Current noise of hydrodynamic electrons

A resistor at finite temperature produces white noise fluctuations of the current called Johnson-Nyquist noise. Measuring the amplitude of this noise provides a powerful primary thermometry technique to access the electron temperature at the nanoscale. In practical situations, one needs to generalize the Johnson-Nyquist theorem to handle spatially inhomogenous temperature profiles. Recent work provided such a generalization for ohmic devices obeying the Wiedemann-Franz law, and showed that Joule heating leads to a geometry-independent increase in Johnson noise. However, there has been great recent interest in strongly-interacting hydrodynamic electron systems which do not admit a local conductivity nor obey the Wiedemann-Franz law. Such systems provide unusual sensitivity for Johnson noise thermometry, but so far there is no theory to describe the current noise they produce. Here we consider low-frequency Johnson noise in the hydrodynamic setting for a rectangular geometry. Unlike in the ohmic setting, we find that the Johnson noise is no longer geometry-independent due to non-local viscous gradients. Nonetheless, ignoring the geometric correction only leads to an error of at most 40% as compared to naively using the ohmic result.