  Connecting short time microscopic dynamics with long time hydrodynamics in
strongly correlated quantum systems is one of the outstanding questions. In
particular, it is very difficult to determine various hydrodynamic coefficients
like the diffusion constant or viscosity starting from a microscopic model:
exact quantum simulations are limited to either small system sizes or to short
times, which are insufficient to reach asymptotic behavior. In this Letter, we
show that these difficulties, at least for a particular model, can be
circumvented by using the cluster truncated Wigner approximation (CTWA), which
maps quantum Hamiltonian dynamics into classical Hamiltonian dynamics in
auxiliary high-dimensional phase space. We apply CTWA to a XXZ
next-nearest-neighbor spin 1/2 chain and find behavior consisting of short time
spin relaxation which gradually crosses over to emergent diffusive behavior at
long times. For a random initial state we show that CTWA correctly reproduces
the whole spin spectral function. Necessary in this construction is sampling
from properly fluctuating initial conditions: the Dirac mean-field
(variational) ansatz, which neglects such fluctuations, leads to incorrect
predictions.
