Quasicrystallization by stochastic flips

Olivier Bodini (1) , Thomas Fernique (2) , Damien Regnault (2)

1 APR - Algorithmes, Programmes et Résolution
2 LIF - Laboratoire d'informatique Fondamentale de Marseille - UMR 6166

Tilings are often used as a toy model for quasicrystals, with the ground states corresponding to the tilings satisfying some local properties (matching rules). In this context, a challenging problem is to provide a theory for quasicrystals growth. One of the proposed theories is the relaxation process. One assumes that the entropy of a tiling increases with the number of tilings which can be formed with the same tiles, while its energy is proportional to the ratio of satisfied matching rules. Then, by starting from an entropically stabilized tiling at high temperature and by decreasing the temperature, the phason flips which decrease (resp. increase) the energy would become more and more favoured (resp. inhibited). Ideally, the tiling eventually satisfies all the matching rules, and thus shows a quasicrystalline structure. This paper describes a stochastic process inspired by this and discusses its convergence rate.

Mots clés: quasicrystal; flip; markov chain

Domaines: Mathématique discrète [cs.DM]

Olivier Bodini, Thomas Fernique, Damien Regnault. Quasicrystallization by stochastic flips. 2009. ⟨hal-00436517⟩