Mean Direct-Transit and Looping Times as Functions of the Potential Shape

Abstract

Any trajectory of a diffusing particle making a transition between two end points of an interval can be divided into two segments, which we call direct-transit and looping parts. The former is the final segment of the trajectory, when the particle goes from one end point of the interval to the opposite end point, without retouching the starting point. The rest of the trajectory is the looping part. We study mean durations of the two parts in the presence of a symmetric linear cusp potential which, depending on the parameter values, forms either a barrier or a well between the end points. For the cusp barrier, we find that the mean direct-transit time decreases as the barrier height increases at a fixed interval length. This happens because the increase in the barrier height results in the increase of the magnitude of the force acting on the particle on both sides of the barrier. Interestingly, though the mean looping and direct-transit times are different in the case of the barrier and well potentials with equal height and depth, respectively, the mean first-passage times for the two cases are identical.

Figure 1

Double-well linear cusp potential U_dw(x), defined in eq 6.

Figure 2

Single-well linear cusp potential U_sw(x), defined in eq 17.

Figure 3

Dependences of the mean direct-transit (dtr), looping (l), and first-passage (FP) times, given in eqs 7, 9, 14, 19, and 20, on the parameter βΔU = βFL for the double-well (dw) and single-well (sw) potentials, defined in eqs 6 and 17, respectively. The mean first-passage times for the two potentials are identical. All mean times are normalized to the mean first passage time at ΔU = 0.

Figure 4

Mean direct-transit times in the moderate friction regime for the parabolic and cusp potential barriers, eqs 29 and 33, normalized to their values at ΔU = 0.