastname: Tarlie