Pattern forming instabilities leading to the formation of hexagonal clusters

It is well known amongst physicists that seemingly distinct systems can sometimes exhibit strikingly similar behaviors under some circumstances. In this talk, I will draw a parallel between two comparable cases of clustering observed in very different models : on the one hand, Brownian particles interacting via a "soft potential", purely repulsive but allowing overlaps between particles (two characteristics typical of various types of polymeric macromolecules like polymer chains or dendrimers for example). On the other hand, a model of population dynamics with neighborhood dependent birth and death rates, accounting for the spatial heterogeneity of the competition between individuals. Both of these systems exhibit a similar phase transition leading to a segregation of their particles into clusters arranged in hexagonal patterns. Macroscopic equations can be derived for the two models and their structures are quite comparable, despite the fact that the number of particles is not conserved in the second case. Their linear stability analysis shows that this clustering is induced by an instability of the homogeneous state, triggered by the non-local mutual interactions.