Confirmation of the random tiling hypothesis by polar calculus

The random tiling hypothesis, first proposed by Elser and Henley in the 1980s, states that quasicrystals are entropy-stabilized and, hence, are high-temperature phases. We confirm the hypothesis for a two-dimensional T\"ubingen triangle tiling, which arises in molecular dynamics simulations with a Lennard-Jones-Gau{} potential, by investigating the temperature dependence of its two phason elastic constants λ₆ and λ₈. These are the second derivatives of the free energy F(χ₆,χ₈,T) with respect to symmetrized phason strain modes χ₆ and χ₈. At $T=0, F$ has a saddle point by descending along the χ₈ direction. Therefore λ₈<0 characterizes the quasicrystal's initial instability. The configurational entropy due to phason flips turns F upwards at higher temperatures, reverses the sign of λ₈, and leads to a stable quasicrystal. We obtain this result by applying geometric methods in the form of the polar calculus, where the projection window W is divided into atomic domains Cᵢ for each vertex environment. We extend the calculus to a dynamic one by separating the window into areas Pₖ that characterize the different kinds of phason flips. By phasonic deformation of the window, we can determine the types of flips and their frequency of occurrence in dependence on phason strain, perform energy relaxations by flips, and compute the configurational free energy. Previously, flips therein had been dealt with as uncorrelated and described as Ising spins. We consider nearest neighbor correlations between flips by extending the Ising to a Potts model and find that they constitute an important mechanism supporting the quasicrystal stability.