Statistical Physics — Computational Models in Julia

Classical models of statistical physics and complex systems implemented in Julia: equilibrium phase transitions, out-of-equilibrium collective motion, synchronization, and stochastic processes — with Monte Carlo, ensemble averaging, and finite-size critical analysis.

Developed for the graduate course Física Estadística Computacional (MSc in Physics). Author: José Ake · github.com/ak3sit0

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Models

Module	Model	Method	Key result
spin-models/ising	2D Ising	Metropolis and Wolff Monte Carlo, Binder cumulant	Ferro–para transition; recovers Onsager's exact Tc = 2/ln(1+√2) ≈ 2.269
spin-models/xy	2D XY	Metropolis, spatial correlations G(r)	Quasi-long-range order (algebraic vs. exponential decay)
vicsek	Vicsek	Self-propulsion, threaded ensembles	Order–disorder transition (out of equilibrium), φ(η)
kuramoto	Kuramoto	Coupled phase oscillators	Spontaneous synchronization, order parameter r(K)
random-walks	Brownian, Ornstein–Uhlenbeck, Run-and-Tumble, resetting	Langevin dynamics, MSD	Diffusion, persistence, stochastic resetting

Highlighted Results

Ising Model: Critical Temperature via Binder Cumulant

The 2D Ising critical point is found via the Binder cumulant crossing across lattice sizes (finite-size scaling), recovering the exact Onsager value Tc = 2/ln(1+√2) ≈ 2.2692.

Also computed: magnetization, specific heat, and spatial correlations.

Vicsek Model: Order-Disorder Phase Transition

Self-propelled particles exhibit a sharp phase transition from ordered (aligned flocking) to disordered (random motion) as noise increases.

Low Noise (Ordered)	High Noise (Disordered)	Order Parameter φ(η)

Gallery

Ising at Tc	XY (low T)	Vicsek (ordered)

Engineering

• vicsek/SimulationUtils.jl — a reusable module: metaprogramming (@pbc, a zero-overhead @debug, @timed) and a higher-order parallel_ensemble that runs realizations across threads with an independent RNG per thread, keeping results reproducible.
• Observables are estimated by averaging over independent realizations with reported SEM.

Reproducibility

Requires Julia 1.9+. Dependencies are pinned in Project.toml:

using Pkg
Pkg.activate(".")
Pkg.instantiate()

Notebooks run with IJulia (or VS Code + the Julia extension). For threading: export JULIA_NUM_THREADS=8.

Usage

Using modules directly

Each model is available as a reusable Julia module in src/:

using Pkg
Pkg.activate(".")

# Load a model
using Ising

# Create and simulate
model = Ising.IsingModel(L=20, T=2.269)
Ising.equilibrate!(model, 1000)
M = Ising.compute_magnetization(model)

Available modules: Ising, XY, Vicsek, Kuramoto, RandomWalks, SimulationUtils.

Running tests

Physics-based tests validate key properties (e.g., Onsager critical temperature, MSD exponents):

julia --project -e 'using Pkg; Pkg.test()'

Tests run automatically on every push via GitHub Actions (see .github/workflows/test.yml).

Structure

statistical-physics-computational/
├── spin-models/      ising/ · xy/
├── vicsek/           vicsek.ipynb · SimulationUtils.jl
├── kuramoto/         kuramoto.ipynb
├── random-walks/     brownian, ou, run-and-tumble, resetting notebooks
├── assets/           GIFs and animations
└── Project.toml      reproducible environment

References

Onsager (1944), Crystal Statistics I; Landau & Binder, A Guide to Monte Carlo Simulations in Statistical Physics; Vicsek et al. (1995); Kuramoto (1975). Per-model references live in each module's README.

License

Code under the MIT license. © 2026 José Ake.