Variance Sum Rule for Entropy Production

Entropy production is the hallmark of nonequilibrium physics, quantifying irreversibility, dissipation, and the efficiency of energy transduction processes. Despite many efforts, its measurement at the nanoscale remains challenging. We introduce a variance sum rule for displacement and force variances that permits us to measure the entropy production rate $\sigma$ in nonequilibrium steady states. We first illustrate it for directly measurable forces, such as an active Brownian particle in an optical trap. We then apply the variance sum rule to flickering experiments in human red blood cells. We find that $\sigma$ is spatially heterogeneous with a finite correlation length and its average value agrees with calorimetry measurements. The VSR paves the way to derive $\sigma$ using force spectroscopy and time-resolved imaging in living and active matter.