  Anomalous heat transport observed in low dimensional classical systems is
associated to super-diffusive spreading of space-time correlation of the
conserved fields in the system. This leads to non-local linear response
relation between the heat current and the local temperature gradient in
non-equilibrium steady state. This relation provides a generalisation of
Fourier's law of heat transfer and is characterised by a non-local kernel
operator which is related to fractional operators describing super-diffusion.
The kernel is essentially proportional, in appropriate hydrodynamic scaling
limit, to the time integral of the space-time correlations of local currents in
equilibrium. In finite size systems, the time integral of correlation of
microscopic currents at different locations over infinite duration is
independent of the locations. On the other hand the kernel operator is
space-dependent. We demonstrate that the resolution of this apparent puzzle
appears through taking appropriate combination of limits of large system size
and large integration time duration. Our study shows the importance of taking
the limits in proper way even for (open) systems connected to reservoirs. In
particular we reveal how to extract the kernel operator from simulation data of
microscopic current-current correlation. For two model systems exhibiting
anomalous transport, we provide direct and detailed numerical verification of
the kernel operators.
