FFT of velocity-auto-correlation to get DOS

[tommya]

I'm trying to find out how to "properly" fourier-transform a velocity-auto-correlation function (VAC), for calculation of phonon density of states (DOS), from molecular dynamics simulation.

The problem I'm running into is that my calculated DOS:
a) doesn't fall of to zero as the frequency goes to zero (i.e. as ω → 0)
b) doesn't have a sharp fall-off at some cut-off frequency.

I've tried to read up and found two ways to proceed:
1) Pad the signal, i.e. add suitable number of zeros to the end of the velocity data. This didn't influence a) or b) but only decreased the amplitude of the transformed VAC.
2) DFFT assumes periodic signal. So I to replicated and "reflected" my velocity array in two ways to obtain a periodic signal:
2.1) First doing it "oddly" so that the end-point and the starting point are continuous and have continuous first derivative. --> seem to give nice ω → 0 behavior but influences high frequency in unsatisfactory way.
2.2) Secondly doing it "evenly" so that the end-point and the starting point are continuous but dosen't have continuous first derivative. --> better result for high frequencies but no improvement for ω → 0.

I think the origin to my problem is due to that a VAC have VAC(t→0)=1, VAC(t→∞)=0, and matching these two conditions with the assumption of periodicity in DFFT gives raise to a lot of overtones to try to capture the steplike jump.

Calculating VAC and FFTs thereof is very well established but I haven't been able to find any similar problem description.

Any suggestion for resources to read?
Experience from similar problems?

I'm grateful for all input!

 

[eljose79]

Hi there,

Thank you for reaching out and sharing your problem. I am also a scientist working in the field of molecular dynamics simulation and I have encountered similar challenges with Fourier transforming velocity auto-correlation functions for phonon density of states calculations.

Firstly, let me address your two proposed methods for dealing with the issue. Padding the signal with zeros at the end of the velocity data is not an ideal solution as it only decreases the amplitude of the transformed VAC without addressing the underlying problem.

The second method of replicating and reflecting the velocity array to obtain a periodic signal is a more common approach. However, as you have observed, it can lead to undesirable results at high frequencies and does not improve the behavior at ω → 0.

To address your specific concerns, I would suggest looking into windowing functions. These functions are used to reduce the effects of discontinuities in the data, such as the step-like jump in the VAC. Some commonly used windowing functions include Hann, Hamming, and Blackman. By multiplying your velocity data with a suitable windowing function before performing the FFT, you can improve the behavior at ω → 0 and reduce the overtones at high frequencies.

Additionally, I would recommend checking the time-step and length of your simulation. A longer simulation with smaller time-steps can provide more accurate results for the VAC and improve the behavior at ω → 0.

As for resources to read, I would suggest looking into textbooks on molecular dynamics simulation and Fourier analysis. I have also found some helpful discussions and examples on online forums and research papers.

I hope this helps and I wish you all the best in your research. If you have any further questions or would like to discuss this further, please do not hesitate to reach out.