For a dense gas, the time constant is determined by fitting an exponential decaying function to the velocity autocorrelation function of an MD simulation, but what form is used for liquid VACFs where caging results in back scattering? Is the envelope of the VACF used instead? Thank you.
Cameron, what do you mean by the time constant? Presumably the inverse of the decay rate for the exponential decay? However, even for dense gases the asymptotic decay of the vacf is known to follow a t^{-3/2} power law, for which no decay rate can be defined, but the contribution to the diffusion coefficient (the time integral of the vacf) of this "long time tail" mostly is very small.
In the presence of a cage effect the vacf develops a negative tail preceding the t^{-3/2} power law, which is still the asymptotic long time behavior.
Heuristically one could define a decay time as the time t_D for which the integral of the vacf up to t_D reaches a given large fraction of the diffusion coefficient For the inverse decay rate of exponential decay this fraction would be (1-1/e), but of course in general there is some arbitrariness in this choice. Near the glass transition, where the diffusion constant becomes extremely small, it makes more sense comparing the contribution of the remaining part of the long time tail for t>t_D to the size of the positive initial part of the vacf, or alternatively that of the almost opposite part resulting from the cage effect.
Henk van Beijeren , Thank you for the excellent and detailed response. I should have been more specific when I wrote time constant and I realized this is somewhat arbitrary. I am simply trying to determine a good metric for how frequently I can sample the ensemble without samplings being closely correlated. The reference to time constant, tau, came from literature in which the VACF of a LJ Ar gas was fit using exp(~t/tau). However, to your points, it seems a more robust method is the comparison of the contributions of the VACF area from the initial positive section to the tail/cage effect.
Hi @Cameron, that is an entirely different question, indeed. A few mean free times between subsequent measurements should suffice for that. You could take as requirement a decay of the vacf to, say, 10% of its initial value. In case of a cage effect you should impose this on the absolute value of the vacf.
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