Einstein-Kubo-Helfand and McQuarrie relations for transport coefficients

The formal equivalence of the Green-Kubo and Einstein-Kubo-Helfand (EKH) expressions for transport coefficients is well known. For finite systems subject to periodic boundary conditions, the EKH relations are ambiguous as to whether the toroidal or infinite-checkerboard descriptions should be used for the coordinates. We first describe qualitatively the application of both descriptions to the calculation of the self-diffusion and shear viscosity coefficients. We then show that the calculation of the self-diffusion coefficient using the infinite-checkerboard EKH relation is equivalent to the Green-Kubo calculation, while the toroidal calculation is not. For shear viscosity, we find that neither the toroidal nor infinite-checkerboard calculation from the EKH relation is equivalent to the Green-Kubo calculation, even though the formal theory presumably suggests that each is correct when the long-time limit is taken after the limit of large-system size. An alternative relation for the shear viscosity of finite periodic systems is derived from the Green-Kubo formula, consisting of the infinite-checkerboard expression plus correction terms having a fundamentally more complicated dependence on the coordinates and momenta. A simple qualitative analysis of the system-size dependence of the difference between the time-dependent Green-Kubo and the infinite-checkerboard EKH shear viscosities [η(t;N) and ${\mathrm{\eta }}_{\mathit{E}}^{\left(\mathit{C}\right)}$(t;N), respectively] shows this difference to be of O(${\mathit{N}}^{1/3}$) (N being the number of particles) at early times.