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 $P\left(s\right)$ 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 $P\left(s\right)\sim C{s}^{-\kappa }{z}^s$ 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.

• Received 21 September 2005

DOI:https://doi.org/10.1103/PhysRevE.73.031101