ScalingCollapse

This is a package for automatic finite size scaling. Finite size scaling is a method to determine the critical parameters of a phase transition by exploiting, so called, finite size effects. This package implements an automatic optimization algorithm to find the best parameters for a finite size scaling collapse.

It is still in early development, so expect bugs and breaking changes. Feel free to create issues in case you stumble upon any problems!

For installation run the following command:

julia> using Pkg; Pkg.add("https://github.com/maltepuetz/ScalingCollapse.jl.git")

To find the best parameters for a finite size scaling analysis, we need to create a ScalingProblem. Lets say we have data for the binder cumulant of the 2D Ising model binder for different temperatures Ts and different system sizes Ls. We want to find the critical temperature and the critical exponent of the correlation length. Let's create a ScalingProblem as follows:

using ScalingCollapse
sp = ScalingProblem(Ts, binder, Ls;
    sf = ScalingFunction(:x),
    dx = [-1.0, 1.0],
)

The kwarg sf is used to set the scaling function. In this case we use the scaling function preset :x which only rescales the x-axis as x -> (x - p1)/p1 * L^(1/p2). We can also give the parameters custom names by passing the kwarg p_names::Vector{String} to the ScalingFunction constructor. We use the kwarg dx to limit the optimization to a fixed interval because powerlaw dependece is usually only accurate close to the critical point. The default value is dx = [-Inf, Inf]. Other kwargs are:

• p_space = [0.1:0.1:3.0 for _ in 1:n_parameters(sf)]] This sets the parameter space that will be scanned for local minima of the cost function. The optimization algorithm will start from the determined local minima to find the global minimum. Note that the the optimal parameters should lie within the given parameter space. Also note, that the initial scan can take a while for too small stepsizes. If you have a good idea where the optimal parameters should lie, you can set the parameter space accordingly. You can also skip the local minima search completely, by setting the kwarg starting_ps.
• starting_ps::Vector{Float64} This kwarg can be used to skip the local minima search. If starting_ps is set to a specific parameter set, the optimization algorithm will start from this point. This can be useful if you already have a good guess for the optimal parameters. Note that you might not find the optimal parameters if you start from a bad guess.
• error::Bool=false This kwarg can be used to turn the error calculation on or off. If error=true, the error of the parameters will be calculated using a 2S method. This means that we fix all but one parameter and vary the free paramter, such that we double the cost function value. The error is then the maximum of the variation in plus and minus direction.
• verbose::Bool=false This kwarg can be used to turn the verbose output on or off. If verbose=true, the algorithm will print some information about the optimization process.

There are also other preset ScalingFunctions:

• :x Rescales the x-axis as x -> (x - p1)/p1 * L^(1/p2).
• :xy This is the default scaling function. It rescales the x-axis as x -> (x - p1)/p1 * L^(1/p2) and the y-axis as y -> y * L^(p3/p2).
• :xny This scaling function is similar to :xy but it rescales the y-axis as y -> y * L^(-p3/p2). In this case the critical exponent p3 has a negative sign.

Note that you can also create your own scaling function! We can make our scaling function looking a little bit cooler by giving our parameters custom names with the kwarg p_names::Vector{String}.

Now that we know the basics, let us solve another scaling problem. This time we know that the critical temperature is somewhere between T_c = 2.0 and T_c = 2.4 and the critical exponent is somewhere between nu = 0.5 and nu = 1.5. We set the parameter space accordingly:

using ScalingCollapse
sp = ScalingProblem(Ts, binder, Ls;
    sf = ScalingFunction(:x; p_names = ["T_c", "nu"]),
    dx = [-1.0, 1.0],
    p_space = [2.0:0.1:2.4, 0.5:0.1:1.5],
    error = true,
)

We limited the parameter space so the algorithm runs faster, calculated some error bars and made everything look cooler by giving the parameters custom names!

Let's say that we know from our statistical mechanics course that the critical exponent of the correlation length is nu = 1. We can fix this parameter by passing it as a kwarg to the ScalingFunction constructor.

using ScalingCollapse
sp = ScalingProblem(Ts, binder, Ls;
    sf = ScalingFunction(:x; p_names = ["T_c", "nu"], nu=1),
    dx = [-1.0, 1.0],
    p_space = [2.0:0.1:2.4],
    error = true,
)

Now the algorithm will only optimize the critical temperature T_c and the critical exponent of the correlation length will be fixed to nu = 1.

To be written...