Hyperuniformity on the Sphere

Transcription

1 Hyperuniformity on the Sphere Johann S. Brauchart 14. June 2018 Week 2 Workshop MATRIX Research Program "On the Frontiers of High Dimensional Computation" MATRIX, Creswick [ 1 / 56 ]

2 PROLOG [ 2 / 56 ]

3 Joe Corbo, MD, PhD stared into the eye of a chicken and saw something astonishing. Professor, Pathology & Immunology at Washington University School of Medicine in St. Louis [ 3 / 56 ]

4 2014 A Bird s-eye View of Nature s Hidden Order, Natalie Wolchover, July 12, 2016 Olena Shmahalo/Quanta Magazine; Photography: MTSOfan and Matthew Toomey [ 4 / 56 ]

5 THE NON-COMPACT SETTING [ 5 / 56 ]

6 Hyperuniformity in R d Torquato and Stillinger [Physical Review E 68 (2003), no. 4, ]: A hyperuniform many-particle system in d-dimensional Euclidean space is one in which normalized density fluctuations are completely suppressed at very large lengths scales. [ 6 / 56 ]

7 Implications / Refining Hyperuniformity Scattering Experiments The structure factor S(k) = lim B R d 1 #(B X) x,y B X e i k,x y (thermodynamic limit) tends to zero as k k 0. When S(k) k α as k 0, where α > 0, more can be said. [ 7 / 56 ]

8 Equivalently, a hyperuniform many-particle system is one in which the number variance Var[N R ] of particles within a spherical observation window of radius R grows more slowly than the window volume in the large-r limit; i.e., slower than R d. [ 8 / 56 ]

9 Tossing observation windows [ 9 / 56 ]

10 THE COMPACT SETTING [ 10 / 56 ]

11 ESI Programme on "Minimal Energy Point Sets, Lattices, and Designs", 2014 [ 11 / 56 ]

12 [ 12 / 56 ]

13 HYPERUNIFORMITY ON THE SPHERE [ 13 / 56 ]

14 (X N ) N A is a.u.d. on S d if # {k : x k,n B} lim N N N A = σ d (B) for every Riemann-measurable set B in S d. Informally: A reasonable set gets a fair share of points as N becomes large. (X N ) N A is a.u.d. on S d if lim N N A 1 N N f (x k ) = f d σ d S d k=1 for every f C(S d ). [ 14 / 56 ]

15 LOW-DISCREPANCY SEQUENCES ON THE SPHERE [ 15 / 56 ]

16 Spherical cap L -discrepancy Skip DL C (X N ) := sup X N C σ d (C) C N [ 16 / 56 ]

17 Examples Skip Near optimal Coloumb energy points (Hardin & Saff, 2004, Notices of AMS) vs. i.i.d. random points (courtesy of Rob Womersley) [ 17 / 56 ]

18 Motivated by classical (up to log N optimal) results of J. Beck ( 1984), a sequence (X N ) is of low-discrepancy if log N DL C (X N ) c 1 N 1/2+1/(2d). Unresolved Question: Unlike in the unit cube case, there are no known explicit low-discrepancy constructions on the sphere. [ 18 / 56 ]

19 Spherical cap L Discrepancy Theorem (Aistleitner-JSB-Dick, 2012 ) D C L (Z Fm ) 44 8/ Fm and numerical evidence that for some 1 2 c 1, D C L (Z Fm ) = O((log F m ) c F 3/4 m ) as F m. RMK: A. Lubotzky, R. Phillips and P. Sarnak (1985, 1987) have DL C (XN LPS) (log N)2/3 N 1/3 with numerical evidence indicating O(N 1/2 ). [ 19 / 56 ]

20 ln-ln plot of spherical cap L -discrepancy of point set families. [ 20 / 56 ]

21 HYPERUNIFORMITY ON THE SPHERE [ 21 / 56 ]

22 Setting Infinite sequence of N-point sets (X N ) N A, X N S d, N A N. Spherical caps } C(x, φ) := {y S d y, x > cos(φ). Asymptotic behavior of number variance V (X N, φ) := V x [ # ( XN C(x, φ) )]. [ 22 / 56 ]

23 Number Variance & Uniform Distribution V (X N, φ) = S d ( N 2 1 C(x,φ) (x n ) N σ(c(, φ))) d σ d (x) n=1 appears in classical measure of uniform distribution, spherical cap L 2 -discrepancy of X N ( π 1/2 DL C 2 (X N ) := V (X N, φ) sin(φ) d φ), where a.u.d. is equivalent to 0 lim N 1 N DC L 2 (X N ) = 0. [ 23 / 56 ]

24 Heuristics Heuristically, hyperuniformity in the compact setting should mean that the number variance V (X N, φ N ) is of lower order than in the i.i.d. case. [ 24 / 56 ]

25 i.i.d. Case Number Variance for i.i.d. points on S d : ( ) N σ d (C(, φ)) 1 σ d (C(, φ)) which has order of magnitude large caps: N small caps: N σ d (C(, φ N )) threshold order: t d if φ N = t N 1/d [ 25 / 56 ]

26 Definition (three regimes of hyperuniformity) (X N ) N A is hyperuniform for large caps if V (X N, φ) = o(n) as N for all φ (0, π 2 ). [ 26 / 56 ]

27 (X N ) N A is hyperuniform for small caps if V (X N, φ N ) = o(n σ d (C(, φ N ))) as N for all sequences (φ N ) N A s.t. (1) lim φ N = 0, N (2) lim N σ d (C(, φ N )) N }{{} φ d N =. [ 27 / 56 ]

28 (X N ) N A is hyperuniform for caps at threshold order if lim sup N as t. V (X N, t N 1 d ) = O(t d 1 ) analogous to non-compact Euclidean case [ 28 / 56 ]

29 Remark Integrating out the angular radius, π 0 V (X N, φ) sin(φ) d φ, gives back the square of the spherical cap L 2 -discrepancy of X N. [ 29 / 56 ]

30 Stolarsky s Invariance Principle, discrepancy form Local discrepancy function D(X N ; C) := #(X N C) σ d (C). N Spherical cap L 2 -discrepancy: D C L 2 (X N ). Theorem (Stolarsky,1973; JSB-Dick, 2013;) 1 N 2 N j,k=1 x j x k + 1 ] 2 [DL C C 2 (X N ) d = x y d σ d (x) d σ d (y). [ 30 / 56 ]

31 NUMBER VARIANCE: TECHNICAL ASPECTS. Skip [ 31 / 56 ]

32 Laplace-Fourier series of the indicator function 1 C(x,φ) 1 C(x,φ) (y) = σ d (C(x, φ)) + a n (φ) Z (d, n) P (d) n ( x, y ), }{{} n=1 where for n 1, n+λ λ C(λ) n ( x,y ), λ=d 1 2 a n (φ) = γ d d sin(φ)d P (d+2) n 1 (cos(φ)). [ 32 / 56 ]

33 Laplace-Fourier expansion of V (X N, φ) V (X N, φ) N = 1 C(xj,φ)(x) N σ d (C(, φ)) = S d N j=1 i,j=1 n=1 a n (φ) 2 Z (d, n) P (d) n ( x i, x j ), } {{ } g φ ( x i,x j ) 2 d σ d (x) where ( sin(φ) a n (φ) 2 d 1) = O, Z (d, n) = O(n d 1 ). n d+1 [ 33 / 56 ]

34 Necessary condition for hyperuniformity for large caps Theorem If (X N ) N N hyperuniform for large caps, then for every n 1 Proof: 1 s(n) := lim N N 0 = lim N V (X N, φ) N a n (φ) 2 }{{} >0 lim sup N N i,j=1 1 N N i,j=1 P (d) n ( x i, x j ) = 0. Z (d, n) P (d) n ( x i, x j ). [ 34 / 56 ]

35 Uniform distribution of hyperuniform sequences (X N ) N N uniformly distributed iff #(X N C) σ d (C) N as N for all caps C 1 N iff Z (d, n) P (d) N 2 n ( x i, x j ) 0 i,j=1 as N for all n 1 (Weyl criterion). Corollary Hyperuniform (X N ) N N uniformly distributed. NOT obvious in the small caps and threshold order regimes! [ 35 / 56 ]

36 HYPERUNIFORMITY OF QMC DESIGN SEQUENCES [ 36 / 56 ]

37 QMC design sequences for H s (S d ), s > d/2 Definition (JSB-Saff-Sloan-Womersley, 2014 ) (X N ) is a QMC design sequence for H s (S d ) if there is a c(s, d) > 0 independent of N s.t. 1 c(s, d) f (x) f d σ d f N x X S N s/d H s d N for f H s (S d ). c(s, d) may depend on H s (S d )-norm and (X N ). [ 37 / 56 ]

38 Example: Maximal Sum-of-Distances Points A sequence (XN sum) of maximal sum-of-distance N-point sets define QMC rules that satisfy Q[XN sum ](f ) I(f ) csum (s, d) f H s N s/d for all f H s (S d ) and all d 2 < s d+1 2. The order of N cannot be improved. Open: Determine strength of (X sum N ). [ 38 / 56 ]

39 Numerics Spherical Fibonacci Points 10 0 Worst case error in H s for s = 3/ Pseudo random N Fibonacci Log potential (Riesz s = 0) Spherical Design Generalized spiral (Bauer) Maximum sum of distances N Number of points N (cf. (JSB-Saff-Sloan-Womersley, 2014)) [ 39 / 56 ]

40 Hyperuniformity of QMC design sequences Theorem A QMC design sequence for H s (S d ) with s d+1 2 is hyperuniform for large caps, small caps, and caps at threshold order. Lemma The number variance satisfies [ ] 2 V (X N, φ) (sin φ) d 1 N 2 wce(q[x N ]; H d+1 2 (S d )) for any N-point set X N S d and opening angle φ (0, π 2 ). [ 40 / 56 ]

41 HYPERUNIFORMITY: PROBABILISTIC ASPECTS Skip Joint work with Peter J. Grabner (Graz University of Technology) Wöden Kusner (Vanderbilt University) Jonas Ziefle (University of Tübingen) [ 41 / 56 ]

42 Given: point process X N by joint densities (X 1,..., X N ) ρ (N) invariant under permutation (exchangeable particles) and isometries of the sphere. Then: Number variance VX N (B) = E(X N (B)) 2 (EX N (B)) 2 = Nσ d (B) (1 σ d (B)) ( ) + N(N 1) ρ (N) 2 (x, x ) 1 B B d σ d (x) d σ d (x ). [ 42 / 56 ]

43 We study, in particular: Spherical ensembles on S 2 and S 2d hyperuniform Harmonic ensemble on S d hyperuniform, except treshold order Jittered sampling on S d hyperuniform, determinantal; [ 43 / 56 ]

44 Thank You!

45 APPENDIX [ 45 / 56 ]

46 Return [ 46 / 56 ]

47 Structure Factor a Return a proportional to the scattered intensity of radiation from a system of points and thus is obtainable from a scattering experiment Scattering pattern for a crystal vs disordered stealthy hyperuniform material. J. Phys.: Condens. Matter 28 (2016) [ 47 / 56 ]

48 Spherical Cap L -Discrepancy Return J. Beck, 1984 To every N-point set Z N on S d there exists a spherical cap C S d s.t. c 1 N 1/2 1/(2d) < Z N C σ d (C) N and (by a probabilistic argument) there exist an N-point sets ZN on Sd s.t. ZN C σ d (C) N < c 2 N 1/2 1/(2d) log N for every spherical cap C. [ 48 / 56 ]

49 Return [ 49 / 56 ]

50 SPHERICAL FIBONACCI LATTICE POINTS [ 50 / 56 ]

51 Fibonacci sequence (OEIS: A000045): F 0 := 0, F 1 := 1, F n+1 := F n + F n 1, n 1. Fibonacci lattice in [0, 1] 2 F n : ( { k, k F }) n 1, 0 k < F n, F n F n has optimal order star-discrepancy bounds: D(F n ; ) n log F n. {x} is fractional part of real x. [ 51 / 56 ]

52 n = 14: F n = 377. [ 52 / 56 ]

53 Area preserving Lambert transformation Φ : [0, 1] 2 S 2 Φ(x, y) = 2 cos(2πy) x x 2 2 sin(2πy) x x 2 1 2x [ 53 / 56 ]

54 Illumination Integrals Return [ 54 / 56 ]

55 Sum of distances for Spherical Fibonacci points Return 1 F 2 n F n 1 j=0 F n 1 k=0 8 3 z j z k = l=1 1 2l 1 l m=1 l=1 (l m)! (l + m)! 1 2l 1 1 F n 1 F n 1 k=0 F n F n 1 k=0 P l (1 2k ) F n Pl m (1 2k 2 ) e 2πi m kf n 1/F n. F n On the right-hand side one has (the error of) the numerical integration rule 0 = 1 1 P l (x) d x 1 F n 1 F n k=0 P l (1 2k F n ), l 1, with equally spaced nodes in [ 1, 1] for the Legendre polynomials P l (x) and the Fibonacci lattice rule 0 = Pl m (1 2x) e 2πi m y d x d y 1 F n 1 F n k=0 P m l (1 2k F n ) e 2πi m kf n 1/F n based on the Fibonacci lattice points in the unit square [0, 1] 2 for functions f m l (x, y) := P m l (1 2x) e 2πi m y, l 1, 1 m l. 2 [ 55 / 56 ]

56 Estimates for s for d = 2 s := sup { s : (X } N) is QMC design sequence for H s (S d. ) Return Table: Estimates of s for d = 2 Point set s Fekete 1.5 Equal area 2 Coulomb energy 2 Log energy 3 Generalized spiral 3 Distance 4 Spherical designs [ 56 / 56 ]