Volume 129, number 5,6 PHYSICS LETTERS A 30 May 1988 ON THE MEAN FIRST PASSAGE TIME IN A BISTABLE SYSTEM: SOME RECENTLY COMPUTED DATA R. MANNELLA a and V. PALLESCHI b Dipartimento di Fisica dell’Università di Pisa, Piazza Torricelli 2, 56100 Pisa, Italy and Physics Department, University of Lancaster, Bailrigg, Lancaster LA] 4YB, UK b Istituto di Fisica Atomica e Molecolare del CNR, Via del Giardino 7, 56100 Pisa, Italy Received 4 January 1988; revised manuscript received 3 March 1988; accepted for publication 25 March 1988 Communicated by A.P. Fordy In this Letter we report a new complete set of computed data on the mean first passage time in a bistable system. The results are compared with some recent theoretical predictions, Recent years witnessed an increasing interest in 16,23—34]. Clearly, an even more open question is the study of the first passage time (FPT) in systems the problem of the distribution of the FPTs or, more driven by stochastic forcings (for example, refs. simply, the derivation of the MFPT. To further corn- [1—22] and references quoted therein). plicate the problem, still focusing our attention on Generally speaking, the theory is developed for a the system described by eq. (2), the existing theories prototype bistable system such as for the MFPT are in agreement only when the white noise limit (i.e. ‘r—~0) is taken. It should be clear that, x=a(t)x—b(t)x 3+~(t), (1) . . ~, in order to single out the best theoretical ap- where ~( t) is a stochastic forcing of given statistics proach, some sort of simulation of eq. (2), carried and well defined correlations, and a ( t) and b ( t) are out under controlled environmental conditions, given time-dependent (deterministic) functions. The would suffice. Remarkably, given the importance, system is set at an initial position (x 0, ~) and the only two simulations of eq. (2) are to be found in time taken to leave a volume inside a boundary the literature (see refs. [10,11], the latter with only B = {XB, ~ is derived (FPT). Generally, the FPTs few data points). The aim of this Letter is to fill the are random variables, characterized by a distribu- existing gap, giving the researchers in the field a tion. The first moment (the so-called mean first pas- complete set of recently computed MFPTs that we sage time, MFPT) is the theoretical quantity most hope will be used for comparison with present and often derived, future theories. Herein we will compare our data only Let us point out that even in the simple case when with the theoretical results of refs. [14,121 the rea- 3 son being that this work was actually prompted and X X X + i~( t) stimulated by the authors of those references. It is = (Dir) exp( — si It) , beyond the limit and scope of this Letter, and a mat- ter of future publications, to carry out a full corn- ~(r) gaussian, (2) parison with all the existing theories. We have there is still discussion regarding the equilibrium dis- simulated eq. (2) on a digital computer. The basic tribution P(x, ~) attained by x and ~ (a steady state algorithm we used is the one described in ref. [23], quantity) in the large time limit [1—3,5—7,10,12, but for some of the data points it was improved as for ref. [35] to a higher order in h/r. Let us first dis- Present address. pense with some technicalities. For each run, the in- 0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. 317 (North-Holland Physics Publishing Division)