Definition of correlation time is uncommon

[zjing7]

"the correlation time (denoted here as τ ) is the longest separation time ∆t over which x(t)
and x(t+∆t) remain (linearly) correlated."
This definition is very intuitive and easy to calculate, but in practice, it is rarely used and inconsistent with popular definitions.

For example, the rotational correlation time τ is defined such that R(t) = R(0) exp(-t/τ), where R is analagous to the correlation coefficient. It can be seen that R(τ) > 0. In fact, R(t) > 0 for any t, although τ is finite.
The above definition is the exponential correlation time. Another definition is the integrated correlation time. In many trivial cases, we also have R(τ) != 0. In the case of exponential decay, the two definitions are equivalent.
More proper definitions are: (1) the time scale at which the correlation is, on average, reduced by some factor (e.g. e). (2) When the statistical error calculated with all samples over a time period of T is equal to the statistical error of N independent samples, the (integrated) correlation time τ is T/N.
There are certainly other definitions, but in the context of this review, I think the integrated correlation time is the most relevant.

[dmzuckerman]

Thanks for your comment @zjing7 . We did debate this quite a bit and probably should have made a comment in the paper to indicate common procedures/ideas. Our main concern was practical difficulties in calculating \tau except in cases where you have such good sampling that you probably don't need to calculate it! You probably know that the exponential decay is only expected for the simplest systems. If there is complicated relaxation + noisy data on top of that, none of the methods are that great, though perhaps the simplest approach you name (time til decrease by a factor) might still be robust. Anyway, we should probably expand on these concerns in a subsequent version. Maybe @mangiapasta has more to add.