Finite-range kernel decompositions and asymptotic Optimal Transport - PowerPoint PPT Presentation

Finite-range kernel decompositions and asymptotic Optimal Transport between configurations Mircea Petrache, PUC Chile February 27, 2018

Density Functional Theory Cystein molecule simulation , (from Walter Kohn’s Nobel prize laudation page)

D ENSITY F UNCTIONAL T HEORY ◮ The chemical behavior of atoms and molecules is captured by quantum mechanics via Schr¨ odinger’s eq. (Dirac ’29) ◮ Curse of dimensionality : ◮ The Schr¨ odinger equation is of the form H Ψ = E Ψ , a second order PDE on R 3 N , 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 ⇒ 10 18 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 normalized one-particle density ρ .

D ENSITY F UNCTIONAL T HEORY ◮ 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.

D ENSITY F UNCTIONAL T HEORY How to devise faster methods for the full model at large N ? Simulation of heavy-metal pump in E. Coli (Su & al., Nature ’11 )

N-marginal Optimal Transport

A N N - MARGINAL O PTIMAL T RANSPORT PROBLEM :   � �   N � � γ N ∈ P sym (( R d ) N ) , 1 F OT � N ( ρ ) := min | x i − x j | s d γ N ( x 1 , . . . , x N )  . � γ N �→ ρ  ( R d ) N i � = j Optimal γ N (radial part) for N = 2 , s = 1 and d = 3 and different ρ from Cotar-Friesecke-Kl¨ upperberg ’13

Optimal γ N for N = 3 , 4 , 5 points, and s = 1 , d = 1 (projected from R N to R 2 ) , from Di Marino-Gerolin-Nenna ’15

A BOUT N - MARGINAL O PTIMAL T RANSPORT PROBLEMS :   � �   N � � γ N ∈ P sym (( R d ) N ) , F OT � N , c ( ρ ) := min c ( x i − x j ) d γ N ( x 1 , . . . , x N )  . � γ N �→ ρ  ( R d ) N i � = j ◮ Problem appeared naturally in OT theory (for tame c ( x − y ) ) Gangbo-Swiech ’98, Carlier ’03, Carlier-Nazaret ’08 ◮ Optimal transport community Colombo-De Pascale-Di Marino ’13, Colombo-Di Marino ’15, Di Marino-Gerolin-Nenna ’15, De Pascale ’15, Buttazzo-Champion-De Pascale ’17, .. ◮ Link between OT and DFT/math physics Cotar-Friesecke-Kl¨ uppelberg ’13, ’17, Cotar-Friesecke-Pass ’15,.. ◮ Regularity-type results Pass ’13, Moameni ’14, Moameni-Pass ’17, Kim-Pass ’17..

DFT AND MULTIMARGINAL OT ◮ Hohenberg-Kohn functional: energy of N electrons of density ρ (Hohenberg-Kohn ’64, Levy ’79, Lieb ’83) Ψ N ∈A N , Ψ N �→ ρ � Ψ N , ( � 2 � T + � F HK N [ ρ ] := min V ee )Ψ N � . ◮ � T = − 1 2 ∆ R Nd quantum mechanical kinetic energy ◮ V ee ( x 1 , .., x N ) = � 1 ≤ i < j ≤ N 1 / | x i − x j | d − 2 , ◮ � � � Ψ N : ( R d × Z 2 ) N → C , � Ψ N � L 2 = 1 , ∇ | Ψ N | ∈ L 2 , Antisymm. A N = ◮ Ψ N �→ ρ means � � | Ψ( x 1 , s 1 , .., s N , x N ) | 2 dx 2 . . . dx N = ρ ( x 1 ) . s 1 ,.., s N ∈ Z 2 ◮ lim � → 0 F HK N [ ρ ] = F OT N ( ρ ) (Cotar-Friesecke-Kl¨ uppelberg ’13,’17, Lewin ’17, De Pascale-Bindini ’17)

M INIMUM ENERGY AND MULTIMARGINAL OT ◮ If ω N = { x 1 , . . . , x N } ⊂ N and V : R d → R “confining” potential, N � � 1 E V ( ω N ) := | x i − x j | s + N V ( x i ) . i � = j i = 1 ◮ Let ω ∗ N be minimum E V -energy configurations. Then �� � d µ ( x ) d µ ( y ) � � � 1 lim δ p = µ V ∈ argmin + V ( x ) d µ ( x ) | x − y | s N N →∞ p ∈ ω ∗ N ◮ γ ∗ N := uniform on { permutations of ω ∗ N } . Then γ ∗ N ∈ P sym (( R d ) N ) and we find   � �  1 F OT N ( µ V ) ≥ F OT  = E V ( ω ∗ δ p N ) − Vd µ V . N N p ∈ ω ∗ N

Results and Open Problems

L EADING - ORDER ASYMPTOTICS , 0 ≤ s < d ◮ First-order “mean field” functional: Cotar-Friesecke-Pass ’15. ◮ Petrache ’15: generalization by convexity + De Finetti Theorem � N � − 1 � � F N , c ( ρ ) = R d c ( x − y ) ρ ( x ) ρ ( y ) dx dy lim 2 N →∞ R d if and only if c ( x − y ) is balanced positive definite, i.e. � � � ρ ( x ) ρ ( y ) c ( x − y ) ≥ 0 whenever ρ = 0 .

N EXT - ORDER TERM , 0 < s < d ◮ d = 1, general kernels: unpublished note by Di Marino ◮ s = 1 , d = 3: Lewin-Lieb-Seiringer ’17, using Graf-Schenker ’95 ◮ Improving upon the different strategy Fefferman ’85, we get: Theorem (Cotar-Petrache ’17) If d ≥ 1 , 0 < s < d and ρ s.t. the following integrals are finite, then � � ρ ( x ) ρ ( y ) F OT N , s ( ρ ) = N 2 | x − y | s dx dy R d R d � � � + N 1 + s R d ρ 1 + s d ( x ) dx + o ( 1 ) C UG ( d , s ) as N → ∞ . d We can interpret C UG ( d , s ) = min energy of an “Uniform Riesz Gas” (special case: “Uniform Electron Gas” from DFT, for s = d − 2).

N EXT - ORDER TERMS : OT VS . E NERGY Theorem (Cotar-Petrache ’17) If 0 < s < d and d µ ( x ) = ρ ( x ) dx then as N → ∞ � � � N 2 E ( µ ) + N 1 + s R d ρ 1 + s F OT d ( x ) dx + o ( 1 ) N , s ( µ ) = C UG ( d , s ) . d We can interpret C UG ( d , s ) = min E UG ( ν ) “uniform gas” energy on microscale configurations. Recall: Theorem (Petrache-Serfaty ’15) If max { 0 , d − 2 } ≤ s < d under suitable assumptions on V, as N → ∞ � � � 1 + s ( s � = 0 ) N 2 E ( µ V ) + N 1 + s min H N = C Jel ( d , s ) µ d ( x ) dx + o ( 1 ) . d V We can interpret C Jel ( d , s ) = min E Jel ( ν ) , “Riesz Jellium” energy on microscale configurations.

N EXT - ORDER TERMS : OPEN PROBLEMS Theorem (Cotar-Petrache ’17) For d ≥ 2 and d − 2 < s < d there holds C Jel ( d , s ) = C UG ( d , s ) . ◮ The above asymptotic microscale problems are then equivalent for d ≥ 2 and d − 2 < s < d . ◮ Heuristics for s = 1 , d = 3 in Lewin-Lieb ’15: C Jel ( d , d − 2 ) � = C UG ( d , d − 2 ) , questioning the physicists’ conjecture that C Jel ( d , d − 2 ) = C UG ( d , d − 2 ) . ◮ Open problem : prove or disprove C Jel ( d , d − 2 ) � = C UG ( d , d − 2 ) . ◮ Open problem : Sharp asymptotics of min H N for 0 < s < d − 2. ◮ Possible ideas: ◮ s �→ C Jel ( d , s ) might have a jump at s = d − 2. ◮ The (analytic continuation in α of the) fractional laplacian ( − ∆) α from Petrache-Serfaty has a residue at s = d − 2 , d − 4 , . . . .

N EXT - ORDER TERMS : OPEN PROBLEMS “Exchange-correlation” energy = Part of the energy not encoded in 1-particle density � � N ( ρ ) − N 2 ρ ( x ) ρ ( y ) E xc N ( ρ ) := F OT � | x − y | s dx dy . R d ρ R d R d In Cotar-Petrache ’17 we prove � N →∞ N − 1 − s R d ρ 1 + s d E xc d ( x ) dx . lim N ( ρ ) = C UG ( d , s ) C UG ( 3 , 1 ) = asymptotic Lieb-Oxford constant (cf. Dirac ’30, Lieb-Oxford ’81, Lewin-Lieb ’15). ◮ Open questions : Let d ≥ 2, let 0 < s < d . What are the precise values of (any of) N ∈ N N − 1 − s d E xc inf N ( 1 [ 0 , 1 ] d ) or C UG ( d , s ) ? (most physically relevant for s = 1 , d = 3)

Proof strategy

S HARP NEXT - ORDER TERM : PROOF FOR UNIFORM ρ � � M 1 ρ 1 + M 2 ρ 2 1. E xc ≤ E xc M 1 ( ρ 1 ) + E xc M 2 ( ρ 2 ) . M 1 + M 2 M 1 + M 2 2. E xc N ( α d ρ α ) = α − s E xc N ( ρ ) if ρ α ( x ) = ρ ( α x ) . 3. This and a subadditivity argument (Robinson-Ruelle) proves that for ρ = 1 A , | A | = 1, there holds: N →∞ N − 1 − s d E xc lim N ( 1 A ) = C UG ( d , s ) This will give also the interpretation of C UG ( d , s ) as an energy on microscopic blow-up configurations. (Note that C UG ( d , s ) < 0.)

S HARP NEXT - ORDER TERM : PIECEWISE CONSTANT ρ k � ρ ( x ) = α i µ i , µ i uniform prob. on a hyperrectangle i = 1 ◮ Upper bound: subadditivity* ◮ Lower bound: 1. An ensemble (Ω , P ) of packings { F ω } ω ∈ Ω 2. each F ω consisting of balls of sizes 0 < R 1 < · · · < R M in a geometric series, 3. if Σ = spt ( µ ) then | Σ \ ∪ A ∈ F ω A | → 0 as R M → 0 4. at fixed F ω , decompose the kernel + average: N � � � | x i − x j | − s = | x i − x j | − s + err ω ( x 1 , . . . , x d ) , i � = j A ∈ F ω 1 ≤ i � = j ≤ N i , j = 1 xi , xj ∈ A 5.  � � ρ | A �  � E xc  d P ( ω ) ≤ E xc � N ( ρ ) + err , N A A ρ Ω A ∈ F ω

S HARP NEXT - ORDER TERM : PIECEWISE CONSTANT ρ k � ρ ( x ) = α i ρ i , ρ i uniform prob. on a hyperrectangle i = 1 ◮ Upper bound: subadditivity* ◮ Lower bound: 1. A family of packings { F ω } ω ∈ Ω 2. each F ω consisting of balls of sizes 0 < R 1 < · · · < R M in a geometric series, 3. if Σ = spt ( ρ ) then | Σ \ ∪ A ∈ F ω A | → 0 as R M → 0 4. at fixed F ω , decompose the kernel + average: � c ( x − y ) = 1 A ( x ) 1 A ( y ) c ( x − y ) + err ω ( x − y ) , A ∈ F ω 5.  � � ρ | A �  � E xc  d P ( ω ) ≤ E xc � N ( ρ ) + err , N A A ρ Ω A ∈ F ω