TY - JOUR

T1 - Long-range geometrical correlations in two-dimensional foams

AU - Dubertret, B.

AU - Rivier, N.

AU - Peshkin, M. A.

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 1998/1/23

Y1 - 1998/1/23

N2 - The statistical properties of two-dimensional, space-filling random cellular structures (foams, or their dual, random triangulations) in statistical equilibrium are obtained by maximum entropy inference and topological simulations We show by maximum entropy inference that for a broad class of foams (shell-structured, including three-sided cell inclusions), all two-cell topological correlators Aj(k, n) (average number of pairs of k-cell and n-cell at a topological distance j) are linear in n and k, the numbers of neighbours of the cells. This generalizes a correlation known for neighbouring cells (j = 1) which implies the linearity of Aboav's relation (between the total number of neighbours of the cells adjacent to a n-neighboured cell and n). Our results, verified by simulations, also build up Gauss's theorem for cellular structures. Any additional restriction in exploring local cell configurations, besides the constraints of filling space at random, will manifest itself through a deviation from linearity of the correlators Aj(k, n) and the Aboav relation. Notably, foams made of Feynman diagrams have additional, context-dependent restrictions and their Aboav relation is slightly curved. It is essential that the local random variable n denotes the number of neighbours of the cell and not that of its sides, whenever the two are different.

AB - The statistical properties of two-dimensional, space-filling random cellular structures (foams, or their dual, random triangulations) in statistical equilibrium are obtained by maximum entropy inference and topological simulations We show by maximum entropy inference that for a broad class of foams (shell-structured, including three-sided cell inclusions), all two-cell topological correlators Aj(k, n) (average number of pairs of k-cell and n-cell at a topological distance j) are linear in n and k, the numbers of neighbours of the cells. This generalizes a correlation known for neighbouring cells (j = 1) which implies the linearity of Aboav's relation (between the total number of neighbours of the cells adjacent to a n-neighboured cell and n). Our results, verified by simulations, also build up Gauss's theorem for cellular structures. Any additional restriction in exploring local cell configurations, besides the constraints of filling space at random, will manifest itself through a deviation from linearity of the correlators Aj(k, n) and the Aboav relation. Notably, foams made of Feynman diagrams have additional, context-dependent restrictions and their Aboav relation is slightly curved. It is essential that the local random variable n denotes the number of neighbours of the cell and not that of its sides, whenever the two are different.

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U2 - 10.1088/0305-4470/31/3/005

DO - 10.1088/0305-4470/31/3/005

M3 - Article

AN - SCOPUS:0032559157

VL - 31

SP - 879

EP - 900

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 3

ER -