With the following programs one can analyze the 2D-Ising model with monte-carlo algorithms. In addition, calculate critical exponents with reweighting in MATLAB. Furthermore, analyze the autocorrelations and compare metropolis with the heatbath algorithm. With little programming effort also every other calculation with the 2D-Ising model can be achieved.
This project includes the following runnable programs, in the following called {program-name}
Every program runs with 100% CPU load on every core.
- ising-with-plots: generates plots directly after simulation with CV-Plot
- ising-headless: only status updates from console, saves all data to file
- ising-live: you can view the whole configuration of the spin-field in live while simulation. Furthermore, calculates autocorrelation of energies and plots it with CV-Plot.
Furthermore, this repository includes matlab-scripts for post-calculations of the generated values.
Project is based on C++20
(optional for project 2):
- OpenCV from: https://docs.opencv.org/4.5.2/df/d65/tutorial_table_of_content_introduction.html
- CV-Plot from: https://github.com/Profactor/cv-plot
This is a CMake-Project. Compilation and building is pretty easy. Simply run in main folder via terminal:
$ cmake .
$ cmake --build .
$ cmake --build --target {program-name}
The executable can be found in ./bin/{program-name}.
I've done a long headless run with N=16,32,64,128,256 spins per edge with 32 temperatures equally spaced between T=2-3.
Sweeps per temperature-step were 5E5, but I stored only every 10th sweep due to lower autocorrelations. Raw data one can
find in ./result/IsingResultsNew4.zip.
There are too few data for 256^2 spins, and the tremendous autocorrelations to perform a good analysis. Thus, we neglect this data-sample.
The following data are already reweighted around T_c, like described in the literature below.
To estimate the heat capacity and magnetic susceptibility I followed the literature and estimated variances in the data and furthermore reweighted those.
One approach to estimate the critical temperature T_c is by using binder-cumulants U_L like described in the literature below.
The exact value from the literature is T_c=2/log(1+sqrt(2))=2.2691... This is within our confidence interval.
Following the literature further one can estimate the critical exponents of the 2D-Ising-model by fitting
- deviation of finite size critical temperature
- magnetization
- maxima of heat capacities
- maxima of susceptibility
against the number of spins in a double logarithmic plot.
Let's compare the critical exponents with the literature:
| rel. observable | critical exponent | calculated | literature |
|---|---|---|---|
| heat capacity | alpha | 0.13+-0.03 | 0 (logarithmic singularity) |
| magnetization | beta | 0.223+-0.142 | 0.125 |
| susceptibility | gamma | 1.777+-0.038 | 1.75 |
| correlation length | nu | 1.095+-0.138 | 1 |
Overall the values are quite well. However, we only used data up to N=128^2 spins.
One could increase the achieved accuracy by increasing N.
One can observe the monte-carlo process live via a fast GUI built with OpenCV. A small impression:
Furthermore, a plot of autocorrelations is shown after simulation. Here we compare the metropolis-algorithm with the heatbath-algorithm.
Autocorrelations: https://www.statlect.com/fundamentals-of-statistics/Markov-Chain-Monte-Carlo-diagnostics
Metropolis-algorithm and general analysis of ising-model, finite size scaling:
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http://216.92.172.113/courses/phys39/simulations/Simulation.html
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http://216.92.172.113/courses/phys39/simulations/AsherIsingModelReport.pdf
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http://216.92.172.113/courses/phys39/simulations/advancedlab1_Part1.pdf
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http://farside.ph.utexas.edu/teaching/329/lectures/node110.html#eiter
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really nice: https://sites.psu.edu/stm9research/files/2017/09/Chapter-8-122ps83.pdf
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Reweighting: https://www.mv.helsinki.fi/home/rummukai/lectures/montecarlo_oulu/lectures/mc_notes4.pdf
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Finite-Size-Scaling: ./literature/finite-size-scaling.pdf
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Critical exponents: https://de.wikipedia.org/wiki/Kritischer_Exponent