RSA Monte Carlo: mathematics and assumptions

Random sequential adsorption (RSA), also called random sequential addition, builds an explicit packing by random insertion. Here, Monte Carlo means that proposals are drawn randomly. It does not imply equilibrium sampling. The defining RSA postulate is irreversible deposition: an accepted sphere stays at its centre forever. See Monte-Carlo packings for the Python workflow, Metropolis Monte Carlo: mathematics and assumptions for movable spheres, and Percus–Yevick: mathematics and assumptions for the analytical equilibrium reference.

Model and insertion rule

A sphere has centre \(\mathbf{x}_n\) and radius \(a_n>0\). The orthorhombic box has side lengths \(L_x,L_y,L_z\) and volume \(V=L_xL_yL_z\). Spheres are rigid, additive, and impenetrable: two surfaces can touch but their interiors cannot intersect. The exclusion (or contact) distance between centres is \(a_n+a_m\).

For a proposed radius \(a\) and centre \(\mathbf{x}\), the acceptance indicator is

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

where \(\Theta(z)=1\) for \(z\geq0\) and zero otherwise. \(B\) checks boundary admissibility, and \(d\) is the appropriate centre distance. Acceptance adds one sphere; rejection leaves the packing unchanged. Every acceptance changes the exclusion region for later trials, so the result depends on proposal history. Gravity, attraction, particle deformation, and rearrangement are absent.

With walls, \(a\leq x_\alpha\leq L_\alpha-a\) for each Cartesian coordinate \(\alpha\). PackLab draws wall-bounded centres from this admissible interval, rather than counting centres outside it as trials. containment_padding adds an optional wall margin; minimum_center_separation_addition adds an optional pair clearance. These settings change the geometric constraints from ideal touching spheres.

Periodic images and the minimum-image convention

Periodic boundaries identify opposite faces of a box. A periodic image is a translated copy of a particle, with centre

\[\mathbf{x}_n+(m_xL_x,m_yL_y,m_zL_z),\qquad m_\alpha\in\mathbb{Z}.\]

Images describe a repeated configuration; they are not additional independent particles in the stored box. Wrapping a centre into the primary box uses \(x_\alpha\bmod L_\alpha\).

The minimum-image convention selects the closest translated copy when computing a separation. For an orthorhombic box,

\[\Delta x_\alpha^{\mathrm{MI}}= \Delta x_\alpha-L_\alpha\operatorname{round} \!\left(\frac{\Delta x_\alpha}{L_\alpha}\right),\qquad d_{\mathrm{MI}}=\left\|\Delta\mathbf{x}^{\mathrm{MI}}\right\|.\]

For example, centres at \(0.1\,\mathrm{m}\) and \(9.9\,\mathrm{m}\) along an axis of length \(10\,\mathrm{m}\) are separated by \(0.2\,\mathrm{m}\) across the boundary. A particle must also avoid its own images: \(2a\leq\min(L_x,L_y,L_z)\). Equality permits contact. RSA rejects a trial violating this condition; Metropolis initialization and packing archive validation reject such a configuration.

Radius probabilities and realised composition

The proposal distribution describes radii attempted, whereas the realised distribution describes radii accepted. Larger spheres can be harder to place, so these distributions may differ. With enforce_radii_distribution=True, PackLab retries a drawn radius at new centres until it succeeds or a stopping condition is reached. This changes the deposition protocol; it does not create equilibrium samples.

NormalRadiusSampler draws an underlying normal variable \(X\sim\mathcal{N}(\mu,\sigma^2)\), conditions on \(X>0\), then clips at \(a_{\max}\). For \(\sigma>0\), with \(\Phi\) the standard normal cumulative distribution function, its density below the cap is

\[f(a)=\frac{\exp[-(a-\mu)^2/(2\sigma^2)]} {\sigma\sqrt{2\pi}\,\Phi(\mu/\sigma)},\quad 0<a<a_{\max}.\]

There is an atom (a finite probability at one radius) at the upper cap of mass \([1-\Phi((a_{\max}-\mu)/\sigma)]/\Phi(\mu/\sigma)\). This is conditioning at zero and clipping at the upper bound, not truncation at both ends. The mean parameter must be non-negative; when \(\sigma=0\), it must be positive and the radius is \(\min(\mu,a_{\max})\). Optional binning maps draws to bin midpoints; to_bins integrates the same probabilities, including the upper atom. The resulting mean and standard deviation generally differ from the input parameters. Earlier zero-clipped normal sampling could produce zero radii; positive conditioning changes that distribution and seeded trajectories.

Packing fraction, correlations, and stopping

The occupied volume fraction is

\[\eta=\frac{1}{V}\sum_{n=1}^{N}\frac{4\pi a_n^3}{3}.\]

For a homogeneous mixture, the partial pair correlation \(g_{ij}\) describes the expected number of species \(j\) centres in a shell around a species \(i\) centre:

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

Here \(\rho_j\) is number density, not volume fraction. PackLab counts unordered pairs, contributes both ordered class orientations to the histogram, and accounts for the finite number of same-class pairs. If \(T=N(N-1)/2\) pairs are available and \(M<T\) are sampled uniformly without replacement, each has inclusion probability \(M/T\); multiplying sampled counts by \(T/M\) gives an unbiased estimate of full counts. maximum_pairs=0 examines all pairs. random_seed controls capped pair sampling and defaults to deterministic zero; this differs from simulation seeds, where zero requests a random seed. Pair-distance extraction caps the number examined before filtering by radius, so it can return fewer distances.

The correlation range ends at half the shortest box side, within which a periodic spherical shell is unambiguous. With walls, bulk shell normalization has boundary bias; box-size checks are necessary. Sampling noise decreases with more pairs and independent packings. Reordering particles leaves full pair counts unchanged; capped estimates agree statistically, not necessarily for the same seed.

maximum_attempts limits the outer insertion loop; maximum_consecutive_rejections, maximum_spheres, and target_packing_fraction provide other stopping conditions. When radius retries are enabled, one outer iteration can make several placement attempts. A rejection limit is not a proof of saturation, meaning no admissible insertion remains. Report the seed, stopping rule, realised composition, volume fraction, and box dimensions. RSA is neither close packing nor an equilibrium fluid calculation.

Further reading

Zhang and Torquato discuss RSA and rigorous saturation detection in Precise algorithm to generate random sequential addition of hard hyperspheres at saturation (2013). PackLab uses finite proposal budgets rather than their saturation-certifying algorithm.