  A fundamental principle of chaotic quantum dynamics is that local subsystems
eventually approach a thermal equilibrium state. Large subsystems thermalize
slower: their approach to equilibrium is limited by the hydrodynamic build-up
of large-scale fluctuations. For classical out-of-equilibrium systems, the
framework of macroscopic fluctuation theory (MFT) was recently developed to
model the hydrodynamics of fluctuations. We perform large-scale quantum
simulations that monitor the full counting statistics of particle-number
fluctuations in hard-core boson ladders, contrasting systems with ballistic and
chaotic dynamics. We find excellent agreement between our results and MFT
predictions, which allows us to accurately extract diffusion constants from
fluctuation growth. Our results suggest that large-scale fluctuations of
isolated quantum systems display emergent hydrodynamic behavior, expanding the
applicability of MFT to the quantum regime.
