  We study dynamics of a locally conserved energy in ergodic, local many-body
quantum systems on a lattice with no additional symmetry. The resulting
dynamics is well approximated by a coarse grained, classical linear functional
diffusion equation for the probability of all spatial configurations of energy.
This is equivalent to nonlinear stochastic hydrodynamics, describing the
diffusion of energy in physical spacetime. We find the absence of
non-hydrodynamic slow degrees of freedom, a nonlinear fluctuation-dissipation
theorem, and the emergence of a (weakly interacting) kinetic theory for
hydrodynamic modes near thermal equilibrium. The observable part of the
microscopic entropy obeys the local second law of thermodynamics, and
quantitatively agrees with the phenomenological predictions of hydrodynamics.
Our approach naturally generalizes to ergodic systems with additional
symmetries, may lead to numerical algorithms to calculate diffusion constants
for lattice models, and implies sufficiency conditions for a rigorous
derivation of hydrodynamics in quantum systems.
