Europhys. Lett., 54 (1), pp. 112–117 (2001) EUROPHYSICS LETTERS 1 April 2001 The equilibrium state of 2D foams M. F. Miri 1,2 and N. Rivier 2 1 Institute for Advanced Studies in Basic Sciences - P.O. Box 45195-159, Zanjan, Iran 2 LDFC, Universit´ e Louis Pasteur - 3, rue de l’Universit´ e, 67084 Strasbourg, France (received 5 July 2000; accepted in final form 22 January 2001) PACS. 82.70.Rr – Aerosols and foams. PACS. 64.60.Cn – Order-disorder transformations; statistical mechanics of model systems. PACS. 05.90.+m – Other topics in statistical physics, thermodynamics, and nonlinear dynam- ical systems. Abstract. – The dynamics of two-dimensional cellular networks (foams) is written in terms of coupled rate equations, which describe how the population of s-sided cells is affected by cell disappearance or coalescence and division. In these equations, the effect of the rest of the foam in statistical equilibrium on the disappearing or dividing cell is treated as a local mean field. The rate equations are asymptotically integrable; the equilibrium distribution Ps of cells is essentially unique, driven and controlled by the topological transformations for cells with s< 6+ √ μ2. Asymptotic integrability of the equations, and unique distribution, are absent in a global mean-field treatment. Thus, short-ranged topological information is necessary to explain the evolution and stability of foams. Introduction. – Two-dimensional random cellular networks (“2D foams”) are widespread innature(soapfroths,fragmentationpatterns,biologicalepidermis,etc.)[1]. Theyarerandom partitions of the plane by cells, which are topological polygons [1–3]. Disorder or absence of specific adjustment imposes minimal incidence numbers (3 edges incident on a vertex). Foams evolve into a stationary state of statistical equilibrium, with an invariant distribution of cell shapes P s , where s,thenumberofsidesofacell,istheonlytopologicalrandomvariable[1,4]. Statistical equilibrium is established through local, elementary topological transformations (ETT), which can be an edge flip (T1 transformation) or the disappearance of a 3-sided cell (T2 transformation). In biological tissues, combinations of these transformations constitute cell division (m, for mitosis) or its inverse, cell disappearance (d) (coalescence of two cells by removal of their interface). It turns out that the asymptotic behaviour of the distribution P s is “universal”. We show here that this universality is due to asymptotic integrability of the equations describing the variations of the cell population P s under ETT. Rate equations in the local mean field approximation. – The stationary distribution P s forafoamisthesolutionofcoupledrateequations,whichaccountforthelocal,butcorrelated c  EDP Sciences