Ultimate Kinetic Uncertainty Relation and Optimal Performance of Stochastic Clocks

For Markov processes over discrete configurations, an asymptotic bound on the uncertainty of stochastic fluxes is derived in terms of the harmonic mean of decay rates with respect to the stationary distribution. This bound is necessarily tighter than the bound in terms of the arithmetic mean, i.e., the activity, known as the kinetic uncertainty relation. What is more, it can always be saturated. In turn, an exact limit for the uncertainty of first-passage times as well as the optimal long-time performance of stochastic clocks are established. The results generalise to semi-Markov processes, including quantum reset processes, where it can be determined when clock performance improves thanks to coherent driving.