First-passage times and hysteresis in multivariable stochastic processes: The two-mode ring laser

The decay of metastable states of a system obeying an n-variable Fokker-Planck equation is considered by evaluating the mean first-passage time Tₚ using the asymptotic method of Schuss and Matkowsky. The statistics of the first-passage time τ in the small-noise high-barrier limit is shown to follow 〈τʳ〉r!〈τ〉ʳ, where Tₚ=〈τ〉, independent of the number of degrees of freedom, n. The time Tₚ in the low-barrier, high-noise limit is also calculated. The hysteresis window for the control-parameter sweep rate is generalized to multivariate systems. These general results are applied to a model of a two-mode laser, where $n=4$. A comparison with recent results of Mandel and co-workers is made, and experimental tests of the predictions are suggested.