Time autocorrelation function and Green-Kubo formula: Study on a 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 autocorrelation function in the nonequilibrium steady state (NESS) is obtained in terms of nonequilibrium phonon Green's functions. The time integral of the correlation function gives expected result, both in nonequilibrium 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.