First-passage times in renewal and nonrenewal systems

TL;DR

This paper develops a method to calculate first-passage time distributions in Markovian systems, revealing how correlations and violations of fluctuation theorems can uncover internal dynamics and multicyclic structures.

Contribution

It introduces a novel approach to compute first-passage time distributions for arbitrary transitions in Markovian networks, linking them to full counting statistics and system dynamics.

Findings

Uncorrelated first-passage times relate to cumulants of full counting statistics.

Correlated first-passage times can reveal internal system dynamics.

Breaking fluctuation theorem indicates multicyclic network structure.

Abstract

Fluctuations in stochastic systems are usually characterized by the full counting statistics, which analyzes the distribution of the number of events taking place in the fixed time interval. In an alternative approach, the distribution of the first-passage times, i.e. the time delays after which the counting variable reaches a certain threshold value, is studied. This paper presents the approach to calculate the first-passage time distribution in systems in which the analyzed current is associated with an arbitrary set of transitions within the Markovian network. Using this approach it is shown that when the subsequent first-passage times are uncorrelated there exist strict relations between the cumulants of the full counting statistics and the first-passage time distribution. On the other hand, when the correlations of the first-passage times are present, their distribution may provide additional information about the internal dynamics of the system in comparison to the full counting statistics; for example, it may reveal the switching between different dynamical states of the system. Additionally, I show that breaking of the fluctuation theorem for first-passage times may reveal the multicyclic nature of the Markovian network.