Try an exponential function with shifted y-axis and starting with y=0.6.

If the asymptotic value is zero, you can try fitting to a sum of exponentials. If the asymptotic value is not zero, try using an exponential plus a constant. There are tools for fitting to a sum of exponentials, but there may be conditioning problems, especially if the time-scales are very similar for the terms in the series. Search for "fit prony series" and you will find some methods.

I would plot log(C(t)+1) and let me inspire by that graph. It probably will look like piecewise linear and thus suggest a piecewise defined fit.

The answer depens on type of correlation function. For autocorrelator its Fourier-tranform should be non-negative. Some known examples and referencies can be found in attached files.

Sorry. I expected a possibility to attach files to the comment, but it is absent. See some known examples and referencies in the lecture "Phase relaxation in magnetically diluted crystals" at my page https://www.researchgate.net/publication/258050744_Phase_relaxation_in_magnetically_diluted_crystals

Conference Paper Phase relaxation in magnetically diluted crystals

@Fridrikh

The curve is given and the answer depends just on it. What kind of information do you expect from your 'type of correlation function'?

Autocorrelation function must have nonnegative Fourier-transform, that is very strong restriction. Arbitrary correlation function can be fitted without the restriction. I see no problem here. In any case we should know asymptotical value for large "t". For standard definition C(t=infinity)=0. Is it fulfilled here? Errorbars are necessary for fitting as well. In any case it is useful to know primary definition: correlation of what with what?

A rough estimate of your data (since no errorbars are plotted) shows that the curve is compounded by at least sum of three exponents with a kind of negative shift.

Dear Ulrich & Fridrikh,

The property that I am calculating is the correlation time for the angle made by a vector representing two atoms in a molecule with the Z-axis of the periodic box of a bilayer. This plot is not treated by Fourier-Transform.

In my understanding the negative region, after positive correlation time of the system, shows that the dynamics of the system to be of the opposite direction compare to the direction at C(t)=0.

@Fridrikh; Thank you for the paper.

I have used function:

f(x) = -A*log(x/B)

to fit the plot.

Dear Colleague,

standard correlation function is of the form with =0 and =0. Autocorrelation function is realized if A=B. If evolution has finite memory time T, then for large t we have by definition >T)>==0. Symmetric forms, as F(t)=/(2) for example, are preferable as a rule. Standard autocorrelation function F(t) can be negative some times, but its Fourier transform is nonnegative. Initial evolution (both classical and quantum) is of the form F(t)=1-b*t^2+O(t^4). If your aim consists in formal fitting of the curve from your graph disregarding all these exact properties, then sum of functions of the form k*exp(-(d*t)^q) with different k, d and q can be useful.

Dear Fridrikh Dzheparov,

Thank you for your explanations. By the way, for the plot that I showed, which is the time correlation function for the angle of the lipid with Z-axis, how does the Fourier transform could help in extracting the correlation time of the angle behavior. Your explanation might be very useful.

Dear Colleague,

your data are incomplete, because we see no long time asymptotics, where C(t) go to zero. In any case we see that unique correlation time is absent. There exist fast part with very short relaxation (=correlation) time(s), much more slow motions at scale of order 10 ns, and may be more slow processes at longer time, which are not presented at your picture.

In order to give physical explanation for your picture you should check that Fourier transform of your C(t) is positive. If it is not positive, the picture does not show the autocorrelation function and, if necessary, you can fit it by any suitable functions. If the Fourier-transform is positive you can fit it (F-T) by functions, which you like, for example by Lorenz and Gauss. Fourier-transform of exp(-(b*t)^q) is positive as well.

Simplest function for your picture is

C(t)=a1*exp(-(b1*t)^q1)+a2*cos(d*t)exp(-(b2*t)^q2),

where d=(pi/2)/(4000ps). But b2 and q2 still unknown by your data. This function has positive Fourier transform if a1>0 and a2>0. Of course the short time evolution, where

C(t)=1-(b*t)^2+O(t^4)

is omitted here.

@Vijay

You seem to have confused your colleagues completely. As long as the question was how to fit these data, things are easy and you probably got useful ideas from your colleagues.

Your hints concerning the nature of the curve leave too many interpretations. Is C(t) not simply the cosine of a single angle at time t (describing the direction associated with a pair of atoms) as obtained from a simulation run? Or similar, but the mean value taken over many atom pairs in the model, or over many runs? After all, in this case we would not call this a correlation time.

If you would consider two atoms, each providing a direction vector, you could speak about a true correlation time describing the spin dynamics (or of any other meaningful direction) of these atoms. It would not come as a surprise to me if a thorrough analysis of what you did would reveal to you a fundamental flaw or misinterpretation.

Dear Ulrich,

Thanks for your kind reply.

It is true that at fist I indent to fit the plot. But, at the same time I have some difficulty on interpreting the plot.

As you mentioned, the plot is for the cos(theta) where theta is the angle between a vector of two atoms from lipid chain and bilayer normal.

If I describe briefly of how I calculated the auto-correlation function ( ACF): at first I calculated the angle theta for, say lipid-no-1 from frame-no-1 throughout all frames (40000). And I calculated the ACF using those 40000 data. Than, I go on with lipid-no-2 and all the lipids as I did for lipid-no-1. This means I now have ACFs for each lipid of the system. Finally I average them.

According to your suggestion, what we call this type of calculation if not correlation time?

Appreciate if you could kindly give some explanations for this. I may get some info for better understanding.

Thank you.

Dear Vjiay,

what you have computed is an averaged autocorrelation function. The correlation time (the word seems no longer to be in wide-spread use) is the point in time in which the autocorrelation function has fallen to some conventional value (such as 1/e) < 1.

Your data seem to indicate that you lost most of autokorrelation already in the first few frames. Should this be actually be the case, you should repeat your computation with a smaller step size. If not, you should set up your graphical representation in a way that the structure of curve near zero becomes resolved.

Dear colleagues,

correlation time is used for qualitative aims presumably. It is defined frequently as integral from 0 to infinity of C(t).

In reality we should understand where and haw the result will be used.

For example, you can expect receive an estimation of the Fourier-spectrum G(w) of C(t) via the relation

G(w)=(1/pi)*Tc/(1+(w*Tc)^2),

where Tc is correlation time.

But the function C(t) is too complex for such simple estimation. Application of two correlation times (at least) will be more adequate.

Better understanding of the curve can be obtained if the time will be plotted in log scale.