Tightening the uncertainty principle for stochastic currents

TL;DR

This paper refines the uncertainty principle for stochastic currents in nonequilibrium systems, establishing a tighter bound involving partial entropy production and analyzing its implications for thermodynamic consistency.

Contribution

It provides a new, tighter uncertainty relation for thermodynamically consistent currents, connecting large deviation techniques with symmetry analysis and entropy measures.

Findings

Derived a tighter uncertainty bound involving partial entropy production.

Proved the bound's optimality through quadratic bounds analysis.

Linked the Fano factor of entropy production to the tightness of the uncertainty bound.

Abstract

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.