It is of major interest to find out the two-time Lagrangian autocorrelation function of a given property A. This function has generally a decaying behavior giving a characteristic correlation time for the considered quantity. I would like to know whether there is a good method to extract this characteristic time of a function obtained numerically. Please see the plot in attach there. Thanks!
The answer is signal dependent and some common criteria are eg assessed in the following paper. http://sydney.edu.au/engineering/aeromech/15afmc/proceedings/papers/AFMC00064.pdf
If you plot the logarithm of the function at issue, you will be able to read off the characteristic time from the plot directly (relaxation time is minus the slope of the linear part of the plot on the logarithmic scale); for this you either use a linear regression (plotting software, such as Grace, have relevant facility for this; in the case of Grace: Data --> Transformations --> Regression), or plot A exp(-α τ) along with your relevant function (both on the logarithmic scale) and determine the relevant A and α by trial and error (in the region of large τ the two functions must coincide). Incidentally, plotting software (and certainly Grace) allow for the logarithm of positive functions to be plotted (that is, one does not need to evaluate the logarithm of the data explicitly and subsequently plot them).
If your correlation function has more than one relation time, this will show up as piece-wise linear behaviour of this function on the logarithmic scale over different intervals. The same procedure as above applies here, except that in determining the linear regression line, you will have to restrict the region for determining the linear regression line to the relevant region (you repeat this procedure as many times as there are distinct linear pieces in your data plotted on the logarithmic scale).
Thanks Ramis / Behnam for your answers. I have put in attach the log-lin plot of my autocorrelation function. It seems that there is not a clear exponential decay, that's why I would like to know if there is a generalization of the autocorrelation characteristic time. Thanks!
You are welcome Abdallah. You will have to do some analytic work. If you know the time-Fourier transform of your auto-correlation function to frequency domain, then by asymptotic analysis of that transform to time domain you will be able to extract the relaxation times. Of course, strictly speaking, relaxation times correspond to the imaginary parts of the poles of the function in frequency domain, and in general (unless one has to do with very simple models) this function does not have simple poles, but branch points. By the knowledge of the nature of these branch points one can extract the relaxation behaviour of your function (the relevant techniques are described in standards texts on asymptotic analysis).
One further point, and that is that you should consider your function over a longer time span. It is possible that the time interval you are considering does not cover the relaxation tail of your function; instead, the interval you are considering may be dominated by substantial transient contributions.
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