  We study transport in a one-dimensional lattice system with two conserved
quantities -- `volume' and energy. Considering a slowly evolving local
equilibrium state that is slightly deviated from an underlying global
equilibrium, we estimate the correction to the local equilibrium distribution.
This correction arises mainly through the space-time correlations of some local
currents. In the continuum limit, we show that the local equilibrium
distribution along with the correction yields drift-diffusion equation for the
`volume' and super-diffusion equation for the energy in the linear response
regime as macroscopic hydrodynamics as one would obtain from non-linear
fluctuating hydrodynamic theory. We find explicit expression of the
super-diffusion equation. Further, we find diffusive correction to the
super-diffusive evolution. Such a correction allows us to study a crossover
from diffusive to anomalous transport. We demonstrate this crossover
numerically through the spreading of an initially localized heat pulse in
equilibrium as well as through the system size scaling of the stationary
current in non-equilibrium steady state.
