autoScale

This archive contains autoScale.py, a standalone python tool for performing an automated finite-size scaling (FSS) analysis. FSS is done in a model independent manner, based on the minimization of an objective function measuring the quality of the data collapse, as detailed in

J. Houdayer, A. K. Hartmann, "Low-temperature behavior of two-dimensional Gaussian Ising spin glasses", Phys. Rev. B 70, 014418 (2004), DOI: 10.1103/PhysRevB.70.014418

The archive further includes raw data, allowing a user to perfom an exemplary scaling analysis for the two-dimensional site percolatin problem.

Prerequisites

autoScale has been under development since 2007. It was implemented under Python 2.3.4 (and later versions of Python 2) and it imports only basic Python modules that come with any python installation.

Included materials

The repository contains:

autoScale/
├── LICENSE.md
├── README.md
├── ORDER_PARAMETER
├── autoScale.py
├── autoScale_guide.pdf
├── main_fig_FSS.py
├── fig_FSS.png
├── inputFiles.dat
├── scaled_L512_256_128.out
├── error.out
├── scalingPlot.sh
└── scalingScript.sh

Subfolder ORDER_PARAMETER/ contains the raw data for the order parameter, i.e.\ the relative size of the largest cluster, for the two-dimensional site-percolation problem on two-dimensional lattices.

autoScale.py is the stanalone python script that performs the FSS analysis.

autoScale_guide.pdf is a user guide.

inputFiles.dat is a file serving as an easy interface between autoScale.py and the raw data.

scalingPlot.sh is a bash script that allows to display the scaled data using gnuplot.

scalingScript.sh is a bash script that runs a sequence of FSS analyses on the provided data.

main_fig_FSS.py is a python script that generates the figure below (it requires the functionality of numpy and pythons matplotlib).

scaled_L512_256_128.out contains the results produced by scalingScript.sh, and err.out contains the error analysis.

The repository further contains

• LICENSE, a license file.
• Readme.md, this file.

For a more detailed description of functions, defined in the above modules, their parameters and return values we refer to the example cases and documentation provided within the code.

Exemplary results for two-dimensional site percolation

Finite-size scaling analysis using autoScale

Performing a FSS analysis for the enclodes order parameter data for the two-dimensional site percolation problem within the region [-1.5:1.0] of the scaled control parameter is done in the following way:

python autoScale.py -f inputFiles.dat -xc 0.5927 -a 0.75 -b 0.104 -xr -1.5 1.
dx = [-1.500000:1.000000]  xc = 0.592708  a = 0.747566  b = 0.104459  S = 1.076691 

Upon termination, autoScale.py lists the parameter for which the best data collapse was achieved. In this case, the quality of the data collapse is approximately S=1.08. The quality function S measures the mean-squared distance of the rescaled quantities at finite system size to their master curve in units of the standard error, similar to a chi-square-test. Error bars for each of the parameter can be obtained by invoking the additional flag -getErrors, resulting in

# S+1 error analysis yields:
# Scaling analysis restricted to
  xr = [-1.500000 : 1.000000]
# <scalePar>  <-Err>  <+Err>
  xc = 0.592708 0.000075 0.000075
   a = 0.747566 0.005424 0.005419
   b = 0.104459 0.000971 0.000964

For a given parameter, say, parameter xc, the other parameters (a and b) are kept fixed, and the value of xc is first increased and then decreased until the quality S increases its value to S+1. The larger of the two xc-values is then quoted as the error on xc. This is referred to as a S+1 analysis.

Brief explanation of the above figure

Illustration of the data collapse technique for the order parameter of the site percolation problem in 2d. (a) unscaled data for different system sizes close to the critical point (each data point represents an average over 12800 individual occupied/empty configurations). (b) data collapse after the scaling parameter optimization using autoScale.py.

Further information

For further information about program options and file-formats look up autoScale.py directly, or see the users guide enclosed as autoScale_guide.pdf, also available under

O. Melchert, "autoScale.py - A program for automatic finite-size scaling analyses: A user's guide," arXiv:0910.5403 (2009).

Availability of the software

autoScale is provided as a (system-)local software tool. There is no need to install it once you got a local clone of the repository, e.g. via

$ git clone https://github.com/omelchert/autoScale.git

License

This project is licensed under the MIT License - see the LICENSE.md file for details.