Collective behavior of particles in fluids

Paris, December 14-17, 2020

**Salvatore Torquato** (Princeton University)

*Hyperuniform States of Matter*

Hyperuniform many-particle systems in d-dimensional Euclidean space are characterized by an unusual suppression of density fluctuations at large lengths scales. All perfect crystals, perfect quasicrystals and exotic disordered patterns are hyperuniform [1]. Thus, the hyperuniformity concept generalizes the traditional notion of long-range order, and provides a unified theoretical framework to categorize a large class of ordered and disordered systems. Disordered hyperuniform many-particle systems can be regarded to be new states of disordered matter in that they behave more like crystals or quasicrystals in the manner in which they suppress large-scale density fluctuations, and yet are also like liquids and glasses because they are statistically isotropic structures with no Bragg peaks. Such exotic amorphous states of matter exist as both equilibrium and nonequilibrium phases and come in quantum-mechanical and classical varieties. I will provide an overview of the hyperuniformity concept and its generalizations. Topics covered include how optimal hyperuniform point configurations can be posed as energy-minimization problems; rank order of crystals, quasicrystals and disordered hyperuniform systems via a hyperuniformity index; disordered classical ground states that are hyperuniform and highly degenerate; fermionic point processes; zeros of the Riemann zeta function, directional hyperuniformity and novel physical properties [2]. I will also discuss how the hyperuniformity concept motivated a recent study that has uncovered previously unknown multiscale order in the prime numbers [3].

*References:*

[1] S. Torquato and F. H. Stillinger. Local Density Fluctuations, Hyperuniform Systems, and Order Metrics. *Phys. Rev. E* 68:041113, 2003.

[2] S. Torquato. Hyperuniform States of Matter. *Physics Reports* 745:1, 2018.

[3] S. Torquato, G. Zhang, and M. de Courcy-Ireland. Hidden Multiscale Order in the Primes. *J. Physics A: Math. & Theoretical* 52:135002, 2019.

See slides and record.