Interrelation between precisions on integrated currents and on recurrence times in Markov jump processes

TL;DR

This paper derives explicit formulas linking the precision of integrated currents and recurrence times in Markov jump processes, extending previous results to finite times and reversible transitions, with applications in biochemistry.

Contribution

It provides a new explicit expression for the variance of net transitions in Markov jump processes and explores their relation to recurrence timing, including finite-time and reversible cases.

Findings

Explicit formula for squared coefficient of variation of net transitions

Connection between current precision and recurrence timing in long and finite times

Derived kinetic and thermodynamic inequalities

Abstract

For Markov jump processes on irreducible networks with finite number of sites, we derive a general and explicit expression of the squared coefficient of variation for the net number of transitions from one site to a connected site in a given time window of observation (i.e., an `integrated current' as dynamical output). Such expression, which in itself is particularly useful for numerical calculations, is then elaborated to obtain the interrelation with the precision on the intrinsic timing of the recurrences of the forward and backward transitions. In biochemical ambits, such as enzyme catalysis and molecular motors, the precision on the timing is quantified by the so-called randomness parameter and the above connection is established in the long time limit of monitoring and for an irreversible site-site transition; the present extension to finite time and reversibility adds a new dimension. Some kinetic and thermodynamic inequalities are also derived.