Data and analysis tools concerning the article
"Invariance principles in the theory and computation of transport coefficients"
by Federico Grasselli and Stefano Baroni, https://arxiv.org/abs/2105.02137

This folder contains data and Jupyter Notebooks with which the figures of 
the manuscript were generated. The starting point are the time series of the
- electrical charge flux obtained via classical oxidation numbers: "elclass.dat"
- electrical charge flux obtained via Born effective tensors: "elquant.dat"
- the difference between the two definitions above: "diff.dat"

Units are angstrom/fs (i.e., the fluxes are not normalised to the simulation
cell volume, and the elementary charge is set equal to 1). 

To perform the block analysis on the time series of the currents we have 
employed the open-source code "analisi" written by Riccardo Bertossa and 
downloadble from https://github.com/rikigigi/analisi. For our analysis we used
version 0.2.1.

The syntax to generate the file "GKHE_class_40B.dat", whose content is displayed
in the first commented lines, from a 40-block analysis, using all the components
after the first one (included), 'X', is the following:

analisi -l elclass.dat -H -a 'X' -B 40 > GKHE_class_40B.dat

Once such files are generated, one can play and plot the results via the 
Jupyter Notebook "AnalysisPlot.ipynb".

The cepstral analysis have been performed with the SporTran code, downloadable
from https://github.com/lorisercole/sportran (and pip installable) whose main
contributors are Loris Ercole, Riccardo Bertossa and Sebastiano Bisacchi.
The notebook "Cepstral.ipynb" shows how to use it to see the behaviour of
cepstral coefficients and estimator of the log-spectrum at zero frequency, as
in Fig. 3 and 4 of the manuscript.