Equidistribution and microscale results for Coulomb and Riesz gases - PowerPoint PPT Presentation

Next-order asymptotics Hohenberg-Kohn Equidistribution and microscale results for Coulomb and Riesz gases Mircea Petrache, MPI Bonn April 21, 2017

Next-order asymptotics Hohenberg-Kohn A SYMPTOTICS OF PARTICLES WITH C OULOMB AND R IESZ INTERACTIONS ◮ n � � x i ∈ R d , d ≥ 1 H n ( x 1 , . . . , x n ) = g ( x i − x j ) + n V ( x i ) , i � = j i = 1 ◮ g ( x ) = | x | − s interaction potential, V : R d → ] − ∞ , + ∞ ] confining potential growing at infinity ◮ d − 2 ≤ s < d : Riesz gas, ◮ s = d − 2: Coulomb gas, ◮ s = 0: means g ( x ) = − log | x | , log gas. ◮ Gibbs measure: 1 e − β H n ( x 1 ,..., x n ) dx 1 . . . dx n , x i ∈ R d d P n ,β ( x 1 , · · · , x n ) = Z n ,β Z n ,β = partition function, 0 < β = inverse temperature.

Next-order asymptotics Hohenberg-Kohn A SYMPTOTICS OF PARTICLES WITH C OULOMB AND R IESZ INTERACTIONS ◮ n � � x i ∈ R d , d ≥ 1 H n ( x 1 , . . . , x n ) = g ( x i − x j ) + n V ( x i ) , i � = j i = 1 ◮ g ( x ) = | x | − s interaction potential, V : R d → ] − ∞ , + ∞ ] confining potential growing at infinity ◮ d − 2 ≤ s < d : Riesz gas, ◮ s = d − 2: Coulomb gas, ◮ s = 0: means g ( x ) = − log | x | , log gas. ◮ Gibbs measure: 1 e − β H n ( x 1 ,..., x n ) dx 1 . . . dx n , x i ∈ R d d P n ,β ( x 1 , · · · , x n ) = Z n ,β Z n ,β = partition function, 0 < β = inverse temperature.

Next-order asymptotics Hohenberg-Kohn A SYMPTOTICS OF PARTICLES WITH C OULOMB AND R IESZ INTERACTIONS ◮ n � � x i ∈ R d , d ≥ 1 H n ( x 1 , . . . , x n ) = g ( x i − x j ) + n V ( x i ) , i � = j i = 1 ◮ g ( x ) = | x | − s interaction potential, V : R d → ] − ∞ , + ∞ ] confining potential growing at infinity ◮ d − 2 ≤ s < d : Riesz gas, ◮ s = d − 2: Coulomb gas, ◮ s = 0: means g ( x ) = − log | x | , log gas. ◮ Gibbs measure: 1 e − β H n ( x 1 ,..., x n ) dx 1 . . . dx n , x i ∈ R d d P n ,β ( x 1 , · · · , x n ) = Z n ,β Z n ,β = partition function, 0 < β = inverse temperature.

Next-order asymptotics Hohenberg-Kohn M OTIVATION 1: F EKETE POINTS ◮ In log gas case minimizers of H n are maximizers of n � � e − nV ( x i ) | x i − x j | i < j i = 1 → weighted Fekete sets (approximation theory) Saff-Totik, Rakhmanov-Saff-Zhou. ◮ Fekete points and equilibrium configurations on spheres and other closed manifolds M (Borodachov-Hardin-Saff, Brauchart-Dragnev-Saff) � � 1 x 1 ,..., x n ∈M − log | x i − x j | min or min | x i − x j | s x 1 ... x n ∈M i � = j i � = j → Smale’s 7th “problem for the next century” related to computational complexity theory: find an algorithm that provides an almost minimizer in polynomial time.

Next-order asymptotics Hohenberg-Kohn Minimal s -energy points on a torus, s = 0 , 1 , 0 . 8 , 2 (from Rob Womersley’s webpage)

Next-order asymptotics Hohenberg-Kohn M OTIVATION 2: STATISTICAL MECHANICS � � � β 1 e − n β � n i = 1 V ( x i ) dx 1 . . . dx n d P n ,β ( x 1 , · · · , x n ) = | x i − x j | Z n ,β i < j β = 2 � Vandermonde determinant: determinantal processes; ◮ connection to random matrices , first noticed by Wigner, Dyson ( C ⊃ { x 1 , . . . , x n } = eigenvalues of n × n matrices) ◮ d = 1, Coulomb gas: completely solvable Lenard, Aizenman-Martin, Brascamp-Lieb ◮ more recently: microscopic rigidity and universality principles Valko-Virag ’09, Bourgade-Erd¨ os-Yau ’12, Scherbina ’14, Beckerman-Figalli-Guionnet ’14... ◮ Statistical mechanics of Coulomb and (recently) Riesz gasses : d = 1 log gas or d ≥ 2 Coulomb gas Lieb-Narnhofer, Penrose-Smith, Frohlich-Spencer, Jancovici-Lebowitz-Manificat, Kiessling, Kiessling-Spohn,..., general s < d Chafai-Gozlan-Zitt

Next-order asymptotics Hohenberg-Kohn M OTIVATION 2: STATISTICAL MECHANICS � � � β 1 e − n β � n i = 1 V ( x i ) dx 1 . . . dx n d P n ,β ( x 1 , · · · , x n ) = | x i − x j | Z n ,β i < j β = 2 � Vandermonde determinant: determinantal processes; ◮ connection to random matrices , first noticed by Wigner, Dyson ( C ⊃ { x 1 , . . . , x n } = eigenvalues of n × n matrices) ◮ d = 1, Coulomb gas: completely solvable Lenard, Aizenman-Martin, Brascamp-Lieb ◮ more recently: microscopic rigidity and universality principles Valko-Virag ’09, Bourgade-Erd¨ os-Yau ’12, Scherbina ’14, Beckerman-Figalli-Guionnet ’14... ◮ Statistical mechanics of Coulomb and (recently) Riesz gasses : d = 1 log gas or d ≥ 2 Coulomb gas Lieb-Narnhofer, Penrose-Smith, Frohlich-Spencer, Jancovici-Lebowitz-Manificat, Kiessling, Kiessling-Spohn,..., general s < d Chafai-Gozlan-Zitt

Next-order asymptotics Hohenberg-Kohn Eigenvalues of 1000-by-1000 matrix with i.i.d Gaussian entries ( β = 2, V ( x ) = | x | 2 ) (from Benedek Valk´ o’s webpage)

Next-order asymptotics Hohenberg-Kohn M OTIVATION 3: MODELS IN SUPERCONDUCTIVITY ◮ (H. Kamerlingh Onnes* 1911) Cooper* pairs, Meissner effect ◮ Rotating superfluids and Bose-Einstein* condensates, modelled by Ginzburg*-Landau* energy: � |∇ A ψ | 2 + | curl A − h ex | 2 + ( 1 − | ψ | 2 ) 2 G ε ( ψ, A ) = 1 2 2 ε 2 Ω ◮ Formation of Abrikosov* structured vortices. ◮ The functional H n describes the interaction of vortices under G ε -energy (Sandier-Serfaty). (*: Nobel prize winner)

Next-order asymptotics Hohenberg-Kohn Abrikosov lattices: superconducting vortices on Mg B 2 and Nb Se 2 crystals (Vinnikov, Phys.Rev.B ’03, Hess, Phys.Rev.Lett. ’89)

Next-order asymptotics Hohenberg-Kohn T HE LEADING ORDER TO min H n ( OR “ CONTINUUM / MEAN FIELD LIMIT ”) ◮ Assume V → ∞ at ∞ (faster than log | x | in the log cases). For ( x 1 , . . . , x n ) minimizing H n = � i � = j g ( x i − x j ) + n � n i = 1 V ( x i ) , one has � n i = 1 δ x i min H n = µ V = E ( µ V ) lim lim n n 2 n →∞ n →∞ where µ V is the unique minimizer of � � E ( µ ) = R d × R d g ( x − y ) d µ ( x ) d µ ( y ) + R d V ( x ) d µ ( x ) . among probability measures. ◮ E has a unique minimizer µ V among probability measures: the equilibrium measure Frostman 30’s

Next-order asymptotics Hohenberg-Kohn N EXT ORDER EXPANSION OF min H n ◮ Sandier-Serfaty, ’10-’12: d = 1 , 2, g ( x ) = − log | x | ◮ Rougerie-Serfaty, ’13-’16: g ( x ) = 1 / | x | d − 2 ◮ Petrache-Serfaty, ’15-’16: all previous cases plus Riesz cases max ( 0 , d − 2 ) ≤ s < d Theorem (ground state energy, Petrache-Serfaty) Under suitable assumptions on V, as n → ∞ we have min H n =  � � �  µ 1 + s / d  n 2 E ( µ V ) + n 1 + s / d ξ s , d ( x ) dx + o ( 1 )  V � � � n 2 E ( µ V ) − n ξ 0 , d − 1   d log n + n µ V ( x ) log µ V ( x ) dx + o ( 1 )  d where ξ s , d = min W is a functional on microscopic configurations .

Next-order asymptotics Hohenberg-Kohn L OCALIZATION PROCEDURE ◮ When g is the Coulomb kernel, then for a measure µ , � � � ( − ∆ − 1 µ ) µ = |∇ (∆ − 1 µ ) | 2 ( g ∗ µ ) µ = The energy is a single integral, if expressed in terms of the potential g ∗ µ = − ∆ − 1 µ . ◮ If d − 2 < s < d , then g ( x ) = | x | − s is the kernel of a fractional Laplacian ∆ α , α ∈ ] 0 , 1 [ , a nonlocal operator. ◮ Caffarelli-Silvestre extension method: | X | − s is the kernel of a local operator − div ( | y | γ ∇· ) (elliptic, with a good theory) when the space R d is extended by one dimension to R d + 1 = { X = ( x , y ) , x ∈ R d , y ∈ R } . It suffices to take γ = s − d + 1 .

Next-order asymptotics Hohenberg-Kohn ◮ Let k be extension dimension. Identify R d with R d × { 0 } ⊂ R d + k . ◮ In the Coulomb cases k = 0 and γ = 0, In the Riesz cases and d = 1 logarithmic case, then k = 1. ◮ δ R d the uniform measure on R d × { 0 } ⊂ R d + k ◮ Then � R d + k g ( X − X ′ ) ( µδ R d )( X ′ ) h := g ∗ ( µδ R d ) = is the solution in R d + k of − div ( | y | γ ∇ h ) = c d , s µδ R d where in all cases γ := s − d + 2 − k ◮ In terms of h , we find a single integral: � � R d + k | y | γ |∇ h | 2 . ( g ∗ µ ) µ = c d , s

Next-order asymptotics Hohenberg-Kohn S PLITTING FORMULA Define δ ( η ) = measure of mass 1 on ∂ B ( 0 , η ) , such that x − div ( | y | γ ∇ g η ) = c d , s δ ( η ) g η := min ( g , g ( η )) 0 �� � � g ( x − y ) δ ( ℓ ) x i ( x ) δ ( ℓ ) g ( x i − x j ) = lim x j ( y ) ℓ → 0 i � = j i � = j �� �� � � � � n n � � δ ( ℓ ) δ ( ℓ ) g ( x − y ) δ ( ℓ ) 0 ( x ) δ ( ℓ ) = lim g ( x − y ) ( x ) ( y ) − n 0 ( y ) x i x j ℓ → 0 � �� � i = 1 j = 1 � �� � constant self-interaction term = c d , s g ( ℓ ) total interaction between smeared-out charges � � n � Insert the splitting � n i = 1 δ ( ℓ ) i = i δ ( ℓ ) = n µ V + x i − n µ V and x i Frostman’s characterization of µ V : g ∗ µ V + 1 2 V − c =: ζ ≥ 0 , ζ = 0 in spt ( µ V ) , � V where c = E ( µ V ) − 2 d µ V .