Disordered Hyperuniform Point Patterns in Physics, Mathematics and Biology Salvatore Torquato Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied & Computational Mathematics Princeton University http://cherrypit.princeton.edu . – p. 1/50
States (Phases) of Matter Source: www.nasa.gov . – p. 2/50
States (Phases) of Matter Source: www.nasa.gov We now know there are a multitude of distinguishable states of matter, e.g., quasicrystals and liquid crystals, which break the continuous translational and rotational symmetries of a liquid differently from a solid crystal. . – p. 2/50
What Qualifies as a Distinguishable State of Matter? Traditional Criteria Homogeneous phase in thermodynamic equilibrium Interacting entities are microscopic objects, e.g. atoms, molecules or spins Often, phases are distinguished by symmetry-breaking and/or some qualitative change in some bulk property . – p. 3/50
What Qualifies as a Distinguishable State of Matter? Traditional Criteria Homogeneous phase in thermodynamic equilibrium Interacting entities are microscopic objects, e.g. atoms, molecules or spins Often, phases are distinguished by symmetry-breaking and/or some qualitative change in some bulk property Broader Criteria Reproducible quenched/long-lived metastable or nonequilibrium phases, e.g., spin glasses and structural glasses Interacting entities need not be microscopic, but can include building blocks across a wide range of length scales, e.g., colloids and metamaterials Endowed with unique properties . – p. 3/50
What Qualifies as a Distinguishable State of Matter? Traditional Criteria Homogeneous phase in thermodynamic equilibrium Interacting entities are microscopic objects, e.g. atoms, molecules or spins Often, phases are distinguished by symmetry-breaking and/or some qualitative change in some bulk property Broader Criteria Reproducible quenched/long-lived metastable or nonequilibrium phases, e.g., spin glasses and structural glasses Interacting entities need not be microscopic, but can include building blocks across a wide range of length scales, e.g., colloids and metamaterials Endowed with unique properties New states of matter become more compelling if they Give rise to or require new ideas and/or experimental/theoretical tools Technologically important . – p. 3/50
Pair Statistics in Direct and Fourier Spaces For particle systems in R d at number density ρ , g 2 ( r ) is a nonnegative radial function that is proportional to the probability density of pair distances r . The nonnegative structure factor S ( k ) ≡ 1 + ρ ˜ h ( k ) is obtained from the Fourier transform of h ( r ) = g 2 ( r ) − 1 , which we denote by ˜ h ( k ) . Poisson Distribution (Ideal Gas) 2 2 1.5 1.5 S(k) g 2 (r) 1 1 0.5 0.5 0 0 0 1 2 3 0 1 2 3 r k Liquid 3 3 2.5 2.5 2 2 S(k) g 2 (r) 1.5 1.5 1 1 0.5 0.5 0 0 0 5 10 15 20 25 30 0 1 2 3 4 5 r k Crystal (Angularly Averaged) S(k) g 2 (r) 0 4 π/√3 8 π/√3 12 π/√3 0 1 2 3 r k . – p. 4/50
Curiosities Disordered jammed packings 8 40,000−particle random jammed packing 6 φ=0.632 S(k) 4 2 0 0 4 8 12 16 20 24 kD S ( k ) appears to vanish in the limit k → 0 : very unusual behavior for a disordered system. Harrison-Zeldovich spectrum for density fluctuations in the early Universe: S ( k ) ∼ k for sufficiently small k . . – p. 5/50
HYPERUNIFORMITY A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales. . – p. 6/50
HYPERUNIFORMITY A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales. Disordered hyperuniform many-particle systems can be regarded to be new distinguishable states of disordered matter in that they (i) 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 ; (ii) can exist as both as equilibrium and quenched nonequilibrium phases ; (iii) and, appear to be endowed with unique bulk physical properties . Understanding such states of matter, which have technological importance, require new theoretical tools. . – p. 6/50
HYPERUNIFORMITY A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales. Disordered hyperuniform many-particle systems can be regarded to be new distinguishable states of disordered matter in that they (i) 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 ; (ii) can exist as both as equilibrium and quenched nonequilibrium phases ; (iii) and, appear to be endowed with unique bulk physical properties . Understanding such states of matter, which have technological importance, require new theoretical tools. All perfect crystals (periodic systems) and quasicrystals are hyperuniform. . – p. 6/50
HYPERUNIFORMITY A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales. Disordered hyperuniform many-particle systems can be regarded to be new distinguishable states of disordered matter in that they (i) 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 ; (ii) can exist as both as equilibrium and quenched nonequilibrium phases ; (iii) and, appear to be endowed with unique bulk physical properties . Understanding such states of matter, which have technological importance, require new theoretical tools. All perfect crystals (periodic systems) and quasicrystals are hyperuniform. Thus, hyperuniformity provides a unified means of categorizing and characterizing crystals, quasicrystals and such special disordered systems. . – p. 6/50
Local Density Fluctuations for General Point Patterns Torquato and Stillinger, PRE (2003) Points can represent molecules of a material, stars in a galaxy, or trees in a forest. Let Ω represent a spherical window of radius R in d -dimensional Euclidean space R d . R R W W Denote by σ 2 ( R ) ≡ � N 2 ( R ) � − � N ( R ) � 2 the number variance. . – p. 7/50
Local Density Fluctuations for General Point Patterns Torquato and Stillinger, PRE (2003) Points can represent molecules of a material, stars in a galaxy, or trees in a forest. Let Ω represent a spherical window of radius R in d -dimensional Euclidean space R d . R R W W Denote by σ 2 ( R ) ≡ � N 2 ( R ) � − � N ( R ) � 2 the number variance. For a Poisson point pattern and many disordered point patterns, σ 2 ( R ) ∼ R d . We call point patterns whose variance grows more slowly than R d (window volume) hyperuniform . This implies that structure factor S ( k ) → 0 for k → 0 . All perfect crystals and quasicrystals are hyperuniform such that σ 2 ( R ) ∼ R d − 1 : number variance grows like window surface area. . – p. 7/50
Hyperuniformity and Crystals Diffraction Pattern for a Square Lattice Angular-averaged S(k) for a Square Lattice Dots denote Bragg peaks at δ( k-Q) (Lattice spacing = 2 π/ a ) 20 0.10 15 10 5 S(k) 0 0.05 2 π/ a -5 -10 -15 0.00 0 5 10 15 20 25 -20 Wavenumber, k -25 -20 -15 -10 -5 0 5 10 15 20 25 All perfect crystals are trivially hyperuniform. But the degree to which they suppress large-scale density fluctuations varies. Which crystal in R d minimizes such fluctuations? . – p. 8/50
Hidden Order on Large Length Scales Which is the hyperuniform pattern? . – p. 9/50
Outline Hyperuniform Systems “Inverted” Critical Phenomena Minimizing Variance: Special Ground State Problem Variance as an Order Metric at Large Length Scales Examples of Disordered Hyperuniform Systems: Natural and Synthetic . – p. 10/50
ENSEMBLE-AVERAGE FORMULATION For an ensemble in d -dimensional Euclidean space R d : Z h i σ 2 ( R ) = � N ( R ) � 1 + ρ R d h ( r ) α ( r ; R ) d r α ( r ; R ) - scaled intersection volume of 2 windows of radius R separated by r 1 Spherical window of radius R 0.8 0.6 R d=1 α (r;R) 0.4 r d=5 0.2 0 0 0.2 0.4 0.6 0.8 1 r/(2R) . – p. 11/50
ENSEMBLE-AVERAGE FORMULATION For an ensemble in d -dimensional Euclidean space R d : Z h i σ 2 ( R ) = � N ( R ) � 1 + ρ R d h ( r ) α ( r ; R ) d r α ( r ; R ) - scaled intersection volume of 2 windows of radius R separated by r 1 Spherical window of radius R 0.8 0.6 R d=1 α (r;R) 0.4 r d=5 0.2 0 0 0.2 0.4 0.6 0.8 1 r/(2R) For large R , we can show „ R „ R „ R « d « d − 1 « d − 1 i h σ 2 ( R ) = 2 d φ A + B + o , D D D where A and B are the “volume” and “surface-area” coefficients: Z Z A = S ( k = 0 ) = 1 + ρ R d h ( r ) d r , B = − c ( d ) R d h ( r ) rd r , D : microscopic length scale, φ : dimensionless density Hyperuniform : A = 0 , B > 0 . – p. 11/50
INVERTED CRITICAL PHENOMENA h ( r ) can be divided into direct correlations, via function c ( r ) , and indirect correlations: ˜ h ( k ) c ( k ) = ˜ 1 + ρ ˜ h ( k ) . – p. 12/50
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