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

Abstract

The decay of metastable states of a system obeying an n-variable Fokker-Planck equation is considered by evaluating the mean first-passage time T_p 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 < Τ^r > ≃r! < Τ > ^r, where T_p= < Τ >, independent of the number of degrees of freedom, n. The time T_p 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.