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 ηE(C)(t;N), respectively] shows this difference to be of O(N1/3) (N being the number of particles) at early times.Monte Carlo molecular dynamics calculations of ηE(C)(t;N) for an equimolar binary mixture of hard spheres (diameter ratio of 0.4 and mass ratio of 0.03) confirm these large differences at a few mean free times, but suggest a long-time plateau value having the magnitude of the Green-Kubo result, but the values at 70 mean free times do not approach {η}(t;N) with increasing N. Finally, we consider the one-particle, EKH-like, McQuarrie expression for shear viscosity, showing that the Chialvo-Cummings-Evans [Phys. Rev. E 47, 1702 (1993)] ``proof'' is defective. Moreover, we demonstrate through molecular dynamics calculations for the same hard-sphere mixture that the two-particle contribution to the time-dependent viscosity, which must vanish at long times for the McQuarrie formula to be valid, in fact contributes roughly 40% of the shear viscosity at a volume of (5 {}2 /2)tsumₐ{N}ₐσₐ³, where Nₐ is the number of particles of species a having diameter σₐ.