Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

TL;DR

This paper establishes simple bounds on the fluctuations and uncertainty relations for first-passage times of counting observables in stochastic systems, linking fluctuations to dynamical activity.

Contribution

It introduces bounds for first-passage times of counting observables, extending fluctuation and uncertainty relations beyond currents to non-decreasing trajectory observables.

Findings

Fluctuations are bounded by a Conway-Maxwell-Poisson distribution.

Bounds depend only on average observable and dynamical activity.

Dynamical activity controls fluctuation bounds for counting observables.

Abstract

Recent large deviation results have provided general lower bounds for the fluctuations of time-integrated currents in the steady state of stochastic systems. A corollary are so-called thermodynamic uncertainty relations connecting precision of estimation to average dissipation. Here we consider this problem but for counting observables, i.e., trajectory observables which, in contrast to currents, are non-negative and non-decreasing in time (and possibly symmetric under time reversal). In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson distribution dependent only on the averages of the observable and of the dynamical activity. We show how to obtain the corresponding bounds for first-passage times (times when a certain value of the counting variable is first reached) and their uncertainty relations. Just like entropy production does for currents, dynamical activity controls the bounds on fluctuations of counting observables.