Hyperuniformity on the Sphere Peter Grabner (joint work with J. - PowerPoint PPT Presentation

Hyperuniformity on the Sphere Peter Grabner (joint work with J. Brauchart, W. Kusner, and J. Ziefle) Institute for Analysis and Number Theory Graz University of Technology Optimal Point Configurations and Orthogonal Polynomials 2017 P. Grabner Hyperuniformity on the Sphere

Motivation: Hyperuniformity in R d Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. P. Grabner Hyperuniformity on the Sphere

Motivation: Hyperuniformity in R d Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ R d according to a point process with joint density ρ ( N ) being V P. Grabner Hyperuniformity on the Sphere

Motivation: Hyperuniformity in R d Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ R d according to a point process with joint density ρ ( N ) being V (a) invariant under permutation of the particles P. Grabner Hyperuniformity on the Sphere

Motivation: Hyperuniformity in R d Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ R d according to a point process with joint density ρ ( N ) being V (a) invariant under permutation of the particles (b) invariant under Euclidean motion (for V ր R d ) P. Grabner Hyperuniformity on the Sphere

Motivation: Hyperuniformity in R d Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ R d according to a point process with joint density ρ ( N ) being V (a) invariant under permutation of the particles (b) invariant under Euclidean motion (for V ր R d ) Hence, a single particle is distributed with density � 1 V N − 1 ρ ( N ) ( r 1 , . . . , r N ) d r 2 · · · d r N = V | V | Assume N | V | → ρ ( thermodynamic limit ). This means that the distribution is asymptotically uniform with density ρ . P. Grabner Hyperuniformity on the Sphere

Hyperuniformity in R d Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls B R N where ( X 1 , . . . , X N ) ∼ ρ ( N ) � N R := ✶ B R ( X i ) , V i =1 P. Grabner Hyperuniformity on the Sphere

Hyperuniformity in R d Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls B R N where ( X 1 , . . . , X N ) ∼ ρ ( N ) � N R := ✶ B R ( X i ) , V i =1 P. Grabner Hyperuniformity on the Sphere

Hyperuniformity in R d Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls B R N where ( X 1 , . . . , X N ) ∼ ρ ( N ) � N R := ✶ B R ( X i ) , V i =1 The expected number of points in B R is E [ N R ] th. → ρ | B R | . P. Grabner Hyperuniformity on the Sphere

Hyperuniformity in R d The variance measures the deviation over all test sets under consideration. Example: ( X i ) i i.i.d. ⇒ V [ N R ] th. → ρ | B R | . P. Grabner Hyperuniformity on the Sphere

Hyperuniformity in R d The variance measures the deviation over all test sets under consideration. Example: ( X i ) i i.i.d. ⇒ V [ N R ] th. → ρ | B R | . Definition ( ρ ( N ) ) N ∈ N hyperuniform ⇐ ⇒ lim th. V [ N R ] ∼ | ∂B R | for large R Remarks: P. Grabner Hyperuniformity on the Sphere

Hyperuniformity in R d The variance measures the deviation over all test sets under consideration. Example: ( X i ) i i.i.d. ⇒ V [ N R ] th. → ρ | B R | . Definition ( ρ ( N ) ) N ∈ N hyperuniform ⇐ ⇒ lim th. V [ N R ] ∼ | ∂B R | for large R Remarks: If ( ρ ( N ) ) N ∈ N hyperuniform, i.e. R d -term of lim th. V [ N R ] vanishes ⇒ R d − 1 -term cannot vanish. P. Grabner Hyperuniformity on the Sphere

Hyperuniformity in R d The variance measures the deviation over all test sets under consideration. Example: ( X i ) i i.i.d. ⇒ V [ N R ] th. → ρ | B R | . Definition ( ρ ( N ) ) N ∈ N hyperuniform ⇐ ⇒ lim th. V [ N R ] ∼ | ∂B R | for large R Remarks: If ( ρ ( N ) ) N ∈ N hyperuniform, i.e. R d -term of lim th. V [ N R ] vanishes ⇒ R d − 1 -term cannot vanish. Hyperuniformity is a long-scale property. P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions ( ρ ( N ) ) N ∈ N on ❙ d satisfying (a) ρ ( N ) ( x σ 1 , . . . , x σN ) = ρ ( N ) ( x 1 , . . . , x N ) for all x i ∈ ❙ d , σ ∈ S N . ”particles are exchangeable” P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions ( ρ ( N ) ) N ∈ N on ❙ d satisfying (a) ρ ( N ) ( x σ 1 , . . . , x σN ) = ρ ( N ) ( x 1 , . . . , x N ) for all x i ∈ ❙ d , σ ∈ S N . ”particles are exchangeable” (b) ρ ( N ) ( τx 1 , . . . , τx N ) = ρ ( N ) ( x 1 , . . . , x N ) for all x i ∈ S d , τ ∈ SO( d + 1) , ”isometry invariance” P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions ( ρ ( N ) ) N ∈ N on ❙ d satisfying (a) ρ ( N ) ( x σ 1 , . . . , x σN ) = ρ ( N ) ( x 1 , . . . , x N ) for all x i ∈ ❙ d , σ ∈ S N . ”particles are exchangeable” (b) ρ ( N ) ( τx 1 , . . . , τx N ) = ρ ( N ) ( x 1 , . . . , x N ) for all x i ∈ S d , τ ∈ SO( d + 1) , ”isometry invariance” P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions ( ρ ( N ) ) N ∈ N on ❙ d satisfying (a) ρ ( N ) ( x σ 1 , . . . , x σN ) = ρ ( N ) ( x 1 , . . . , x N ) for all x i ∈ ❙ d , σ ∈ S N . ”particles are exchangeable” (b) ρ ( N ) ( τx 1 , . . . , τx N ) = ρ ( N ) ( x 1 , . . . , x N ) for all x i ∈ S d , τ ∈ SO( d + 1) , ”isometry invariance” Given any distribution averaging over permutations and isometries yields joint densities with (a) and (b). P. Grabner Hyperuniformity on the Sphere

✶ Hyperuniformity on the sphere Test sets B R are spherical caps, and the point counting function is N � where ( X 1 , . . . , X N ) ∼ ρ ( N ) N φ := ✶ C ( x,φ ) ( X i ) , i =1 P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere Test sets B R are spherical caps, and the point counting function is N � where ( X 1 , . . . , X N ) ∼ ρ ( N ) N φ := ✶ C ( x,φ ) ( X i ) , i =1 The expectation remains N -dependent N � ρ ( N ) � E [ N φ ] = E [ ✶ C ( x,φ ) ( X i )] = N ( r ) d r = Nσ ( C ( · , φ )) . 1 C ( x,φ ) i =1 P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere The variance depends on N and the pair correlation ρ ( N ) 2 V [ N φ ] = Nσ ( C ( · , φ ))(1 − σ ( C ( · , φ ))) � ( ρ ( N ) + N ( N − 1) ( x, y ) − 1) d x d y 2 C ( · ,φ ) P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere The variance depends on N and the pair correlation ρ ( N ) 2 V [ N φ ] = Nσ ( C ( · , φ ))(1 − σ ( C ( · , φ ))) � ( ρ ( N ) + N ( N − 1) ( x, y ) − 1) d x d y 2 C ( · ,φ ) Example: ( X i ) i i.i.d. (i.e. ρ ( N ) = 1 ) gives E [ N φ ] = Nσ ( C ( · , φ )) and V [ N R ] = Nσ ( C ( · , φ ))(1 − σ ( C ( · , φ ))) . P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V [ N φ ] for N → ∞ . P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V [ N φ ] for N → ∞ . P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V [ N φ ] for N → ∞ . Furthermore, we want to switch from random point sets to deterministically contructed point sets. P. Grabner Hyperuniformity on the Sphere

Hyperuniformity on the sphere Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V [ N φ ] for N → ∞ . Furthermore, we want to switch from random point sets to deterministically contructed point sets. We have identified three regimes for the cap size, where we could encounter and partly characterise this phenomenon. P. Grabner Hyperuniformity on the Sphere

Definition of hyperuniformity on the sphere Definition (Hyperuniformity) Let ( X N ) N ∈ N be a sequence of point sets on the sphere S d . The number variance of the sequence for caps of angle φ is given by V ( X N , φ ) = V x # ( X N ∩ C ( x, φ )) . (1) A sequence is called P. Grabner Hyperuniformity on the Sphere