First-passage time and the fluctuation of the quenched disorder in biased media

First passage on disordered intervals

We derive unexpected first-passage properties for nearest-neighbor hopping on finite intervals with disordered hopping rates, including (a) a highly variable spatial dependence of the first-passage time, (b) huge disparities in first-passage times for different realizations of hopping rates, (c) significant discrepancies between the first moment and the square root of the second moment of the first-passage time, and (d) bimodal first-passage time distributions. Our approach relies on the backward equation, in conjunction with probability generating functions, to obtain all moments, as well as the distribution of first-passage times. Our approach is simpler than previous approaches based on the forward equation, in which computing the mth moment of the first-passage time requires all preceding moments.

First-passage time for superstatistical Fokker-Planck models

The first-passage-time (FPT) problem is studied for superstatistical models assuming that the mesoscopic system dynamics is described by a Fokker-Planck equation. We show that all moments of the random intensive parameter associated to the superstatistical approach can be put in one-to-one correspondence with the moments of the FPT. For systems subjected to an additional uncorrelated external force, the same statistical information is obtained from the dependence of the FPT moments on the external force. These results provide an alternative technique for checking the validity of superstatistical models. As an example, we characterize the mean FPT for a forced Brownian particle.

Survival and residence times in disordered chains with bias

We present a unified framework for first-passage time and residence time of random walks in finite one-dimensional disordered biased systems. The derivation is based on the exact expansion of the backward master equation in cumulants. The dependence on the initial condition, system size, and bias strength is explicitly studied for models with weak and strong disorders. Application to thermally activated processes is also developed.