Escape from a metastable well under a time-ramped force

Thermally activated escape of an over-damped particle from a metastable well under the action of a time-ramped force is studied. We express the mean first passage time (MFPT) as the solution to a partial differential equation, which we solve numerically for a model case. We discuss two approximations of the MFPT, one of which works remarkably well over a wide range of loading rates, while the second is easy to calculate and can provide a valuable first estimate. Introduction – Thermally activated escape of a particle from a metastable well, has found numerous applications in a variety of systems [1]. Escape under the additional action of a time-dependent force constitutes a non-trivial generalization. Several studies [2–6] have been devoted to this problem for a sinusoidal force, which is particularly interesting since this system shows stochastic resonance [7]. The topic of the present paper is the effect of a time-ramped force on the escape rate. Apart from its fundamental significance, motivation to study this problem arises from recent work on the dynamical strength of molecular bonds [8–10]. The strength of a single bond can be measured experimentally using atomic force microscopy where one in practice applies a time-ramped force [8,9,11–13]. Evans and coworkers [8,9] pointed out that the rupture strength of such a bond depends on the loading rate. This behavior has been seen not only in experiments but also in simulations of a Langevin equation [9,10] and molecular dynamics simulations at very large loading rates [10,14]. For the general problem of diffusive escape from a metastable well under a force that increases with time, we express the mean first passage time (MFPT) as the solution to a partial differential equation (pde). This exact approach differs from previous work that introduced an approximation based on instantaneous decay rates [9,10]. The numerical solution of the exact equation is then compared with both this and another simple approximation. The first one works very well over a large range of loading rates. The second one can be calculated easily and still gives a reasonable estimate which is never off more than 30% over the entire range of loading rates. MFPT as an exact solution to a pde – We consider the motion of an overdamped particle in a one-dimensional potential V (x) under the action of a time-dependent force f(t). The Langevin equation for this particle reads