Statistical Mechanics of Aperiodic Tilings^{1}

December 18, 1995 — Rutgers University

Steven A. Janowsky

Quasicrystals are a form of matter that exhibit long range order, but are nonperiodic. One attempt to model the properties of quasicrystals has been through the use of aperiodic tiling models such as the Penrose tiles. More accurately, one is really using the tiling models to describe the quasicrystal at zero temperature—at positive temperatures the quasicrystals will have defects that are unaccounted for in the tiling models.

We attempt to remedy this defect by studying tilings at finite temperature. We do this via the standard language of equilibrium statistical mechanics by first translating the tiling model into an equivalent spin system. At zero temperature the system is ordered; at high temperature it is disordered. Numerical simulations indicate that there is no long range order at any finite temperature, but that there may be a Kosterliz-Thouless type phase transition at T ≈ 0.43.

1joint work with Hans Koch