Equilibrium behavior of the tiling model

Abstract

Equilibrium properties of the tiling model, recently introduced by Stillinger and Weber as a means of studying glass phenomena, are investigated. In the two-dimensional model, a square lattice is covered by tiles of all sizes. The tiles represent domains of well-packed particles and the boundaries between the domains have a positive mismatch energy proportional to the length of the wall. The existence of the thermodynamic limit for this model is proved. It is shown how to obtain bounds for the free energy and the transition temperature from the free energy of semi-infinite strips. A transfer-matrix method is developed for calculating the thermodynamic properties of such semi-infinite strips. The best bound obtained is λ/k_BT_c≥0.2459 from a 9^x∞ strip, where λ is the basic energy parameter in the problem. By extrapolation, the transition temperature is estimated as λ/k_BT_c=0.270 02. By direct counting of states for finite squares, the infinite-temperature entropy is obtained by extrapolation as s_∞=0.314. A connection between the tiling model and electrical networks is discussed in Appendix A.