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 [1], I have studied the fractal properties of Random Apollonian Packing (RAP), 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.

Apollonian Gakset started from three equidistant cercles

In this article [1], I examine a related mechanism in which $d$-dimensional spheres are seeded one at a time randomly in space in a finite-sized volume, and take the largest possible radius that avoids overlap.

One realization of a Random Apollonian Packing with the first 1000 spheres

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. However the RAP mechanism can be approached from a completely different angle. Namely, the ABK mechanism is dynamical - multiple spheres are growing at the same time and collide with one another - whereas the RAP mechanism is sequential - spheres are added one at a time in a static environment.

After intensive numerical simulations, I have 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 our numerical simulations in two, three and, to a lower degree, in four four dimensions.

If you want to explore a RAP in depth, feel free to zoom in the picture below representing a RAP in two dimensions containing a million spheres.


  • [1] Auclair, P. “Mean-field approach to Random Apollonian Packing”, arxiv:2211.07509, 2022

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