On the Mean-Field and Semiclassical Limits of the N-Body Schrdinger - PowerPoint PPT Presentation

On the Mean-Field and Semiclassical Limits of the N-Body Schrödinger Equation François Golse Ecole polytechnique, CMLS Moscow Institute of Physics and Technology,September 12-16, 2016 “Quasilinear equations, inverse problems and applications” In memory of G.M. Henkin Work withT. Paul, Arch. Rational Mech. Anal. DOI 10.1007/s00205-016-1031-x François Golse From N -Body Schrödinger to Vlasov

Motivation N →∞ Schrödinger − → Hartree ↓ ↓ � → 0 ց � → 0 ↓ ↓ N →∞ Liouville − → Vlasov Problem: To derive Vlasov equation from quantum N -body problem by a joint semiclassical ( � → 0) + mean field ( N → ∞ ) limit [Graffi-Martinez-Pulvirenti M3AS 2003] [Pezzotti-Pulvirenti Ann IHP 2009] François Golse From N -Body Schrödinger to Vlasov

DISTANCE BETWEEN CLASSICAL AND QUANTUM STATES François Golse From N -Body Schrödinger to Vlasov

Quantum vs classical densities Quantum density operator ρ = ρ ∗ ≥ 0 , tr H ρ = 1 ⇔ ρ ∈ D ( H ) with H := L 2 ( R d ) Classical density =probability density on R d × R d Wigner transform of ρ ∈ D ( H ) � 1 R d e − i ξ · y ρ ( x + 1 2 � y , x − 1 W � [ ρ ]( x , ξ ) := 2 � y ) dy ( 2 π ) d not nonnegative in general Husimi transform W � [ ρ ] := e � ∆ x ,ξ / 4 W � [ ρ ] ≥ 0 ˜ François Golse From N -Body Schrödinger to Vlasov

Coupling quantum and classical densities Following Dobrushin’s 1979 derivation of Vlasov’s equation, seek to measure the difference between the quantum and the classical dy- namics by a Monge-Kantorovich (or Vasershtein) type distance Couplings of ρ ∈ 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 ) , R d × R d Q ( x , ξ ) dxd ξ = ρ The set of all couplings of the densities ρ and p is denoted C ( p , ρ ) François Golse From N -Body Schrödinger to Vlasov

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 Pseudo-distance “à la” Monge-Kantorovich (with exponent 2) � � 1 / 2 �� E � ( p , ρ ) := inf R d × R d tr ( c � ( x , ξ ) Q ( x , ξ )) dxd ξ Q ∈C ( p ,ρ ) François Golse From N -Body Schrödinger to Vlasov

Töplitz quantization • Coherent state with q , p ∈ R d : | q + ip , � � : x �→ ( π � ) − d / 4 e −| x − q | 2 / 2 � e ip · x / � • With the identification z = q + ip ∈ C d � OP T ( µ ) := 1 OP T ( 1 ) = I C d | z , � �� z , � | µ ( dz ) , ( 2 π � ) d • Fundamental properties: � µ ≥ 0 ⇒ OP T ( µ ) ≥ 0 , tr ( OP T ( µ )) = 1 C d µ ( dz ) ( 2 π � ) d • Important formulas: ( 2 π � ) d e � ∆ q , p / 4 µ , 1 ˜ ( 2 π � ) d e � ∆ q , p / 2 µ 1 W � [ OP T ( µ )]= W � [ OP T ( µ )]= François Golse From N -Body Schrödinger to Vlasov

Basic properties of the pseudo-distance E � Thm A Let p = probability density on R d × R d s.t. �� R d × R d ( | x | 2 + | ξ | 2 ) p ( x , ξ ) dxd ξ < ∞ (1) For each ρ ∈ D ( H ) one has E � ( p , ρ ) ≥ 1 2 d � (2) For each µ ∈ P ( R d × R d ) one has � (( 2 π � ) d µ )) 2 ≤ dist MK , 2 ( p , µ ) 2 + 1 E � ( p , OP T 2 d � (3) For each ρ ∈ D ( H ) , one has E � ( p , ρ ) 2 ≥ dist MK , 2 ( p , ˜ W � [ ρ ]) 2 − 1 2 d � (4) If ρ � ∈ D ( H ) and W � [ ρ � ] → µ in S ′ , then µ ∈ P ( R d × R d ) and lim E � ( p , ρ ) ≥ dist MK , 2 ( p , µ ) � → 0 François Golse From N -Body Schrödinger to Vlasov

PSEUDO-DISTANCE AND DYNAMICS François Golse From N -Body Schrödinger to Vlasov

Vlasov and N -body von Neumann equations Vlasov equation for f ≡ f ( t , x , ξ ) probability density ∂ t f = −{ H f , f } = − ξ · ∇ x f + ∇ x V f · ∇ ξ f with � � V f ( t , x ) := R d V ( x − z ) ρ [ f ]( t , z ) dz , ρ [ f ] := R d fd ξ N -body von Neumann equation ∂ t ρ N , � = − i � [ H N , ρ N , � ] where ρ N , � ∈ D ( H N ) , with H N = H ⊗ N = L 2 (( R d ) N ) and N � � 2 � 2 ∆ y j + 1 − 1 H N := V ( y j − y k ) N j = 1 1 ≤ j < k ≤ N François Golse From N -Body Schrödinger to Vlasov

Indistinguishable particles, symmetries and marginals Notation for σ ∈ S N X N := ( x 1 , . . . , x N ) , σ · X N := ( x σ ( 1 ) , . . . , x σ ( N ) ) Quantum symmetric N -body density for all σ ∈ S N U σ ρ N U ∗ σ = ρ N , where U σ ψ ( X N ) = ψ ( σ · X N ) Notation ρ N ∈ D s ( H N ) k -particle marginal of ρ N ∈ D s ( H N ) is ρ k N ∈ D s ( H k ) such that tr H k ( A ρ k N ) = tr H N (( A ⊗ I H N − k ) ρ N ) for all A ∈ L ( H k ) François Golse From N -Body Schrödinger to Vlasov

From N -body von Neumann to Vlasov Thm B Let f in ≡ f in ( x , ξ ) ∈ L 1 (( | x | 2 + | ξ | 2 ) dxd ξ ) be a probability density on R d × R d , an ρ in N , � ∈ D s ( H N ) . Let f and ρ N , � be the solutions of the Vlasov equation and the von Neumann equation resp. with initial data f in and ρ in N , � . e Γ t − 1 � , N ) e Γ t + ( 2 �∇ V � L ∞ ) 2 � , N ( t )) ≤ 1 E � ( f ( t ) , ρ 1 N E � (( f in ) ⊗ n , ρ in N − 1 Γ with Γ = 2 + 4 max ( 1 , Lip ( ∇ ( V )) 2 � , N = OP T � [( 2 π � ) dN ( f in ) ⊗ N ] If moreover ρ in e Γ t − 1 2 d � ( 1 + e Γ t )+( 2 �∇ V � L ∞ ) 2 � , N ( t )]) 2 ≤ 1 dist MK , 2 ( f ( t ) , � W � [ ρ 1 N − 1 Γ François Golse From N -Body Schrödinger to Vlasov

From N -body von Neumann to Vlasov 2 Amplification In fact, one has a quantitative statement on propa- gation of chaos for this problem: for each fixed n ≥ 1 and all N > n 1 � , N ( t )]) 2 ≤ 1 n dist MK , 2 ( f ( t ) ⊗ n , � W � [ ρ n n E � ( f ( t ) ⊗ n , ρ n � , N ( t )) e Γ t − 1 � , N ) e Γ t + ( 2 �∇ V � L ∞ ) 2 ≤ 1 N E � (( f in ) ⊗ n , ρ in N − 1 Γ This follows from (1) the symmetry of the classical and quantum densitie is, and (2) the structure of the cost which is the sum of costs in each variable François Golse From N -Body Schrödinger to Vlasov

SOME IDEAS FOR THE PROOF François Golse From N -Body Schrödinger to Vlasov

Dynamics of couplings N , � ∈ C s (( f in ) ⊗ N , ρ in Let Q in N , � ) ; solve     N �  + i ∂ t Q N , � + H f ( x j , ξ j ) , Q N , � � [ H N , Q N , � ] = 0  j = 1 � � t = 0 = Q in with Q N , � N , � and N � � 2 � 2 ∆ y j + 1 − 1 H N := V ( y j − y k ) N j = 1 1 ≤ j < k ≤ N �� 2 | ξ | 2 + H f ( x , ξ ) := 1 R d × R d V ( x − z ) f ( t , z , ζ ) dzd ζ François Golse From N -Body Schrödinger to Vlasov

The functional D ( t ) Lemma For each t ≥ 0, one has Q N , � ( t ) ∈ C s ( f ( t ) ⊗ N , ρ N , � ( t )) where f is the solution of the Vlasov equation and ρ N , � is the solution of the N -body von Neumann equation • Define �� N � D ( t ) := 1 tr H N ( c � ( x j , ξ j , y j , ∇ y j ) Q N , � ( t )) dX N d Ξ N N ( R d × R d ) N k = 1 ≥ 1 N E � ( f ( t ) ⊗ N , ρ N , � ( t )) François Golse From N -Body Schrödinger to Vlasov

The evolution of D Multiply both sides of the equation for Q N , � and “integrate by parts”: �� ˙ tr H ( { H f ( x 1 , ξ 1 ) , c � ( x 1 , ξ 1 , y 1 , ∇ y 1 ) } Q 1 D = N , � ) dx 1 d ξ 1 �� − 1 tr H ([∆ y 1 , c � ( x 1 , ξ 1 , y 1 , ∇ y 1 )] Q 1 N , � ) dx 1 d ξ 1 2 i � �� + i tr H 2 ([ N − 1 N V ( y 1 − y 2 ) , c � ( x 1 , ξ 1 , y 1 , ∇ y 1 )]) Q 2 N , � ) dX 2 d Ξ 2 � provided that Q N , � is a symmetric coupling (propagated by the dynamics of couplings). François Golse From N -Body Schrödinger to Vlasov

Summarizing... • The stability part of the analysis (leading to the exponential am- plification by Gronwall’s inequality) is seen at the level of the 1st equation in the BBGKY hierarchy • The consistency part of the analysis requires distributing the inter- action term V on all the particles, and because the V term depends on the X N variables only, and the X N marginal of Q N , � is the N -fold tensor power of the Vlasov solution, one concludes by LLN • Because the cost function in D is a sum of quantities depending on x j , y j , ξ j , there is a “ localization in degree ” effect in the BBGKY hierarchy: no Cauchy-Kovalevska effect when estimating D François Golse From N -Body Schrödinger to Vlasov

Other approaches • Same methods gives (1) a quantitative convergence rate for the semiclassical limit Hartree → Vlasov, and (2) a uniform in N quanti- tative convergence rate for the semiclassical limit of the N -body von Neumann equation to the N -body Liouville equation • Uniform in � → 0 convergence rate for the Hartree (mean-field) limit of the quantum N -body problem [F.G., C. Mouhot, T. Paul, CMP, to appear] • Work in preparation with T. Paul and M. Pulvirenti François Golse From N -Body Schrödinger to Vlasov