RATES AND MEAN FIRST PASSAGE TIMES

The relation between mean first passage times T and transition rates G amma in noisy dynamical systems with metastable states is investigated , It is shown that the inverse mean first passage rime to the separatr ix of the noiseless system may deviate from twice the rate not only be cause in general the deterministic separatrix is not the locus in the state space from which a noisy trajectory goes to either side with equ al probability. A further cause of a deviation from the often assumed relation Gamma T = 1/2 between rates and mean first passage times is g iven if the noisy dynamics is discontinuous, i.e. shows jumps with fin ite probability. Then the value of the splitting probability at the se paratrix does not fix the value of TT since the system need not visit the separatrix during a transition from one to the other side. Most im portant, for discontinuous processes the deviation from the TT = 1/2 r ule survives even in the weak noise limit. A mathematical relation for the product of the rate and the mean first passage time is proposed f or Markovian processes and numerically confirmed for a particular one- dimensional noisy map.