Long time-tails, molecular dynamics

We should point out that Eq. (42) indicates that the function G(s) can be obtained from the value of the friction kernel at t = 0. This is a consequence of the fact that the friction kernel is calculated in the clamping approximation. In any case, Eq. (42) allows for the calculation of G(s) without the numerical difficulties that plague the long-time tail of molecular dynamics simulations. 

The behavior of VACF and of D in one-dimensional systems are, therefore, of special interest. The transverse current mode of course does not exist here, and the decay of the longitudinal current mode (related to the dynamic structure factor by a trivial time differentiation) is sufficiently fast to preclude the existence of any "dangerous" long-time tail. Actually, Jepsen [181] was the first to derive die closed-form expression for the VACF and the diffusion coeffident for hard rods. His study showed that in the long time VACF decays as 1/f3, in contrast to the t d 2 dependence reported for the two and three dimensions. Lebowitz and Percus [182] studied the short-time behavior of VACF and made an exponential approximation for VACF [i.e, Cv(f) = e 2 ], for the short times. Haus and Raveche [183] carried out the extensive molecular dynamic simulations to study relaxation of an initially ordered array in one dimension. This study also investigated the 1/f3 behavior of VACF. However, none of the above studies provides a physical explanation of the 1/f3 dependence of VACF at long times, of the type that exists for two and three dimensions. 

Ferrario, M.,Fionino, A., Cicootti, G. Long-time tails in two-dimensional fluids by molecular dynamics. Physica A 240, 268-276 (1997). doi 10.1016/S0378-4371(97)00150-7... 

---