Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry

TL;DR

This study uses Monte Carlo simulations to evaluate the entropy and phason elastic constants of 2D-fold symmetric rhombus tilings, confirming theoretical predictions and analyzing finite-size and finite-D effects.

Contribution

It provides numerical estimates of entropy and elastic constants for tilings with any 2D-fold symmetry, validating and extending existing theoretical results.

Findings

Entropy becomes insensitive to size, boundary conditions, and phason strain at large D.

Large size and large D limits of entropy commute.

Finite D and size scalings of entropy are characterized.

Abstract

We perform Transition matrix Monte Carlo simulations to evaluate the entropy of rhombus tilings with fixed polygonal boundaries and 2D-fold rotational symmetry. We estimate the large-size limit of this entropy for D=4 to 10. We confirm analytic predictions of N. Destainville et al., J. Stat. Phys. 120, 799 (2005) and M. Widom et al., J. Stat. Phys. 120, 837 (2005), in particular that the large size and large D limits commute, and that entropy becomes insensible to size, phason strain and boundary conditions at large D. We are able to infer finite D and finite size scalings of entropy. We also show that phason elastic constants can be estimated for any D by measuring the relevant perpendicular space fluctuations.