  A recently developed non-linear fluctuating hydrodynamics theory has been
quite successful in describing various features of anomalous energy transport.
However the diffusion and the noise terms present in this theory are not
derived from microscopic descriptions but rather added phenomenologically. We
here derive these hydrodynamic equations with explicit calculation of the
diffusion and noise terms in a one-dimensional model. We show that in this
model the energy current scales anomalously with system size $L$ as $\sim
L^{-2/3}$ in the leading order with a diffusive correction of order $\sim
L^{-1}$. The crossover length $\ell_{c}$ from diffusive to anomalous transport
is expressed in terms of microscopic parameters. Our theoretical predictions
are verified numerically.
