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Optimal and Disordered Hyperuniform Point Configurations

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Optimal and Disordered Hyperuniform Point Configurations Optimal and Disordered Hyperuniform Point Configurations 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. 1/40 States (Phases) of Matter Source: www.nasa.gov . – p. 2/40 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. 2/40 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. 3/40 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. 3/40 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. 3/40 HYPERUNIFORMITY A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales. – p. 4/40 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 ideal 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. 4/40 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 ideal 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. 4/40 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 ideal 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. 4/40 Definitions A point process in d-dimensional Euclidean space Rd is a distribution of an d infinite number of points in R with configuration r1, r2,... with a well-defined number density ρ (number of points per unit volume). This is statistically described by the n-particle correlation function gn(r1,,..., rn). A lattice L in d-dimensional Euclidean space Rd is the set of points that are integer linear combinations of d basis (linearly independent) vectors ai, i.e., {n1a1 + n2a2 + ··· + ndad | n1, . , nd ∈ Z} The space Rd can be geometrically divided into identical regions F called fundamental cells, each of which contains just one point. For example, in R2: Every lattice L has a dual (or reciprocal) lattice L∗. A periodic point distribution in Rd is a fixed but arbitrary configuration of N points (N ≥ 1) in each fundamental cell of a lattice. – p. 5/40 Definitions d For statistically homogeneous and isotropic point processes in R at number density ρ , g2(r) is a nonnegative radial function that is proportional to the probability density of pair distances r. We call h(r) g2(r) 1 ≡ − the total correlation function. When there is no long-range order in the system, h(r) 0 [or g2(r) 1 ] in the large-r limit. → → We call a point process disordered if h(r) tends to zero sufficiently rapidly such that it is integrable over all space. The nonnegative structure factor S(k) is defined in terms of the Fourier transform of h(r), which we denote by h˜(k): S(k) 1+ ρh˜(k), ≡ where k denotes wavenumber. When there is no long-range order in the system, S(k) 1 in the large-k limit, the dual-space → analog of the aforementioned direct space condition. In some generalized sense, S(k) can be viewed as a probability density of pair distances of “points” in reciprocal space. – p. 6/40 Pair Statistics for Spatially Uncorrelated and Ordered Point Processes Poisson Distribution (Ideal Gas) 2 2 1.5 1.5 (r) 1 1 2 S(k) g 0.5 0.5 0 0 0 1 2 3 0 1 2 3 r k Lattice (r) 2 S(k) g 0 1 2 3 0 4π/√3 8π/√3 12π/√3 r k . – p. 7/40 Curiosities Disordered jammed packings 8 40,000−particle random jammed packing 6 φ=0.632 4 S(k) 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. Gabrielli et al. (2003) . – p. 8/40 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 Rd. Ω R R Ω d Average number of points in window of volume v1(R): N(R) = ρv1(R) ∼ R Local number variance: σ2(R) ≡N 2(R)−N(R)2 . – p. 9/40 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 Rd. Ω R R Ω d Average number of points in window of volume v1(R): N(R) = ρv1(R) ∼ R Local number variance: σ2(R) ≡N 2(R)−N(R)2 For a Poisson point pattern and many disordered point patterns, σ2(R) ∼ Rd. We call point patterns whose variance grows more slowly than Rd (window volume) hyperuniform . This implies that structure factor S(k) → 0 for k → 0. All perfect crystals and perfect quasicrystals are hyperuniform such that σ2(R) ∼ Rd−1: number variance grows like window surface area. Hyperuniformity is aka superhomogeneity: Gabrielli, Joyce & Sylos Labini, Phys. Rev. E (2002). – p. 9/40 Hidden Order on Large Length Scales Which is the hyperuniform pattern? . – p. 10/40 Scaled Number Variance for 3D Systems at Unit Density 4 disordered non-hyperuniform 3 3 (R)/R 2 2 σ ordered hyperuniform 1 disordered hyperuniform 0 4 8 12 16 20 R . – p. 11/40 Outline Hyperuniform Point Configurations: History and Recent Developments Connections to Sphere Packing, Covering and Quantizer Problems Running themes: 1. All of these problems can be cast as optimization tasks; specifically energy-minimizing point configurations. 2. Optimal solutions can be both ordered and disordered. – p. 12/40 ENSEMBLE-AVERAGE FORMULATION For a translationally invariant point process at number density ρ in Rd: σ2(R) = N(R) 1+ ρ h(r)α(r; R)dr Rd h Z i α(r; R)- scaled intersection volume of 2 windows of radius R separated by r 1 Spherical window of radius R 0.8 R 0.6 d=1 (r;R) r α 0.4 d=5 0.2 0 0 0.2 0.4 0.6 0.8 1 r/(2R) . – p. 13/40 ENSEMBLE-AVERAGE FORMULATION For a translationally invariant point process at number density ρ in Rd: σ2(R) = N(R) 1+ ρ h(r)α(r; R)dr Rd h Z i α(r; R)- scaled intersection volume of 2 windows of radius R separated by r 1 Spherical window of radius R 0.8 R 0.6 d=1 (r;R) r α 0.4 d=5 0.2 0 0 0.2 0.4 0.6 0.8 1 r/(2R) For large R, we can show d d 1 d 1 R R − R − σ2(R)=2dφ A + B + o , D D D h „ « „ « „ « i where A and B are the “volume” and “surface-area” coefficients: A = S(k = 0)=1+ ρ h(r)dr, B = c(d) h(r)rdr, Rd − Rd Z Z D: microscopic length scale, φ: dimensionless density Hyperuniform: A =0, B > 0 .
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