  Recent experimental results suggest that a particular hydrodynamic theory
describes charge fluctuations at long wavelengths in the square-lattice Hubbard
model. Due to the continuity equation, the correlation functions for the charge
and the current are directly connected: the parameters of the effective
hydrodynamic model thus determine the optical conductivity. Here we investigate
the validity of the proposed hydrodynamic theory in the full range of
parameters of the Hubbard model. In the non-interacting case, there is no
effective hydrodynamics, and the charge fluctuations present a rich variety of
non-universal behaviors. At weak coupling, the optical conductivity is
consistent with the hydrodynamic theory: at low frequency one observes a
Lorentzian-shaped Drude peak, but the high-frequency asymptotics is necessarily
different; the high-temperature limit for the product of the two hydrodynamic
model parameters is also in agreement with numerical data. At strong coupling,
we find that a generalization of the proposed hydrodynamic law is consistent
with our quantum Monte Carlo, as well as the finite-temperature Lanczos results
from literature. Most importantly, the temperature dependence of the
hydrodynamic parameters as well as the dc resistivity are found to be very
similar in the weak and the strong-coupling regimes.
