Universality in two-dimensional cellular structures evolving by cell division and disappearance
MirFaez Miri, N. Rivier · Physical Review E · 2006
The dynamics of two-dimensional cellular networks is written in terms of coupled population equations, which describe how the population of s-sided cells is affected by cell division and disappearance. In these equations the effect of the rest of the foam on the disappearing or dividing cell is treated as a local mean field. Under not too restrictive conditions, the equilibrium distribution P(s) of cells satisfies a linear difference equation of order two or higher. The population equations are asymptotically integrable. The asymptotic integrability implies a "universal" distribution P(s) approximately Cs-kZs for large values of s, which is also the Boltzmann distribution associated with the maximum entropy inference. Asymptotic integrability of the population equations is absent in a global mean-field approximation. The importance of short-range topological information to control the evolution of foams is thus confirmed.