Green-Kubo algorithm in the calculation of anomalous heat conduction for models with and without sound mode

Green-Kubo algorithm is an effective method for the calculation of transport coefficients in terms of integral of the current correlation function. In this paper, we investigate two important issues of this algorithm in the calculation of anomalous heat conduction. Firstly, since the correlation function should be calculated in the thermodynamic limit which is never possible in practise, the necessary size of system should be determined. Secondly and more importantly, in the anomalous heat conduction cases, in order to work out a length-N-dependent heat conductivity kappa(N), we need to set a cutoff time tau(N) as a upper limit of the integral. tau(N) is commonly set to be proportional to N. However it has been observed very recently that in a model without sound mode tau(N) is not that simple but grows as N-nu with nu = 1.5 instead. We apply the algorithm to two typical one-dimensional models with and without sound modes, and find that the necessary size is extremely small for the model without sound mode, but relatively large for the model with strong sound modes. By studying the heat diffusion process via the local energy correlation, the value of tau(N) can also be quantitatively determined.