Abstract
A foam is a random partition of space into cells. In two dimensions, it is a planar, regular graph, consisting of trivalent vertices, edges and cells. The number n of neighbours of a cell (which is equal to the number of its edges) is the only, local random topological variable of the problem. One looks for the most probable distribution p n , subject to some information of the system. This information is of two types:
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(i)
Const ra ints (e.g. 〈n〉 = 6), pertaining to filling (topological) space. These are inevitable, but can be made redund ant. The redundancy is expressed as correlations, equations of state, between random variables on the same or different cells. These equations of state have been observed experimentally, and are known as the laws of Aboav-Weaire, Lewis, Peshkin , etc ….
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(ii)
Pri or knowledge on th e system, which can be the prior probability distribution qn, but also some symmet ry or t ransition probability of th e system. It must, however , be a local, and context-independent description of the state of the foam. We show, by simulat ions, that a foam is in statistical equilibrium, i.e. that it has the maxent , most probable distri bution Pn. We will give examples of priors, and also of rest rictions which are context-dependent and lead to foams off statistical equilibrium [1,2].
We will also demonstrate analytically by maxent, including selection of least informative priors, a different kind of equation of state, Lemaitre’s law [3], thereby justifying its surprising universality. Lemaitre et al. noticed that a relation between two different measures (”order parameters”) of disorder, was universal in two-dimensional foams. Universality of the relation may be thought, at first sight, to express their equivalence. This is not so: Lemaitre’s law is the virial equation of state for foams. It is non-linear, thus non-trivial. In a gas, non-triviality reflects the specific interactions between the atoms or molecules of the particular gas. In foams, it is universal [4].
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Rivier, N., Dubertret, B., Aste, T., Ohlenbusch, H. (1999). Universality, Prior Information and Maximum Entropy in Foams. In: von der Linden, W., Dose, V., Fischer, R., Preuss, R. (eds) Maximum Entropy and Bayesian Methods Garching, Germany 1998. Fundamental Theories of Physics, vol 105. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4710-1_5
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DOI: https://doi.org/10.1007/978-94-011-4710-1_5
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