Rate processes in a delayed, stochastically driven, and overdamped system

A Fokker-Planck formulation of systems described by stochastic delay differ ential equations has been recently proposed. A separation of time scales ap proximation allowing this Fokker-Planck equation to br simplified in the ca se of multistable systems is hereby introduced, and applied to a system con sisting of a particle coupled to a delayed quartic potential. In that appro ximation, population numbers in each well obey a phenomenological rate law. The corresponding transition rate is expressed in terms of the noise varia nce and the steady-state probability density. The same type of expression i s also obtained for the mean first passage time from a given point to anoth er one. The steady-state probability density appearing in these formulas is determined both from simulations and from a small delay expansion. The res ults support the validity of the separation of time scales approximation. H owever, the results obtained using a numerically determined steady state pr obability are more accurate than those obtained using the small delay expan sion. thereby stressing the high sensitivity of the transition rate and mea n first passage time to the shape of the steady-state probability density. Simulation results also indicate that the transition rate and the mean firs t passage time both follow Arrhenius' law when the noise variance is small, even if the delay is large. Finally, deterministic unbounded solutions are found to coexist with the bounded ones. Ln the presence of noise, the tran sition rate from hounded to unbounded solutions increases with the delay.