Universal Fluctuations of Local Measurement in Low-Dimensional Systems

Probes perform local measurements by interacting with the surrounding system. This often occurs in a dilute system whose effective macroscopic dynamics are diffusive. In this work, it is analytically shown that, for dimensions two or less, the relative uncertainty of any such measurement converges to an explicit universal power law in the long-time limit. The dynamical exponent depends on the dimension, and the prefactor depends solely on the bulk density and the effective mass diffusivity. Simulations of four distinct microscopic models support the results. The results imply that saturation of thermodynamic uncertainty relations cannot be approached for local measurements. They also suggest an experimental pathway for measuring effective diffusivities using a single local measurement.