A Spatio-Temporal Generalisation of Green Kubo

Abstract

The Green-Kubo (GK) method which is used to obtain the shear viscosity of a model liquid in equilibrium molecular dynamics (MD) often requires long simulation times to obtain acceptable statistics. This work extends the GK approach to include spatial correlations. The total shear stress in the GK expression is split into a grid of its components in contiguous volumes. In doing this, the autocorrelation of the total system stress can be rewritten as the cross-correlation of the subvolume stresses. This provides a novel real-space insight into the liquid structure, something previously only treated in Fourier space by liquid-state theory. A novel travelling wave like structure is exposed that is shown to be well fitted by a leading Gaussian pulse added to a second negative Gaussian for the bounce back. The fitting parameters include the wave position, magnitude and width which show deep insights into the liquid property. The wave moves at the speed of sound over short times before decreasing in speed with the square root of time, while the wave packet spreads out also as the square root of time. The magnitude is decreasing with form $t^{-3/2}$, a result with strong significance in the history of MD simulation. These fitted forms mean the entire spatial temporal response of the liquid can be modelled in closed form by fitting to data, with short term requiring MD and the long time and distances approximated by the Gaussian form. By limiting the correlation to localized interactions the longer range contribution, which essentially only contribute to the noise, can be eliminated. This looks like a promising approach to model viscosity by taking short MD runs and fitting the long time behaviour.