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DY: Fachverband Dynamik und Statistische Physik

DY 10: Statistical Physics I (General)

DY 10.10: Vortrag

Montag, 12. März 2018, 12:30–12:45, BH-N 333

Scaling of density fluctuations and hyperuniformity in one-dimensional substitution tilings — •Erdal C. Oğuz^1,4, Joshua E. S. Socolar², Paul J. Steinhardt³, and Salvatore Torquato⁴ — ¹School of Mechanical Engineering and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel — ²Department of Physics, Duke University, Durham, NC 27708 — ³Princeton Center for Theoretical Science and Department of Physics, Princeton University, Princeton, NJ 08544 — ⁴Department of Chemistry, Princeton University, Princeton, 08540

Substitution tilings include periodic, quasiperiodic, limit periodic, and other self-similar structures generated by iterated subdivision and rescaling of a finite set of tiles. We study the scaling of density fluctuations associated with a broad class of substitution rules in one dimension. We show that a simple, heuristic argument for the rate of decay of the integrated Fourier intensity Z(k) at small values of the wavenumber k correctly predicts the scaling of the variance σ²(R) in the number of points contained in intervals of length 2R. The exponent α, defined by Z∼ k^α+1, is determined by the ratio of the second largest and largest eigenvalues of the substitution matrix and can vary between −1 and 3, where α > 0 implies a hyperuniform distribution of tile vertices. The hyperuniform class includes tilings that are periodic, quasiperiodic, or limit periodic, including a new class of limit-periodic tilings for which Z approaches zero faster than any power law. Tilings with singular continuous diffraction spectra may be hyperuniform or may exhibit stronger fluctuations than a Poisson system.