Time auto-correlation function and Green-Kubo formula: A study on disordered harmonic chain

We have considered heat conduction in a one-dimensional mass disordered harmonic chain of $N$ particles connected to two Langevin type reservoirs at different temperatures. An exact expression for the boundary heat current-current auto-correlation function in the non-equilibrium steady state (NESS) is obtained in terms of non-equilibrium phonon Green's functions. The time integral of the correlation function gives expected result, both in non-equilibrium as well as equilibrium cases. Using the form of this correlation function we show that asymptotic system size dependence of current fluctuation in NESS for a mass disordered harmonic chain is $N^{-\alpha}$ for different boundary conditions. For free and fixed boundary conditions we get $\alpha=1/2$ and 3/2 respectively, while for pinned case the fluctuation decays exponentially with system size.