Tightening the uncertainty principle for stochastic currents

We connect two recent advances in the stochastic analysis of nonequilibrium systems: the (loose) uncertainty principle for the currents, which states that statistical errors are bounded by thermodynamic dissipation; and the analysis of thermodynamic consistency of the currents in the light of symmetries. Employing the large deviation techniques presented in [Gingrich et al., Phys. Rev. Lett. 2016] and [Pietzonka et al., Phys. Rev. E 2016], we provide a short proof of the loose uncertainty principle, and prove a tighter uncertainty relation for a class of thermodynamically consistent currents $J$. Our bound involves a measure of partial entropy production, that we interpret as the least amount of entropy that a system sustaining current $J$ can possibly produce, at a given steady state. We provide a complete mathematical discussion of quadratic bounds which allows to determine which are optimal, and finally we argue that the relationship for the Fano factor of the entropy production rate $\mathrm{var}\, \sigma / \mathrm{mean}\, \sigma \geq 2$ is the most significant realization of the loose bound. We base our analysis both on the formalism of diffusions, and of Markov jump processes in the light of Schnakenberg's cycle analysis.