  We provide a stochastic fractional diffusion equation description of energy
transport through a finite one-dimensional chain of harmonic oscillators with
stochastic momentum exchange and connected to Langevian type heat baths at the
boundaries. By establishing an unambiguous finite domain representation of the
associated fractional operator, we show that this equation can correctly
reproduce equilibrium properties like Green-Kubo formula as well as
non-equilibrium properties like the steady state temperature and current. In
addition, this equation provides the exact time evolution of the temperature
profile. Taking insights from the diffusive system and from numerical
simulations, we pose a conjecture that these long-range correlations in the
steady state are given by the inverse of the fractional operator. We also point
out some interesting properties of the spectrum of the fractional operator. All
our analytical results are supplemented with extensive numerical simulations of
the microscopic system.
