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Berlin 2018 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 10: Statistical Physics I (General)

DY 10.10: Talk

Monday, March 12, 2018, 12:30–12:45, BH-N 333

Scaling of density fluctuations and hyperuniformity in one-dimensional substitution tilings — •Erdal C. Oğuz1,4, Joshua E. S. Socolar2, Paul J. Steinhardt3, and Salvatore Torquato41School 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 Zkα+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.

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