Theoretical background

PackLab offers three complementary hard-sphere workflows. Percus–Yevick (PY) is an analytical equilibrium reference; random sequential adsorption (RSA) constructs explicit, irreversible deposition configurations; and Metropolis Monte Carlo (MC) samples an equilibrium system through particle moves. They can share radii, composition, and volume fraction, but they do not represent the same physical process.

Technique guides

Use the Analytical structure factors guide for the PY Python workflow and the Monte-Carlo packings guide for RSA packing generation and Metropolis MC moves. The References and citation page collects the underlying literature and release citation information.

Percus–Yevick hard-sphere mixtures

The analytical namespace evaluates the multicomponent PY approximation for an equilibrium hard-sphere mixture. Given particle radii, number fractions, and a total volume fraction, it derives species densities and computes partial pair correlations \(g_{ij}(r)\) and structure factors \(S_{ij}(k)\).

PY is useful when a rapid equilibrium reference is needed for parameter sweeps or a structure-aware scattering calculation without generating sphere centres. It is an approximation: its physical accuracy depends on density, composition, and size ratio. Its numerical accuracy also depends on the wavenumber grid used to recover real-space correlations. These are separate questions. The automatic grid is a useful default, while explicit grids allow a resolution study for a particular distance range.

For species number densities \(\rho_i\), number fractions \(x_i\), and total number density \(\rho\), the mixture has

\[\rho_i = x_i\rho, \qquad \phi = \sum_i \rho_i \frac{4\pi a_i^3}{3}.\]

The partial pair correlation is defined by the expected number of species \(j\) centres in a shell around a species \(i\) centre,

\[\mathrm{d}N_j = \rho_j g_{ij}(r)\,4\pi r^2\mathrm{d}r.\]

For hard spheres, \(g_{ij}(r)=0\) inside the excluded core \(r<a_i+a_j\). Defining \(h_{ij}(r)=g_{ij}(r)-1\), the reciprocal-space structure-factor matrix is

\[S_{ij}(k) = \delta_{ij} + \sqrt{\rho_i\rho_j}\,\widetilde{h}_{ij}(k).\]

The Ornstein–Zernike relation couples every species pair through the direct correlations \(c_{ij}\). Percus–Yevick closes that relation by enforcing the hard core and setting the direct correlation to zero outside it. This is why PY is a useful, fast reference but still an approximation.

Random sequential adsorption

Random sequential adsorption (RSA) proposes one sphere at a time. A proposal is retained only when it does not overlap an accepted sphere. Once accepted, the sphere never moves; a rejected proposal is discarded. Early accepted spheres can therefore block positions permanently, making the final packing depend on the random proposal order. RSA produces a physically valid, history-dependent deposition configuration, not an equilibrium hard-sphere sample. In PackLab, maximum_attempts limits rejected proposals and target_packing_fraction provides an optional earlier stopping condition.

For spheres with radii \(r_i\) in a domain of volume \(V\), the packing fraction is

\[\phi = \frac{1}{V}\sum_i \frac{4\pi r_i^3}{3}.\]

The radius sampler is consequently part of the physical model: a monodisperse simulation and a broad polydisperse simulation at the same nominal packing fraction need not have the same structure.

For a candidate centre \(\mathbf{x}\) and radius \(a\), RSA accepts a proposal only when it is outside every excluded volume. Symbolically,

\[A(\mathbf{x}, a) = \prod_{n=1}^{N} \Theta\!\left(\lVert\mathbf{x}-\mathbf{x}_n\rVert-a-a_n\right),\]

where \(\Theta\) is one for a non-overlapping proposal and zero otherwise. Once a proposal is accepted, it becomes part of the product for every later proposal. This simple rule is the source of RSA’s history dependence.

Metropolis Monte Carlo

Metropolis MC starts from any valid non-overlapping configuration, keeps every particle radius and class label fixed, and proposes symmetric single-particle displacements. A proposed move is accepted when it remains in the domain and creates no overlap. Repeated moves target the equilibrium distribution at the chosen particle count and volume; unlike RSA, accepted particles can move and rearrange.

The initial states retain memory of their starting configuration, so discard a burn-in interval before collecting measurements. Successive MC states may also be similar; their autocorrelation indicates how many sweeps should separate approximately independent samples. Finally, repeat a calculation for larger boxes or particle counts when bulk correlations are required, because a finite simulation box can alter the measured result.

For a symmetric proposal distribution, the Metropolis acceptance probability is

\[P_{\mathrm{accept}} = \min\!\left[1, \exp\!\left(-\beta\Delta U\right)\right].\]

The hard-sphere potential is either zero (no overlap) or infinite (overlap), so this reduces to probability one for a valid move and zero for an overlapping move. For an observable \(A\), estimate the autocorrelation \(\rho_A(t)\) across sampling lags. Its integrated value sets the approximate number of independent observations,

\[N_{\mathrm{eff}} \approx \frac{N}{1 + 2\sum_{t=1}^{\infty}\rho_A(t)}.\]

In practice, truncate the sum once the measured autocorrelation has decayed into noise and compare results across larger domains.

Periodic boundaries

With periodic boundaries enabled, opposite faces of the simulation box are identified. A sphere close to one face can therefore overlap a sphere close to the opposite face. This reduces boundary artefacts when estimating bulk properties such as a pair-correlation function. Use non-periodic boundaries when the edges themselves are part of the system being modelled.

Pair correlations and structure factors

RSA and MC configurations can estimate the radial pair-correlation function \(g_{ij}(r)\) from accepted centres. The analytical namespace calculates the corresponding PY pair correlations and structure factors over a user-supplied wavenumber grid. Match radii, number fractions, total volume fraction, boundary conditions, and the definition of each observable before comparing results.

Agreement between an equilibrated MC calculation and PY can test how closely the approximation represents a specified equilibrium mixture. RSA–PY differences are expected even with matched inputs, because irreversible RSA deposition and an equilibrium hard-sphere fluid are different models. Use explicit configurations when their centres, finite size, or deposition history are the quantities of interest; use PY when a fast analytical equilibrium reference is the useful object.