PackLab

PackLab computes structure in three-dimensional hard-sphere systems. Its primary analytical workflow evaluates the Percus--Yevick (PY) approximation for equilibrium mixtures. It also generates explicit random sequential adsorption (RSA) configurations and samples fixed-volume equilibrium configurations with Metropolis Monte Carlo (MC).

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The workflows share physical inputs such as particle radii, number fractions, and volume fraction, but they answer different questions:

• PY is a fast analytical equilibrium reference for pair correlations, structure factors, and structure-aware scattering calculations.
• RSA creates an explicit, non-overlapping deposition configuration. It is irreversible and retains the history of accepted particles.
• Metropolis MC moves particles in a valid configuration to sample an equilibrium hard-sphere system at fixed volume, particle count, and radii.

Installation

Install the core package from PyPI:

pip install packlab

For scattering calculations, install the optional PyMieSim integration:

pip install "packlab[scattering]"

Or install the Conda package:

conda install -c martinpdes packlab

Verify the compiled package with:

python -c "import PackLab; print(PackLab.__version__)"

1. Compute a Percus--Yevick equilibrium reference

Use PY when you need equilibrium mixture correlations without generating an explicit packing. PackLab computes the partial pair correlations g_{ij}(r) and reciprocal-space correlations on an automatically resolved wavenumber grid.

import numpy as np

from PackLab import analytical, ureg

radii = np.array([75, 140]) * ureg.nanometer
domain = analytical.PercusYevickDomain(
    size=50 * ureg.micrometer,
    radii=radii,
    volume_fraction=0.25,
    number_fractions=np.array([0.7, 0.3]),
)
distances = np.linspace(0.0, 1.5, 300) * ureg.micrometer
result = analytical.PercusYevickSolver(
    densities=domain.particle_densities_per_radius,
    radii=domain.radii,
    wavenumber="auto",
).compute(distances)

g_12 = result.g[0, 1]
wavenumber = result.wavenumber

The automatic grid is a useful default. For a resolution study, use analytical.make_wavenumber_grid(...) and compare the resulting curves. PY is an analytical approximation to an equilibrium hard-sphere mixture; it does not create particle centres or reproduce the irreversible RSA process.

2. Generate an explicit RSA packing

RSA proposes particles one at a time and keeps only non-overlapping proposals. Accepted particles never move, so the final configuration is physically valid but history-dependent rather than an equilibrium sample.

from PackLab import monte_carlo, samplers, ureg

domain = monte_carlo.PackingDomain(
    5 * ureg.micrometer,
    5 * ureg.micrometer,
    5 * ureg.micrometer,
    use_periodic_boundaries=True,
)
sampler = samplers.UniformRadiusSampler(
    90 * ureg.nanometer,
    170 * ureg.nanometer,
    bins=8,
)
options = monte_carlo.RSAOptions()
options.random_seed = 42
options.maximum_attempts = 40_000
options.target_packing_fraction = 0.12

rsa_result = monte_carlo.RSASimulator(domain, sampler, options).run()
print(rsa_result.statistics.packing_fraction_geometry)
figure = rsa_result.plot_slice_2d(show=False)

PackingResult provides accepted centres, sampled radii, packing statistics, pair-correlation estimators, and plotting helpers. Radius samplers support constant, uniform, normal, log-normal, and discrete distributions.

3. Equilibrate hard spheres with Metropolis MC

Use Metropolis MC when an equilibrium configuration is needed. It can start from the valid RSA configuration above, but then proposes particle displacements; particle count, radii, and class labels remain fixed.

options = monte_carlo.MetropolisOptions()
options.random_seed = 34
options.number_of_sweeps = 500
options.maximum_displacement = 50 * ureg.nanometer

simulator = monte_carlo.MetropolisSimulator(
    domain,
    rsa_result.sphere_configuration,
    options,
)
mc_result = simulator.run()
print(simulator.statistics.acceptance_rate)
figure = mc_result.plot_slice_2d(show=False)

The number of sweeps alone does not establish equilibration. Discard an initial burn-in interval, assess autocorrelation for the quantity of interest, and compare larger systems when finite-size effects may matter.

Choosing the right workflow

Workflow	Use it when you need	Important limitation
PY analytical	Fast equilibrium pair correlations, structure factors, parameter sweeps, or scattering inputs.	It is an equilibrium approximation, not an explicit packing.
RSA	Particle centres, radius-sampling effects, deposition history, or a finite non-overlapping configuration.	It is irreversible and is not an equilibrium sampler.
Metropolis MC	An explicit equilibrium hard-sphere configuration at fixed volume and composition.	Equilibration, autocorrelation, and finite-size effects require checks.

Scattering

The optional PackLab.scattering workflow computes optical amplitudes with PyMieSim. Combine them with the PY correlation tensor when you need a structure-corrected mixture phase function. See the scattering examples for runnable single-particle and mixture calculations.

Documentation, validation, and citation

The online documentation contains theory, API reference, output conventions, assumptions, and executable galleries for PY, RSA, Metropolis MC, scattering, validation, and benchmarks.

For development:

git clone https://github.com/MartinPdeS/PackLab.git
cd PackLab
pip install -e ".[testing,documentation]"
pytest

If PackLab contributes to academic work, cite the archived Zenodo release you used. Release metadata is included in .zenodo.json.