ScalingCollapse.jl

ScalingCollapse.jlAutomatic finite-size scaling

Find the critical point and exponents that collapse data from many system sizes onto one master curve, in pure Julia.

Getting Started

One call, full fit

Pass your data and system sizes; a parameter scan and a Nelder–Mead refinement find the best collapse.

Flexible scaling forms

Use the presets for the usual observables, fix any parameter by name, or write your own scaling function.

Parameter errors

Profile, conditional or bootstrap errors for every fitted parameter, and the quality landscape around the fit.

What is ScalingCollapse.jl?

Near a continuous phase transition, an observable measured in systems of linear size follows a scaling form such as

Plotted against and multiplied by , the curves of all system sizes then fall onto one master curve , but only for the right values of , and . ScalingCollapse.jl finds these values for you: it measures how well the rescaled curves collapse, minimizes this measure over the parameters, and estimates their errors.

julia

using ScalingCollapse
using Random

# synthetic data with T_c = 2.0, ν = 1.0 and β = 0.125
F(z) = 1 / (1 + exp(2z))
Ts = collect(range(1.8, 2.2; length=41))
Ls = [16, 32, 64, 128]
ys = [L^(-0.125) .* F.((Ts .- 2.0) .* L) .+ 0.002 .* randn(Xoshiro(L), 41) for L in Ls]

sp = ScalingProblem(Ts, ys, Ls;
    sf=ScalingFunction(:xy; p_names=["T_c", "nu", "beta"]),
    p_space=[1.9:0.02:2.1, 0.5:0.1:1.5, 0.0:0.05:0.5],
)

ScalingProblem
    Scaling function:
        x -> (x - T_c) * L^(1/nu)
        y -> y * L^(beta/nu)
    Optimal parameters:
        T_c = 2.0000170402418176 ± 0.00041621320597460946
        nu = 1.0483092977275075 ± 0.0388852776812044
        beta = 0.12956803324166713 ± 0.007245270920234711
    Optimization interval dx:
        -Inf < x < Inf
    Searched parameter space:
        T_c ∈ 1.9:0.02:2.1
        nu ∈ 0.5:0.1:1.5
        beta ∈ 0.0:0.05:0.5

julia

using CairoMakie

colors = cgrad(:Blues)[range(0.35, 1.0; length=length(Ls))]
fig = Figure(size=(760, 320))
ax1 = Axis(fig[1, 1]; xlabel=L"T", ylabel=L"Y", title="data")
ax2 = Axis(fig[1, 2]; xlabel=L"(T - T_c)\,L^{1/\nu}", ylabel=L"Y\,L^{\beta/\nu}", title="collapse")
xs_scaled, ys_scaled, _, _ = scaled_data(sp)
for i in eachindex(Ls)
    scatterlines!(ax1, Ts, ys[i]; color=colors[i], markersize=6, label="L = $(Ls[i])")
    scatter!(ax2, xs_scaled[i], ys_scaled[i]; color=colors[i], markersize=7)
end
xlims!(ax2, -6, 6)
Legend(fig[1, 3], ax1)
fig

Installation

ScalingCollapse.jl is a registered Julia package (Julia 1.9 or later):

julia

using Pkg
Pkg.add("ScalingCollapse")

Where to go next

Getting Started walks through a first collapse step by step.
The Manual explains the input formats, the scaling functions, the fit, the error analysis and the quality functions.
The Examples apply the package to real data: the 2D Ising model, site percolation, a custom scaling form and more.
The API Reference documents every exported function.

Development status

ScalingCollapse.jl is still young. Expect breaking changes between minor versions, and please open an issue if you run into a problem.

Citing

If you use ScalingCollapse.jl in your research, please cite it via its Zenodo record, doi:10.5281/zenodo.10712427.