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

ASYMPTOTICS OF PARTICLES WITH COULOMB AND RIESZ INTERACTIONS

◮

Hn(x1, . . . , xn) =

• i=j

g(xi − xj) + n

n

• i=1

V(xi), xi ∈ Rd, d ≥ 1

◮ g(x) = |x|−s interaction potential, V : Rd →] − ∞, +∞] 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:

dPn,β(x1, · · · , xn) = 1 Zn,β e−βHn(x1,...,xn)dx1 . . . dxn, xi ∈ Rd Zn,β = partition function, 0 < β = inverse temperature.

Next-order asymptotics Hohenberg-Kohn

ASYMPTOTICS OF PARTICLES WITH COULOMB AND RIESZ INTERACTIONS

◮

Hn(x1, . . . , xn) =

• i=j

g(xi − xj) + n

n

• i=1

V(xi), xi ∈ Rd, d ≥ 1

◮ g(x) = |x|−s interaction potential, V : Rd →] − ∞, +∞] 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:

dPn,β(x1, · · · , xn) = 1 Zn,β e−βHn(x1,...,xn)dx1 . . . dxn, xi ∈ Rd Zn,β = partition function, 0 < β = inverse temperature.

Next-order asymptotics Hohenberg-Kohn

ASYMPTOTICS OF PARTICLES WITH COULOMB AND RIESZ INTERACTIONS

◮

Hn(x1, . . . , xn) =

• i=j

g(xi − xj) + n

n

• i=1

V(xi), xi ∈ Rd, d ≥ 1

◮ g(x) = |x|−s interaction potential, V : Rd →] − ∞, +∞] 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:

dPn,β(x1, · · · , xn) = 1 Zn,β e−βHn(x1,...,xn)dx1 . . . dxn, xi ∈ Rd Zn,β = partition function, 0 < β = inverse temperature.

Next-order asymptotics Hohenberg-Kohn

MOTIVATION 1: FEKETE POINTS

◮ In log gas case minimizers of Hn are maximizers of

• i<j

|xi − xj|

n

• i=1

e−nV(xi) → weighted Fekete sets (approximation theory) Saff-Totik, Rakhmanov-Saff-Zhou.

◮ Fekete points and equilibrium configurations on spheres and

• ther closed manifolds M (Borodachov-Hardin-Saff,

Brauchart-Dragnev-Saff) min

x1,...,xn∈M −

• i=j

log |xi − xj|

• r

min

x1...xn∈M

• i=j

1 |xi − xj|s → 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

MOTIVATION 2: STATISTICAL MECHANICS

dPn,β(x1, · · · , xn) = 1 Zn,β

i<j

|xi − xj| β e−nβ n

i=1 V(xi)dx1 . . . dxn

β = 2 Vandermonde determinant: determinantal processes;

◮ connection to random matrices, first noticed by Wigner, Dyson

(C ⊃ {x1, . . . , xn} = 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¨

• s-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

MOTIVATION 2: STATISTICAL MECHANICS

dPn,β(x1, · · · , xn) = 1 Zn,β

i<j

|xi − xj| β e−nβ n

i=1 V(xi)dx1 . . . dxn

β = 2 Vandermonde determinant: determinantal processes;

◮ connection to random matrices, first noticed by Wigner, Dyson

(C ⊃ {x1, . . . , xn} = 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¨

• s-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´

• ’s webpage)

Next-order asymptotics Hohenberg-Kohn

MOTIVATION 3: MODELS IN SUPERCONDUCTIVITY

◮ (H. Kamerlingh Onnes* 1911) Cooper* pairs, Meissner effect ◮ Rotating superfluids and Bose-Einstein* condensates, modelled

by Ginzburg*-Landau* energy: Gε(ψ, A) = 1 2

• Ω

|∇Aψ|2 + |curl A − hex|2 + (1 − |ψ|2)2 2ε2

◮ Formation of Abrikosov* structured vortices. ◮ The functional Hn describes the interaction of vortices under

Gε-energy (Sandier-Serfaty). (*: Nobel prize winner)

Next-order asymptotics Hohenberg-Kohn

Abrikosov lattices: superconducting vortices on Mg B2 and Nb Se2 crystals (Vinnikov, Phys.Rev.B ’03, Hess, Phys.Rev.Lett. ’89)

Next-order asymptotics Hohenberg-Kohn

THE LEADING ORDER TO min Hn (OR “CONTINUUM/MEAN FIELD LIMIT”)

◮ Assume V → ∞ at ∞ (faster than log |x| in the log cases).

For (x1, . . . , xn) minimizing Hn =

i=j g(xi − xj) + n n i=1 V(xi),

• ne has

lim

n→∞

n

i=1 δxi

n = µV lim

n→∞

min Hn n2 = E(µV) where µV is the unique minimizer of E(µ) =

• Rd×Rd g(x − y) dµ(x) dµ(y) +
• Rd V(x) dµ(x).

among probability measures.

◮ E has a unique minimizer µV among probability measures: the

equilibrium measure Frostman 30’s

Next-order asymptotics Hohenberg-Kohn

NEXT ORDER EXPANSION OF min Hn

◮ 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 Hn =        n2E(µV) + n1+s/d

• ξs,d
• µ1+s/d

V

(x)dx + o(1)

• n2E(µV) − n

d log n + n

• ξ0,d − 1

d

• µV(x) log µV(x) dx + o(1)
• where ξs,d = min W is a functional on microscopic configurations.

Next-order asymptotics Hohenberg-Kohn

LOCALIZATION PROCEDURE

◮ When g is the Coulomb kernel, then for a measure µ,

• (g ∗ µ)µ =
• (−∆−1µ)µ =
• |∇(∆−1µ)|2

The energy is a single integral, if expressed in terms of the potential g ∗ µ = −∆−1µ.

◮ 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 Rd is extended by one dimension to Rd+1 = {X = (x, y), x ∈ Rd, y ∈ R}. It suffices to take γ = s − d + 1.

Next-order asymptotics Hohenberg-Kohn

◮ Let k be extension dimension. Identify Rd with Rd × {0} ⊂ Rd+k. ◮ In the Coulomb cases k = 0 and γ = 0,

In the Riesz cases and d = 1 logarithmic case, then k = 1.

◮ δRd the uniform measure on Rd × {0} ⊂ Rd+k ◮ Then

h := g ∗ (µδRd) =

• Rd+k g(X − X′) (µδRd)(X′)

is the solution in Rd+k of −div (|y|γ∇h) = cd,sµδRd where in all cases γ := s − d + 2 − k

◮ In terms of h, we find a single integral:

• (g ∗ µ)µ = cd,s
• Rd+k |y|γ|∇h|2.

Next-order asymptotics Hohenberg-Kohn

SPLITTING FORMULA

Define δ(η)

x

= measure of mass 1 on ∂B(0, η), such that −div (|y|γ∇gη) = cd,sδ(η) gη := min(g, g(η))

• i=j

g(xi − xj) = lim

ℓ→0

• i=j
• g(x − y)δ(ℓ)

xi (x)δ(ℓ) xj (y)

= lim

ℓ→0

• g(x − y)
• n
• i=1

δ(ℓ)

xi

• (x)
• n
• j=1

δ(ℓ)

xj

• (y)
• total interaction between smeared-out charges

−n

• g(x − y)δ(ℓ)

0 (x)δ(ℓ) 0 (y)

• constant self-interaction term=cd,sg(ℓ)

Insert the splitting n

i=1 δ(ℓ) xi

= nµV + n

i=i δ(ℓ) xi − nµV

• and

Frostman’s characterization of µV: g ∗ µV + 1 2V − c =: ζ ≥ 0, ζ = 0 in spt(µV), where c = E(µV) − V

2 dµV.

Next-order asymptotics Hohenberg-Kohn

SPLITTING FORMULA

Define δ(η)

x

= measure of mass 1 on ∂B(0, η), such that −div (|y|γ∇gη) = cd,sδ(η) gη := min(g, g(η))

• i=j

g(xi − xj) = lim

ℓ→0

• i=j
• g(x − y)δ(ℓ)

xi (x)δ(ℓ) xj (y)

= lim

ℓ→0

• g(x − y)
• n
• i=1

δ(ℓ)

xi

• (x)
• n
• j=1

δ(ℓ)

xj

• (y)
• total interaction between smeared-out charges

−n

• g(x − y)δ(ℓ)

0 (x)δ(ℓ) 0 (y)

• constant self-interaction term=cd,sg(ℓ)

Insert the splitting n

i=1 δ(ℓ) xi

= nµV + n

i=i δ(ℓ) xi − nµV

• and

Frostman’s characterization of µV: g ∗ µV + 1 2V − c =: ζ ≥ 0, ζ = 0 in spt(µV), where c = E(µV) − V

2 dµV.

Next-order asymptotics Hohenberg-Kohn

SPLITTING FORMULA + BLOW-UP

◮ Then we obtain (case s > 0) the formula

Hn(x1, . . . , xn) = n2E(µV) + 2n

n

• i=1

ζ(xi) + n1+ s

d lim

η→0

1 cd,s 1 n

• Rd+k |y|γ|∇h′

n,η|2 − cd,sg(η)

• ,

◮ where we set µ′ V(x) = µV(n−1/dx)= blown-up equilibrium

measure, x′

i = n1/dxi, ◮ and h′ n,η(x′) = g ∗

n

i=1 δ(η) x′

i

− µ′

V

• .

We can now expect compactness (in Lp

loc spaces for h′ n,η) as n → ∞.

Next-order asymptotics Hohenberg-Kohn

SPLITTING FORMULA + BLOW-UP

◮ Then we obtain (case s > 0) the formula

Hn(x1, . . . , xn) = n2E(µV) + 2n

n

• i=1

ζ(xi) + n1+ s

d lim

η→0

1 cd,s 1 n

• Rd+k |y|γ|∇h′

n,η|2 − cd,sg(η)

• ,

◮ where we set µ′ V(x) = µV(n−1/dx)= blown-up equilibrium

measure, x′

i = n1/dxi, ◮ and h′ n,η(x′) = g ∗

n

i=1 δ(η) x′

i

− µ′

V

• .

We can now expect compactness (in Lp

loc spaces for h′ n,η) as n → ∞.

Next-order asymptotics Hohenberg-Kohn

THE RENORMALIZED ENERGY

Definition

Let m > 0. Call Am the class of ∇h such that for Np ∈ N∗, −div (|y|γ∇h) = cd,s

p∈Λ

Npδp − mδRd

• in Rd+k

m = average density of points = uniform background charge density

Definition (Petrache-Serfaty ’15)

Set KR = [−R/2, R/2]d. For ∇h ∈ Am we let W(∇h) = lim

η→0

• lim

R→∞

1 Rd

• KR×Rk |y|γ|∇hη|2 − mcd,sg(η)
• .

This improves on similar definitions from Sandier-Serfaty, Rougerie-Serfaty, initially inspired from Ginzburg-Landau theory Bethuel-Brezis-Helein

Next-order asymptotics Hohenberg-Kohn

MORE EXPLICITLY FOR s > 0

Hn(x1, . . . , xn) = n2E(µV) + 2n

n

• i=1

ζ(xi) + n1+ s

d lim

η→0

1 cd,s 1 n

• Rd+k |y|γ|∇h′

n,η|2 − cd,sg(η)

• Theorem (ground state energy Petrache-Serfaty ’15)

min Hn = n2E(µV) + n1+s/d

• min

A1 W

• µ1+s/d

V

(x)dx + o(1)

• =

n2E(µV) + n1+s/d

• min

AµV(x) W dx + o(1)

• where

◮ W(∇h) = limη→0

• limR→∞ 1

Rd

• KR×Rk |y|γ|∇hη|2 − cd,sg(η)
• ◮ h ∈ A1 admissible micro-scale blow-up potential.

Next-order asymptotics Hohenberg-Kohn

INGREDIENTS FOR THE RESULT

◮ We really prove a (next order) Γ-convergence result. ◮ The lower bound is based on ergodic theorem following an idea

• f Varadhan + Young measure type of quantity

(cf for the quantum case, for first-order asymptotics Fefferman ’85, Graf-Schenker ’95).

◮ The upper bound (recovery sequence) uses a “screening” or

truncation procedure to “truncate vector fields” and then “copy paste” them to build test-configurations (reminiscent of Alberti-Choksi-Otto ’09).

◮ The fact that the equations for the potentials are local is crucially

used for both.

Next-order asymptotics Hohenberg-Kohn

CRYSTALLIZATION QUESTIONS

Question: What is the minimizer of the next-order functional W?

◮ Abrikosov conjecture: in 2d, the regular triangular lattice is a

minimizing configuration for W.

◮ Rigidity results in 2D: g(x) = − log |x| by Ameur-Ortega-Cerda

’11, Rota-Nodari-Serfaty ’14: discrepancy bounds at the microscale.

◮ Petrache-Rota-Nodari ’16: general s = d − 2 as above, problem

reduces to a simpler bound for s = d − 2.

Theorem (Petrache-Rota-Nodari ’16)

For s = d − 2, let ωn = {xn

1, . . . , xn n} be a sequence of minimizing

configurations for Hn and ω′

n := n1/dωn. Consider cubes Kn of size ℓn,

staying ”well inside“ n1/d(spt(µV)). Then

• #(ω′

n ∩ Kn) −

• Kn

m′

V(x′)dx′

• Cℓd−1

n

Next-order asymptotics Hohenberg-Kohn

CRYSTALLIZATION QUESTIONS

Question: What is the minimizer of the next-order functional W?

◮ Abrikosov conjecture: in 2d, the regular triangular lattice is a

minimizing configuration for W.

◮ Rigidity results in 2D: g(x) = − log |x| by Ameur-Ortega-Cerda

’11, Rota-Nodari-Serfaty ’14: discrepancy bounds at the microscale.

◮ Petrache-Rota-Nodari ’16: general s = d − 2 as above, problem

reduces to a simpler bound for s = d − 2.

Theorem (Petrache-Rota-Nodari ’16)

For s = d − 2, let ωn = {xn

1, . . . , xn n} be a sequence of minimizing

configurations for Hn and ω′

n := n1/dωn. Consider cubes Kn of size ℓn,

staying ”well inside“ n1/d(spt(µV)). Then

• #(ω′

n ∩ Kn) −

• Kn

m′

V(x′)dx′

• Cℓd−1

n

Next-order asymptotics Hohenberg-Kohn

QUESTIONS: HYPERUNIFORMITY

◮ The notion of hyperuniform random point configurations was

introduced by Torquato-Stillinger: intermeditae notion between general ”random“ and ”crystal-like“.

◮ Definition: if X ⊂ Rd random point config. is hyperuniform if

Var(#(X ∩ Kℓ(x)) grows like ℓd−1 (or ℓd−α, α > 0).

◮ In our case: Nℓ,n := ω′ n(Kℓ(n1/dx)) is a random variable on the

probability (Ω, B, P) where Ω = supp(µV), B = Borel sets, P = LdΩ

|Ω| . ◮ Conjecture: Var(Nℓ,n) ∼ ℓd−1. ◮ we proved Nℓ,n − EP(Nℓ,n)L∞(P) ≤ Cℓd−1, not enough! ◮ Strategy: go back to structure function (cf. talk by P. Grabner),

and fit it in our framework.

Next-order asymptotics Hohenberg-Kohn

QUESTIONS: POSITIVE TEMPERATURE

◮ Positive temperature β < ∞ regime: Lebl´

e-Serfaty ’16 found the next-order term of the free energy, Fβ(P) = W(P) + β−1ent[P|Π] is the analogue of W, defining an energy on configurations ensembles.

◮ Analogue to Petrache-Rota-Nodari for the 2d Coulomb gas at

positive temperature.

◮ It allows to conjecture that there is a unique β-Ginibre ensemble

minimizing Fβ (Abrikosov-analogue).

◮ Bauerschmidt-Bourgade-Nikula-Yau ’16 by viscosity-like

approach + loop equation, 2D.

◮ Lebl´

e ’16 + Lebl´ e-Serfaty ’16 by very similar ”screening + bootstrapping“ method, 2D.

◮ Extension of Lebl´

e ’16 towards general dimensions.

Next-order asymptotics Hohenberg-Kohn

QUESTIONS: CURVED CRYSTALS

Colloid charged particles at oil-glycerol interface (Irvine-Vitelli-Chaikin, Nature ’10)

Next-order asymptotics Hohenberg-Kohn

QUESTIONS: CURVED CRYSTALS

◮ Importance: modelling of vesicles, study of defects,

interpolation, Smale’s problem...

◮ Experimentally: Curvature influences the formation of ”scars“

in Hn-minimizing configurations.

◮ How to quantify defects? Error of local point density compared

to the planar case. Precise control possible due to Petrache-Rota-Nodari ’16.

◮ Asymptotics of min Hn with submanifold constraints:

◮ B´

etermin-Sandier ’15: second-order term, without a splitting formula (Coulomb energy on S2).

◮ Method: (stereographic projection) + (conformal invariance of the

energy): unavailable towards more general surfaces/dimensions.

Next-order asymptotics Hohenberg-Kohn

QUESTIONS: MORE GENERAL KERNELS |x|−s?

◮ We had a nice boundary value problem for the PDE

−∆h = charges for s = d − 2

◮ We extended this to (−∆)αh = charges for d − 2 ≤ s < d, α ∈]0, 1].

(by looking at −div(|y|γ∇h) = charges)

◮ What about (−∆)αh = charges for α > 1, i.e. power laws

0 ≤ s < d − 2? (kernels with heavy tails)

◮ What is the sharp class of kernels/operators that allows ∃

next-order term?

Next-order asymptotics Hohenberg-Kohn

T H A N K Y O U !

Next-order asymptotics Hohenberg-Kohn

DENSITY FUNCTIONAL THEORY

◮ The chemical behavior of atoms and molecules is captured by

quantum mechanics via Schr¨

• dinger’s eq. (Dirac ’29)

◮ Curse of dimensionality:

◮ The Schr¨

• dinger equation is of the form HΨ = EΨ, a second
• rder PDE on R3n, Ψ represents the state of the n-particle system.

◮ Chemical behavior ∼ energy differences ≪ total energy

◮ Example: carbon atom: n = 6, spectral gap= 10−4×(total

energy). Discretize R by 10 points ⇒ 1018 total grid points.

◮ A scalable simplified reformulation of the precise equations is

the Hohenberg-Kohn-Sham (HK) model (Levy ’79 - Lieb ’83). It is formulated in terms of the one-particle marginal ρ.

Next-order asymptotics Hohenberg-Kohn

DENSITY FUNCTIONAL THEORY

◮ The chemical behavior of atoms and molecules is captured by

quantum mechanics via Schr¨

• dinger’s eq. (Dirac ’29)

◮ Curse of dimensionality:

◮ The Schr¨

• dinger equation is of the form HΨ = EΨ, a second
• rder PDE on R3n, Ψ represents the state of the n-particle system.

◮ Chemical behavior ∼ energy differences ≪ total energy

◮ Example: carbon atom: n = 6, spectral gap= 10−4×(total

energy). Discretize R by 10 points ⇒ 1018 total grid points.

◮ A scalable simplified reformulation of the precise equations is

the Hohenberg-Kohn-Sham (HK) model (Levy ’79 - Lieb ’83). It is formulated in terms of the one-particle marginal ρ.

Next-order asymptotics Hohenberg-Kohn

DFT computation of the aminoacid cystein, from the web page of Nobel prize press release for Walter Kohn.

Next-order asymptotics Hohenberg-Kohn

SUCCESS AND DRAWBACKS OF DFT

◮ The HK model boomed in computational chemistry since the

1990’s.

◮ More than 15000 papers a year contain the keywords ’density

functional theory’.

◮ There exist ’cheap’ versions which allow computations of large

molecules (e.g. DNA, enzymes), routinely used in comp. chemistry, biochemistry, material science, etc.

◮ Lack of systematic improvability of the computations.

How to devise faster methods for the full model at large n?

Next-order asymptotics Hohenberg-Kohn

QUANTUM PARTICLES WITH MANY-PARTICLE

INTERACTIONS AND OPTIMAL TRANSPORT

The relevant minimization in the HK model is a multimarginal repulsive optimal transport problem: Fn(ρ) := min

γn→ρ

• (R3)n

n 2 −1

i=j

1 |xi − xj|dγn(x1, . . . , xn),

◮ γn varies among symmetric probability measures (γn = |Ψ|2 in

terms of n-particle wavefunction Ψ),

◮ γn → ρ means that the marginals of γn are equal to ρ. ◮ First-order “mean field” functional: Cotar, Friesecke, Pass found

a decorrelation effect: F∞(ρ) =

• R3
• R3

1 |x−y|ρ(x)ρ(y)dx dy. ◮ Petrache ’15: generalization by convexity + De Finetti.

Next-order asymptotics Hohenberg-Kohn

QUANTUM PARTICLES WITH MANY-PARTICLE

INTERACTIONS AND OPTIMAL TRANSPORT

The relevant minimization in the HK model is a multimarginal repulsive optimal transport problem: Fn(ρ) := min

γn→ρ

• (R3)n

n 2 −1

i=j

1 |xi − xj|dγn(x1, . . . , xn),

◮ γn varies among symmetric probability measures (γn = |Ψ|2 in

terms of n-particle wavefunction Ψ),

◮ γn → ρ means that the marginals of γn are equal to ρ. ◮ First-order “mean field” functional: Cotar, Friesecke, Pass found

a decorrelation effect: F∞(ρ) =

• R3
• R3

1 |x−y|ρ(x)ρ(y)dx dy. ◮ Petrache ’15: generalization by convexity + De Finetti.

Next-order asymptotics Hohenberg-Kohn

NEXT-ORDER TERM FOR HOHENBERG-KOHN AND

EXPLICIT LIEB-OXFORD BOUND

◮ The Lieb-Oxford inequality bounds the error between the mean

field and the n-particle minimum. A better formula: Fn(ρ) = n2F∞(ρ) − CLOn4/3 ρ4/3 + o(n4/3).

◮ Very relevant for comput. chemistry: (1) precise optimal

constant CLO, but especially (2) quantitative understanding of the second term Work in progress Cotar-Petrache:

◮ Similar to Petrache-Serfaty, we find that CLO can be quantified.

It can be interpreted as a minimum energy on micropatterns for d − 2 < s < d and new nonlocal effects seem to occur for s = d − 2!

◮ Adaptive fast algorithms for very large n, with error bounds.

Cotar-Petrache-Pokern

Next-order asymptotics Hohenberg-Kohn

NEW MATHEMATICS ARISING IN ASYMPTOTICS FOR

MULTIMARGINAL OT

◮ We have pair interactions only, so only the 2-marginals

γN,2 ∈ Psym(X2) of γN ∈ Psym(XN) “count”, for γN,1 = µ fixed.

◮ Question: How to quantify effectively the gap γN,2 − γ∞,2 for the

• ptimizers? Main tools: De Finetti theorems on representability.

Many generalisations possible, links to ∞-dim. representation theory.

◮ Question: What properties of |x − y|2−d are robust under

N → ∞? Main tool: positive definiteness. Robust version of ellipticity: develop its regularity theory (extend Fefferman ’85).

◮ Active research area in Probability/Optimal

Transport/Statistical Mechanics/(?)Analysis of PDEs: Agueh, Carlier, Colombo, Cotar, De Pascale, Di Marino, Friesecke, Gangbo, Ghoussoub, Gori-Giorgi* Kim, Lewin, Lieb, McCann, Pass, Seidl*, Sweich.. (*: part of physics community)

Next-order asymptotics Hohenberg-Kohn

T H A N K Y O U !