Uncovering Multiscale Order in the Prime Numbers via Scattering

@article{Torquato2018UncoveringMO,
  title={Uncovering Multiscale Order in the Prime Numbers via Scattering},
  author={Salvatore Torquato and G. Zhang and Matthew de Courcy-Ireland},
  journal={arXiv: Statistical Mechanics},
  year={2018}
}
The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call {\it effectively limit-periodic}. In particular, the primes… Expand

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References

SHOWING 1-10 OF 50 REFERENCES
Structure Factor of the Primes
Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes usingExpand
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  • Mathematics
  • Journal of Physics A: Mathematical and Theoretical
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We study the {pair correlations between} prime numbers in an interval $M \leq p \leq M + L$ with $M \rightarrow \infty$, $L/M \rightarrow \beta > 0$. By analyzing the \emph{structure factor}, weExpand
Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory
It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line. Here we analyticallyExpand
Hyperuniformity of quasicrystals
Hyperuniform systems, which include crystals, quasicrystals and special disordered systems, have attracted considerable recent attention, but rigorous analyses of the hyperuniformity of quasicrystalsExpand
Enhanced hyperuniformity from random reorganization
TLDR
This work finds that hyperuniformity of the absorbing state is not only robust against noise, diffusion, or activity, but that such perturbations reduce fluctuations toward their limiting behavior, λ→d+1, a uniformity similar to random close packing and early universe fluctuations, but with arbitrary controllable density. Expand
Local density fluctuations, hyperuniformity, and order metrics.
TLDR
The purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d, and to study the variance in the number of points contained within a regularly shaped window of arbitrary size to further illuminate the understanding of hyperuniform systems. Expand
Hyperuniform States of Matte
Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found inExpand
Ensemble Theory for Stealthy Hyperuniform Disordered Ground States
It has been shown numerically that systems of particles interacting with "stealthy" bounded, long-ranged pair potentials (similar to Friedel oscillations) have classical ground states that are,Expand
Classical disordered ground states: Super-ideal gases and stealth and equi-luminous materials
Using a collective coordinate numerical optimization procedure, we construct ground-state configurations of interacting particle systems in various space dimensions so that the scattering ofExpand
Rational design of stealthy hyperuniform two-phase media with tunable order.
TLDR
This paper presents several algorithms enabling the systematic identification and generation of discrete (digitized) stealthy hyperuniform patterns with a tunable degree of order, paving the way towards the rational design of disordered materials endowed with novel thermodynamic and physical properties. Expand
...
1
2
3
4
5
...