Mean-field approach to Random Apollonian Packing

Bubble nucleation is a phenomenon ubiquitous in physics, with applications ranging from the geometry of tree crowns, the structure of porous media and of sphere packing. Bubbles also find applications in cosmology such as the characterization of cosmic voids in the large scale structure and the signatures of cosmological phase transitions. In a new preprint, Pierre proposes a new method to determine the fractal properties of the so-called Random Apollonian Packing.

Random Apollonian Packing (RAP) is inspired by the better-known Apollonian Gasket. In mathematics, an Apollonian gasket is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three.

gasket

In Ref. [1], Pierre examines a related mechanism in which \(d\)-dimensional spheres are randomly seeded in space, one at a time, within a finite-sized volume, and with the largest possible radius that avoids overlap.

packing

The interest of the RAP mechanism is that it is thought to share universal properties with more general dynamic mechanisms, such as the ABK mechanism (named after Andrienko, Brilliantov and Krapivsky) in which bubbles grow linearly with time.

Pierre has built a model to explain at which rate Random Apollonian Packings grow after the insertion of one sphere after the other. The model’s prediction for the fractal properties of RAP are consistent with numerical simulations made in two, three and four dimensions.

If you want to explore a RAP in depth, feel free to zoom in that picture, it contains more than a million spheres.

References

  • [1] Auclair P 2022 Mean-field approach to Random Apollonian Packing arXiv e-prints arXiv:2211.07509
    Abstract: arXiv:2211.07509