Time correlation function of the shear stress in sheared particle systems

The long time behaviors of current autocorrelation functions are important to understand the macroscopic properties of fluids.1), 2), 3), 4) In is known that the existence of the long-time tail in the correlation functions leads to the anomalous behaviors of the transport coefficients.5), 6) In equilibrium systems, the existence of the long-time tail t−d/2 with the time t and the spatial dimension d is well recognized. However, the long time behaviors of the correlation functions under a steady shear have different feature from those at equilibrium.7), 8) In our previous paper, we find that the velocity autocorrelation function C(t) has the cross-over from t−d/2 to t−d in sheared elastic particles without thermostat, and C(t) obeys t−(d+2)/2 after the known tail t−d/2 in sheared isothermal fluids.8) However, we did not discuss the other correlation functions such as the correlation of the shear stress and the related transport coefficients. In this paper, thus, we theoretically calculate the correlation function of the shear stress. Our theoretical method is based on the classical one developed by Ernst et al.2), 9) In the next section, we will introduce the model we use. In section 3, we will present the details of the analysis. In section 4, we will discuss and conclude our results.