Tight uncertainty relations for cycle currents

Several recent inequalities bound the precision of a current, i.e., a counter of the net number of transitions in a system, by a thermodynamic measure of dissipation. However, while currents may be defined locally, dissipation is a global property. Inspired by the fact that, ever since Carnot, cycles are the unit elements of thermodynamic processes, we prove similar bounds tailored to cycle currents, counting net cycle completions, in terms of their conjugate affinities. We show that these inequalities are stricter than previous ones, even far from equilibrium, and that they allow us to tighten those on transition currents. We illustrate our results with a simple model and discuss some technical and conceptual issues related to shifting attention from transition to cycle observables.

Automated summary

Several recent studies have found inequalities that relate the precision of a current to a measure of dissipation in a system. We prove similar bounds for cycle currents, which are stricter than previous ones and applicable even in far from equilibrium situations. Using a simple model, we illustrate our results and discuss the implications of shifting attention from transition to cycle observables.