Ising

This package implements Markov-chain-Monte-Carlo methods (MCMC) to study the equilibrium thermodynamic behavior of the Ising model on a square lattice. Currently it implements the classical Metropolis algorithm as well as Wolff's cluster algorithm.

To watch the algorithms in action, a visualization app based on Makie is provided.

For now the simulation is restricted to a square lattice with constant couplings and external fields. It should however be fairly straightforward to extend the code to cover more general systems.

Please dig around the source, alter the UI to your liking, introduce new observables, etc!

If you spot an error or miss a feature, please feel free to open an issue or a pull request!

Compatibility and Installation

Julia >= 1.3 is required.

Clone the repository to your computer git clone https://github.com/skleinbo/Ising.git or download the latest release.

The MCMC routines assembled in src/mcmc.jl depend on StaticArrays. If you wish to use those only, consider installing the dependency manually with ] add StaticArrays.

The interactive application requires Makie and a few other packages (see Project.toml). Start Julia from the root of the cloned directory with julia --project. The first time you use the app, install the dependencies from the REPL with ] instantiate. Then start the app via include("bin/main.jl").

Usage

Load the MCMC methods with include(src/mcmc.jl). Make sure all dependencies are installed if you do not use the predefined environment (see above).

Help is available

help?> run_metropolis
search: run_metropolis _run_metropolis!

  run_metropolis(L, beta, h;Tmax=1, sweep=0, sample_interval=1)

  Sets up a random state and runs the Metropolis algorithm for a given set of parameters. Samples in defined intervals along the Markov-Chain.

  An initial thermal sweep to go to equilibrium may be specified.

  Returns an array of averaged observables: [E, E^2, m, m^2, m^4] with total energy E and magnetisation per spin m.

  Arguments
  ≡≡≡≡≡≡≡≡≡≡≡

    •    L::Integer: Linear system size

    •    beta::Float: Inverse temperature

    •    h::Float: External field

    •    Tmax::Integer: Number of steps

    •    sweep::Integer: Length of the initial sweep

    •    sample_interval::Integer: sample interval

  Example
  ≡≡≡≡≡≡≡≡≡

  julia> run_metropolis(50, 0., 0.;Tmax=50*10^3*50^2,sample_interval=10*50^2,sweep=10^3*50^2)
  5-element Array{Float64,1}:
    -39.9664      # <E>
   1753.5168      # <E^2>
      0.010382079999999908 # <m>
      0.00011572633600000021 # <m^2>
      1.6924325969920386e-8 # <m^4>

Look around the source code to see which methods are available. Here's a minimal example to get you going:

julia> include("src/mcmc.jl");

julia> L=128; # _Linear_ system size

help> run_metropolis
[...]

julia> run_metropolis(L, 1/2.26, 0.; Tmax=25*10^3*L^2,sample_interval=10*L^2,sweep=10^3*L^2)
5-element Array{Float64,1}:
  -253.3488
 64566.7776
     0.002934521484375
     9.098881483078003e-6
     1.0010325963705214e-10

If you want full time series, use metropolis_timeseries.

Examples

The example_data/ directory contains a CSV-file with measurements and the julia file used to generate them.

Interactive Visualization

include("bin/main.jl") opens a window with sliders and buttons to adjust parameters, a depiction of the current state, as well as energy and magnetization over time. Zoom in and out with the mouse wheel. "Speed" determines how many sweeps the program does per frame, i.e a value of 1/2 means 1/2*L^2 attempts to flip a spin each frame (default: 60fps; change with BASE_FPS[]=$fps). The cluster update algorithm will always perform one update per frame. Larger values typically exceed the desired frame time easily.

Controls:

• To change the system size (default 128x128) from the REPL: L[]=_linear system size_
• Either use the sliders, or set Temperature and external field from the REPL by assigning T[] and h[].
• Adjust frame rate: BASE_FPS[] = _fps_
• To change colors of the two states, set color1 and color2
• Press p to run/pause the simulation
• Reset the state by pressing r
• Speed up/slow down with = / - (QWERTY layout)
• Reset the view with c
• l performs a sweep of 1.000*N spin flips or 1.000 cluster updates respectively.
• a toggles between spin-flip and cluster updates.

Warning: Choosing BASE_FPS or speed to large will freeze the application, in particular when the system size is large too. You will notice the UI becoming unresponsive, because both the UI and the simulation run on the same thread (TODO: make this multi-threaded).

A Core i5 (2016) @ 3.2GHz caps out above L=512.

Caveats:

• [BUG] Sometimes only part of the array is displayed. Press c to reset the camera.