Scaling Behaviour in Mathematical Models of Aggregation

The rings of Saturn are among the most spectacular structures in our solar system, made up of innumerable fragments of ice and rock ranging in size from metres to millimetres. The stability of the rings is a matter of some debate among planetary scientists. The combined effect of sticky inter-particle collisions and gravity could cause the rings to coalesce into one or more small moons, but this is not observed. Empirical data also suggest that the distribution of particle sizes in the rings follows a power law, but the physical mechanism for this scaling is unknown.
A theory based on the statistical physics of coalescence-fragmentation processes proposes that these two questions are related1. This theory is based on the observation that two particles of ice colliding with high relative momentum are much more likely to shatter than to coalesce. The key idea is that the generation of large particles by coalescence of small particles could be balanced on average by the destruction of large particles by shattering. Theoretically, a statistically stationary distribution of particle sizes could result, without particles coalescing into a small number of large particles, i.e. the formation of new moons. Mathematical analysis of the statistical dynamics of coalescence-shattering processes also shows that this statistically stationary distribution can be a power law.
In a new paper, LML Fellow Colm Connaughton and colleagues offer a detailed theoretical study2 of the kinetic equations for such coalescence-shattering processes, which shows that steady states with power law size distributions are generic. However, they also find that the exponent of this distribution depends in a crucial way on the relative importance of collisions between particles of similar size as compared to collisions between very large and very small particles.
1 Brilliantov, N., Krapivsky, P. L., Bodrova, A., Spahn, F., Hayakawa, H., Stadnichuk, V., & Schmidt, J. (2015). Size distribution of particles in Saturn’s rings from aggregation and fragmentation. Proceedings of the National Academy of Sciences112(31), 9536-9541.
2 Connaughton, C., Dutta, A., Rajesh, R., Siddharth, N., & Zaboronski, O. (2018). Stationary mass distribution and nonlocality in models of coalescence and shattering. Physical Review E97(2), 022137
Links for the latter paper are
and (preprint)

Leave a Reply

Your email address will not be published. Required fields are marked *