Prediction of transport coefficients in one-dimensional systems

A theory is developed which describes the coupling between the momentum transport and the heat transport in a one-dimensional chain of particles. The theory is based on the assumption that heat is carried in one-dimensional chains "ballistically" by sound waves. The other key assumption is that the frequency dependent sound damping coefficient, Gamma(o) is related to the frequency dependent bulk viscosity, zeta(o), and the frequency dependent thermal conductivity, kappa(o) in the same way that the macroscopic sound damping coefficient in a fluid is related to the macroscopic bulk viscosity and thermal conductivity. These assumptions lead to a way to predict the low frequency thermal conductivity, kappa(o) from the higher frequency parts of kappa(o) and zeta(o). In particular, it allows one to predict the low frequency power law dependence of kappa on o for a variety of situations. The case of a system with equal specific heat capacities (cP = cV) is an important special case. In this case the theory predicts that kappa(o) ∼ o -1/2 as o → 0. In the more general case of c P ≠ cV the low frequency limit is predicted to be either kappa(o) ∼ o-1/3 or kappa(o) ∼ o -p*, where p* = 3-5/ 2 . However, if the latter is the case then this will only be seen at frequencies too low to be accessible by computer simulation. Computer simulations of two one-dimensional chain systems---one with cP = c V and one with cP ≠ cV---are reported which demonstrate the ability of this theory to accurately predict kappa(o) for sufficiently low frequency. The predicted low frequency limit for the cP = cV case is strongly supported by the simulation results, but the predicted behaviour in the cP ≠ c V case is not apparent at the lowest frequencies examined. Predictions are also made for the frequency dependent bulk viscosity zeta(o) as o → 0. For the cP = cV case zeta(o) is predicted to be finite in the zero frequency limit. This prediction is supported by simulation results. For the c P ≠ cV case the low frequency limit is predicted to be either zeta(o) ∼ o-1/2 or zeta(o) ∼ o -p*, where p* is the same as for kappa(o). While the simulations support the prediction of an infinite bulk viscosity in the zero frequency limit, neither of the predicted power law behaviours are apparent at the lowest frequencies examined. The predictions and the results of the simulations are compared and contrasted with other results which have been reported in the literature. It is suggested that many of the puzzling aspects of the previously reported results are because earlier work did not probe sufficiently low frequencies to observe the low frequency limits.