Mean-Field Limits for Large Particle Systems Lecture 3: From Schrödinger to Vlasov

Transcription

1 for Large Particle Systems Lecture 3: From Schrödinger to Vlasov CMLS, École polytechnique Frontiers of Applied and Computational Mathematics Shanghai Jiao-Tong University, July

2 The diagram Schrödinger N Hartree 0 0 Liouville N Vlasov Problem (1) Uniformity as 0 of the horizontal limit? (2) Validity of the diagonal limit (1/N + 0)?

3 MEASURING DISTANCES BETWEEN QUANTUM STATES F.G. - T. Paul: C.R. Acad. Sci. Paris, Sér. I (2018)

4 A remark on the topology In the semiclassical limit (i.e. as 0), quantum particles become perfectly localized on trajectories in phase-space. Neighboring trajectories correspond to classical events that are close. Schatten norms fail to capture this; here is the reason why. Key observation (1) If π 1 = π1 and π 2 = π2 are rank-one projections in L(H) and ran(π 1 ) ran(π 2 ) = {0}, one has tr(π 1 π 2 ) = 0 so that π 1 π 2 L 1 (H) = 2 π 1 π 2 L 2 (H) = 2 π 1 π 2

5 Gaussian example Next, one computes explicitly π 1 π 2 L 2 (H) = tr(π 1 π 2 ) 2 = tr(π 1 + π 2 π 1 π 2 π 2 π 1 ) = 2(1 tr(π 1 π 2 )) Example For each q R d, set q (x) := (π ) d/4 e x q 2 /2 Pick q 1, q 2 R d and set π 1 = q 1 q 1, π 2 = q 2 q 2 then π 1 π 2 2 L 2 (H) = 2(1 q 1 q 2 2 ) = 2 (1 ) e q q 2 /2

6 MK distance vs. total variation Example: let q 1, q 2 R d ; then dist MK,p (δ q1, δ q2 ) = q 1 q 2 whereas { 2 if q1 q 2 δ q1, δ q2 TV = 0 if q 1 = q 2 Conclusion (1) Monge-Kantorovich well suited for measuring the proximity of particle trajectories unlike the total variation (2) Schatten norms behave like total variation on rank-one orthogonal projections (3) Therefore, one needs to export Monge-Kantorovich distances to the quantum world in order to measure the proximity between quantum states uniformly in

7 Quantum couplings Couplings between two density operators R 1, R 2 D(H): tr H H ((A I )R) = tr H (AR 1 ) R D(H H) s.t. tr H H ((I A)R) = tr H (AR 2 ) for all A L(H) Set of couplings of R 1, R 2 denoted Q(R 1, R 2 ) Example R 1 R 2 Q(R 1, R 2 ) (not very interesting...)

8 Quantum Monge-Kantorovich pseudo-distance For R 1, R 2 D(L 2 (R d )), define MK 2 (R 1, R 2 ) = inf tr (cr)1/2 R Q(R 1,R 2 ) where c = transportation cost = operator on H H analogous to d ((x m y m ) 2 + (ξ m η m ) 2 ) m=1 Specifically, c is chosen as follows: d c := ((x m y m ) 2 2 ( xm ym ) 2 ) m=1 Remark In general cr L 1 (H H) since c L 1 (H H), in fact ( tr (cr) := tr R 1/2 cr 1/2) [0, + ]

9 The cost operator c and the harmonic oscillator On R d R d, change the original variables (x, y) into { X = 1 2 (x + y) Y := x y so that { Xm = xm + ym Ym = 1 2 ( x m ym ) Since [ Ym, Y m ] = (Heisenberg s uncertainty principle) Then the cost operator c is a sum of harmonic oscillators in Y m Y 2 m Y m 2 = ( 2 Ym + Y m )(2 Ym + Y m ) = (2 Ym + Y m ) (2 Ym + Y m ) 0

10 Properties of MK 2 : separation? Hence c = d j=1 (Y 2 j Y j ) 2d Proposition For all R 1, R 2 D(H), one has MK 2 (R 1, R 2 ) 2 2d > 0 Indeed, if R Q(R 1, R 2 ), then tr H H (R 1/2 cr 1/2 ) 2d tr H H (R) = 2d Remark Even if R 1 = R 2, one has MK 2 (R 1, R 2 ) > 0; therefore MK 2 is not a distance on D(H)

11 Properties of MK2 : triangle inequality? Apart from the obvious case MK 2 (R 1, R 2 ) MK 2 (R 1, R 2 ) + 2d MK 2 (R 1, R 2 ) + MK 2 (R 2, R 2 ) one does not know whether MK 2 satisfies the triangle inequality Quantum analogue of the glueing of couplings procedure? π Π(λ, µ) and ρ Π(µ, ν) ω := π y δ y ρ y µ(dy) with π = π y δ y µ(dy), ρ = δ y ρ y µ(dy)

12 A CRASH COURSE ON TÖPLITZ OPERATORS N. Lerner: Psidos, Spinger LNM 1949

13 Coherent states For p, q R d, define the coherent state (Gaussian wave packet) q, p (x) = (π ) d/4 e x q 2 /2 e ip (x q)/ One easily checks that q, p L 2 (R d ) = 1, 1 (2π ) d R 2d q, p q, p dpdq = I L 2 (R d ) while F ( q, p ) = e ip q/ p, q where F φ(ξ) = 1 (2π ) d/2 e iξ x/ φ(x)dx

14 Z Y X Figure: With ~ = , Z =real part of coherent state centered at q = (0, 0) with momentum p = (1, 0) with space variable (X, Y ) R2

15 Figure: Oscillating structure of a Gaussian coherent state.

16 Töplitz quantization For all positive Borel measure µ on R 2d, define OP T [µ] := 1 (2π ) d R 2d q, p q, p µ(dpdq) with domain Dom(OP T [µ]) {ψ L2 (R d ) s.t. q, p ψ L 2 (µ(dpdq))} Elementary properties (a) OP T [µ] = OP T [µ], tr(opt [µ]) = 1 (2π ) d R 2d µ(dpdq) (b) m(p, q) 0 a.e. OP T [m] m L (R 2d ) (c) µ P(R 2d ) OP T [(2π )d µ] D(L 2 (R d ))

17 Töplitz operators with quadratic symbols If f is a quadratic form on R d, then OP T [f (q)] = f (x) ( f )I OP T [f (p)] = f ( i x) ( f )I Fourier s inversion formula implies that OP T [q kq l ](x, y) = 0, q (x) 2 q k q l dqδ(x y) and =(x k x l δ kl)δ(x y) F ( q, p ) = e ip q/ p, q OP T [p kp l ] = F 1 OP T [q kq l ]F

18 Wigner transform Let T be an operator on L 2 (R d ) with integral kernel T (x, y) Wigner transform of T at scale 1 W [T ](x, ξ) := T (2π ) d (x y, x 1 y)e iξ y 2 dy R d (to be understood in the sense of distributions if T S (R d R d )) Proposition (a) T = T implies that W [T ](x, ξ) R (b) T 0 does not imply that W [T ] 0 (c) R, S L 2 (L 2 (R d ) implies that tr(r S) = (2π ) d R d W [R](x, ξ)w [S](x, ξ)dxdξ

19 Wigner transform and Töplitz operators Important example W [ p, q p, q ](x, ξ) = (π ) d/2 e x q 2 + ξ p 2 Exercise For which ψ L 2 (R d ) does one have W [ ψ ψ ] 0? Wigner transform of a Töplitz operator W [OP T [µ]] = 1 (2π ) d e x,ξ/4 µ Other examples (a) W [I L 2 (R d )] = (2π ) d (b) W [f (x)](x, ξ) = (2π ) d f (x) for f quadratic form, and (c) W [f ( i x )](x, ξ) = (2π ) d f (ξ)

20 Husimi transform Husimi transform at scale of an operator T W [T ](x, ξ) := e x,ξ/4 W [T ](x, ξ) Proposition (a) T 0 implies that W [T ] 0 (b) for µ positive Borel measure on R 2d and T operator on L 2 (R d ) W [T ] L 1 (R 2d, µ) tr(t OP T [µ]) = R 2d W [T ](x, ξ)µ(dxdξ) (c) if µ P 2 (R 2d ) and if f is a quadratic form on R d tr((f (x) + f ( i x )) OP T [(2π )d µ]) = (f (p) + f (q))µ(dpdq) + 1 R 2d 2 f

21 ESTIMATING THE PSEUDO DISTANCE MK2 F. Golse, C. Mouhot, T. Paul Commun. Math. Phys. 343, (2016)

22 Comparing MK 2 and dist MK,2 Theorem (1) For all µ 1, µ 2 P 2 (R 2d ), one has MK 2 (OP T [(2π )d µ 1 ], OP T [(2π )d µ 2 ]) 2 dist MK,2 (µ 1, µ 2 ) 2 + 2d (2) For all R 1, R 2 D(L 2 (R d )), one has dist MK,2 ( W [R 1 ], W [R 2 ]) 2 MK 2 (R 1, R 2 ) 2 + 2d

23 Proof (1) If λ Π(µ 1, µ 2 ), then Q := OP T [(2π )2d λ] Q(OP T [(2π )d µ 1 ], OP T [(2π )d µ 2 ]) Since the cost operator c is a quadratic form in x y and x y, there is an explicit formula for tr(cq).

24 (2) Set µ j = W [R j ] for j = 1, 2; by convex duality ( dist MK,2 (µ 1, µ 2 ) 2 = sup a(x,ξ)+b(y,η) x y 2 + ξ η 2 a,b C b (R 2d ) aµ 1 + bµ 2 ) If a, b C b (R 2d ) satisfies a(x, ξ) + b(y, η) x y 2 + ξ η 2 (by positivity of Töplitz quantization+quantification of quadratic forms) OP T [a] I + I OPT [b] c + 2d I Thus, for all Q Q(R 1, R 2 ) tr(q 1/2 cq 1/2 ) + 2d tr(r 1 OP T [a]) + tr(r 2 OP T [b])) = aµ 1 + bµ 2

25 UNIFORM AS 0 CONVERGENCE RATE FOR THE QUANTUM MEAN FIELD LIMIT F. Golse, C. Mouhot, T. Paul Commun. Math. Phys. 343, (2016)

26 Theorem A (uniform as 0 bound on MK 2 ) Assume that the potential V is even with V Lip(R d ). Let R (t) be the solution of Hartree s equation with initial data D(H), and let R in R,N (t) = e ith N/ R in eith N/ with U σr in,n U σ = R in,n D(H N) for all σ S N. Then, for each n = 1,..., N, and each t 0 1 n MK 2 (R (t) n, R,N:n (t)) 2 1 N MK 2 ((R in ) N, R in,n )2 e Λ 2t + 8 N V 2 L e Λ 2t 1 Λ 2 with Λ 2 := 1 + max(1, 8 Lip( V ) 2 )

27 Evolution of quantum couplings Let Q in,n Q((Rin ) N, R in N ) and let t Q,N(t) be the solution of i t Q,N =[H N R (t) I H N +I HN H N, Q,N ], Q,N t=0 =Q in,n where N H N R (t) = I Hj 1 H R (t) I HN j, I H0 := 1 j=1 Lemma (1) for each t R, one has Q,N (t) Q((R (t)) N, R,N (t)) (2) for σ S N, set U σ Ψ 2N (X N, Y N ) := Ψ 2N (σ X N, σ Y N ). Then U σ Q in,n U σ = Q in,n U σq,n (t)u σ

28 The functional D,N Set D,N (t) := 1 N N tr H2N ((Q,N (t)) 1/2 c j (Q,N (t)) 1/2 ) j=1 where c j = I Hj 1 c I HN j In other words, c j is the cost (differential) operator acting on the variables corresponding to the j-th particle: c j = N ((x j,m y j,m ) 2 ( xj,m yj,m ) 2 ) m=1

29 Computing the evolution of D,N Following the computation of Ḋ N (t) in the classical case, one finds Ḋ,N(t) D,N(t) + IN (t) + J N (t) + 1 N tr H2N ( 2 xjm yjm 2 Q,N (t)) N j=1 Since R D(H), one has ρ (t, x) := R (t, x, x) L t L 1 x and N IN (t):=2 N V x ρ (t, x j ) V µ XN (x j ) 2 ρ (t, x j )dx j N j=1 k=1 JN (t) := 2 N tr H2N ( V µ XN (x j ) V µ YN (y j ) 2 Q,N (t)) N j=1 8 N Lip( V )2 tr H2N ( x j y j 2 Q,N (t))) N j=1

30 Theorem B (convergence rate for Töplitz initial states) Under the same conditions as in Theorem A, assume moreover that R in and R in N, are Töplitz operators, with symbols (2π )d µ in and (2π ) dn µ in N, resp. Then, for each n = 1,..., N, and each t 0 1 n dist MK,2( W [R (t) n ], W [R,N:n (t)]) 2 8 N V e Λ2t 1 2 L Λ N dist MK,2((µ in ) N, µ in,n )2 e Λ 2t + 2d (e Λ 2t + 1) with Λ 2 := 1 + max(1, 8 Lip( V ) 2 )

31 Classical/Quantum Dictionary Monge-Kantorovich dist MK,2 Pseudo-distance MK 2 a div(fu) = (u a)f tr(a[h, R]) = tr([h, A]R) Cauchy-Schwarz inequality and Young s inequality tr((a B + B A)R) tr(( A 2 + B 2 )R)

32 FROM N-BODY SCHRÖDINGER TO VLASOV F. Golse, T. Paul: Archive Rational Mech. Anal. 223, (2017)

33 Coupling quantum and classical densities Problem can one measure the difference between the quantum and the classical dynamics by a Monge-Kantorovich type distance? Couplings of R D(H) and p probability density on R d R d (x, ξ) Q(x, ξ) = Q(x, ξ) L(H) s.t.q(x, ξ) 0 tr(q(x, ξ)) = p(x, ξ), Q(x, ξ)dxdξ = R R d R d The set of couplings of the densities R and p is denoted C(p, R) Example the map p R : (x, ξ) p(x, ξ)r belongs to C(p, R)

34 Pseudo-distance between quantum and classical densities Cost function comparing classical and quantum coordinates (i.e. position and momentum) c (x, ξ) := x y 2 + ξ + i y 2 Definition of a pseudo-distance à la Monge-Kantorovich between classical and quantum densities E (p, R) := Remark ( ) 1/2 inf tr(c (x, ξ)q(x, ξ))dxdξ Q C(p,R) R d R d tr(c (x, ξ)q(x, ξ)) := tr(q(x, ξ) 1/2 c (x, ξ)q(x, ξ) 1/2 ) [0, ]

35 Theorem (From N-body von Neumann to Vlasov) Let f in f in (x, ξ) L 1 (( x 2 + ξ 2 )dxdξ) be a probability density and R in,n D(H N) such that, for all σ S N U σ R in,n U σ = R in,n Let f and R,N be the solutions of the Vlasov and the Heisenberg equation resp. with initial data f in and R in,n. (1) Then, for each t 0 one has E (f (t), R,N:1 (t)) 2 1 N E ((f in ) N, R,N in )2 e Λ2t + 4 V 2 L e Λ2t 1 N Λ 2 (2) If moreover R in,n = OPT [(2π )dn (f in ) N ], then dist MK,2 (f (t), W [R,N:1 (t)]) 2 d (1+e Λ 2t )+ 4 V 2 L N e Λ 2t 1 Λ 2

36 Dynamics of couplings Solve N t Q,N + H f (t, x j, ξ j ), Q,N + i [H N, Q N, ] = 0 j=1 Q,N t=0 = Q in,n with N H N := yj + 1 V (y j y k ) N j=1 1 j<k N H f (t, x, ξ) := 1 2 ξ 2 + V (x z)f (t, z, ζ)dzdζ R d R d Then for all t R and all σ S N, one has U σ Q,N (t, σ X N, σ Ξ N )U σ = Q,N (t, X N, Ξ N ) Q,N (t) C(f (t) N, R,N (t))

37 The functional D,N The proof is based on analyzing the evolution of the functional D,N (t) = 1 N N j=1 tr HN (c(x j, ξ j )Q,N (t, X N, Ξ N ))dx N dξ N following the computation presented in detail in Lecture 1, and the dictionary between the quantum and the classical dynamics

38 REMARKS AND OPEN PROBLEMS

39 Singular potentials (a) The mean-field limit of the N-particle Schrödinger equation to the Hartree equation for the Coulomb potential has been proved by Erdös-Yau [Adv. Theor. Math. Phys. 5, (2001)] (see also Pickl [loc. cit.]); result is not uniform as 0 (b) The mean-field limit of the Newton equations for an N-point particle system to the Vlasov-Poisson equation is still an open problem (c) Significant progress obtained by Hauray-Jabin (interaction force with singularity O(r α ) for α < 1) or for the Coulomb interaction with vanishing regularization as 1/N 0 [Arch. Rational Mech. Anal. 183, (2007)], [Ann. Sci. Ecole Normale Sup. 48, (2015)]; See also Lazarovici-Pickl arxiv:1502:04608 [math-ph] for the regularized Coulomb interaction

40 2nd quantization approach (a) Rodnianski-Schlein [Commun. Math. Phys. 291, (2009)] (b) Bardos-Golse-Gottlieb-Mauser [J. Stat. Phys. 115, (2004)] (c) Fröhlich-Knowles-Schwarz: [Commun. Math. Phys. 288, (2009)] (d) Benedikter-Jaksic-Porta-Saffirio-Schlein [Comm. on Pure Appl. Math., doi: /cpa.21598] Porta-Rademacher-Saffirio-Schlein arxiv:1608:05268 [math-ph], Benedikter-Porta-Saffirio-Schlein [Arch. Ration. Mech. Anal. 221, (2016)]

41 Other mean-field problems/mathematical techniques (a) Mean-field games P.-L. Lions; symmetric functions of infinitely many variables, video of the lectures of November 9 and 16, 2007, fr/site/pierre-louis-lions/ (b) Vortex dynamics in fluid mechanics (Goodman-Hou-Lowengrub [Comm. Pure Appl. Math. 43, (1990)], Hauray [Math. Models Methods Appl. Sci. 19, (2009)]) (c) From Kac s master equation to the space homogeneous Boltzmann equation