Combined mean-field and semiclassical limits of large fermionic - PowerPoint PPT Presentation

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Combined mean-field and semiclassical limits of large fermionic systems Li CHEN Joint work with Jinyeop Lee and Matthew Liew Universit¨ at Mannheim 21.10.2019-25.10.2019, CIRM Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Setting and introduction from many particle Schr¨ odinger to Vlasov 1 Reformulation of the many body Schr¨ odinger equation by using 2 Husimi measures and the main result Proof strategy and Uniform estimates 3 Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic systems

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Setting and introduction for the combined limits from many particle Schr¨ odinger to Vlasov Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Time dependent N -particle Schr¨ odinger equation     N N � �  − � 2  ∆ q j + 1   Ψ N , t , i � ∂ t Ψ N , t = V ( q i − q j ) 2 2 N  j =1 i � = j   Ψ N , 0 = Ψ N , the initial data and the time-dependent states Ψ N , t are both in L 2 a ( R 3 N ) functions (fermionic case) V is the interacting potential � = N − 1 3 (Combined semiclassical and mean field limit) Vlasov equation � � ∂ t m t ( q , p ) + p · ∇ q m t ( q , p ) = ∇ V ∗ ρ t ( q ) · ∇ p m t ( q , p ) , Goal: N → infinity . Schr¨ odinger → Vlasov Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Mean field limit and semiclassical limit Table-1: Picture from Golse, Mouhot, and Paul Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates State of arts (not complete!) 1980’s Narnhofer, Neunzert, Sewell, Spohn ......(for Bosons and Fermions) Mean filed limit Bosons (to Hartree) Bardos, Erd¨ os, Golse, Mauser, Yau, Rodnianski, Schlein, Chen, Lee, Pickl, Petrat...... Fermions (to Hartree Fock) Benedikter, Porta, ,Schlein, Bach, Breteaux, Petrat, Pickl, Tzaneteas...... Semiclassical limit (from HF to Vlasov) Benedikter, Porta, Saffirio, Dietler, Rademacher, Schlein...... Wigner transform Combined mean field and semiclassical limit (to Vlasov), Golse, Mouhot, Paul Husimi measure (Wigner measure convoluted with Gaussian) (for ∇ V Lip) We hope to contribute in the combined limits with less regular V . To propose new approach. Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Motivated by Fournais, Lewin and Solovej’s work (stationary case) Main tool: The Husimi measure. For any real-valued f ∈ L 2 ( R 3 ) with � f � L 2 = 1, the coherent state is � y − q � q , p ( y ) := � − 3 i f � 4 f � p · y √ e � The k -particle Husimi measure is defined as, for any 1 � k � N � � m ( k ) ψ N , a ∗ ( f � q 1 , p 1 ) · · · a ∗ ( f � q k , p k ) a ( f � q k , p k ) · · · a ( f � N ( q 1 , p 1 , . . . , q k , p k ) := q 1 , p 1 ) ψ N where ψ N ∈ F ( N ) is the N -fermionic states. a Remark : Husimi measure measures how many particles are in the k √ semiclassical boxes with length scaled of � centered in its respectively phase-space pair, ( q 1 , p 1 ) , . . . , ( q k , p k ). Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates Reformulation of the many body Schr¨ odinger equation Let ψ N , t ∈ L 2 a ( R 3 N ) be anti-symmetric N -particle state satisfying the Schr¨ odinger equation. Then ∂ t m (1) N , t ( q 1 , p 1 ) + p 1 · ∇ q 1 m (1) N , t ( q 1 , p 1 ) �� 1 d q 2 d p 2 ∇ V ( q 1 − q 2 ) m (2) N , t ( q 1 , p 1 , q 2 , p 2 ) + ∇ q 1 · R 1 + ∇ p 1 · ˜ = (2 π ) 3 ∇ p 1 · R 1 , where the remainder terms R 1 and ˜ R 1 , are given by � � ∇ q 1 a ( f � q 1 , p 1 ) ψ N , t , a ( f � R 1 := � ℑ q 1 , p 1 ) ψ N , t , �� �� �� � 1 � � 1 ˜ R 1 := (2 π ) 3 · ℜ d w d u d y d v d q 2 d p 2 d s ∇ V su + (1 − s ) w − y 0 f � q 1 , p 1 ( u ) f � q 1 , p 1 ( w ) f � q 2 , p 2 ( y ) f � q 2 , p 2 ( v ) � a w a y ψ N , t , a u a v ψ N , t � �� 1 d q 2 d p 2 ∇ V ( q 1 − q 2 ) m (2) − N , t ( q 1 , p 1 , q 2 , p 2 ) . (2 π ) 3 Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates For 1 < k � N , we have the following hierarchy ∂ t m ( k ) q k m ( k ) N , t ( q 1 , p 1 , . . . , q k , p k ) + � p k · ∇ � N , t ( q 1 , p 1 , . . . , q k , p k ) �� 1 d q k +1 d p k +1 ∇ V ( q j − q k +1 ) m ( k +1) = (2 π ) 3 ∇ � p k · ( q 1 , p 1 , . . . , q k +1 , p k +1 ) N , t p k · ˜ R k + � + ∇ � q k · R k + ∇ � R k , where the remainder terms are denoted as � � � � a ( f � qk , pk ) · · · a ( f � ψ N , t , a ( f � qk , pk ) · · · a ( f � R k := � ℑ ∇ � q 1 , p 1 ) q 1 , p 1 ) ψ N , t , qk � � �� 1 � � � ⊗ k 1 ( d w d u ) ⊗ k f � ( ˜ R k ) j := d s ∇ V ( su j + (1 − s ) w j − y ) q , p ( w ) f � q , p ( u ) (2 π ) 3 ℜ d y 0 �� � � � p f � d v f � d ˜ qd ˜ p ( y ) p ( v ) a wk · · · a w 1 a y ψ N , t , a uk · · · a u 1 a v ψ N , t q , ˜ ˜ q , ˜ ˜ �� 1 d q k +1 d p k +1 ∇ V ( q j − q k +1 ) m ( k +1) − ( q 1 , p 1 , . . . , q k +1 , p k +1 ) , (2 π ) 3 N , t � � � � � 2 k � ⊗ k � ( d w d u ) ⊗ k � f � q , p ( w ) f � R k := ℑ V ( u j − u i ) − V ( w j − w i ) q , p ( u ) 2 j � = i � � a wk · · · a w 1 ψ N , t , a uk · · · a u 1 ψ N , t Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy

Setting and introduction from many particle Schr¨ odinger to Vlasov Reformulation of the many body Schr¨ odinger equation by using Husimi measures and the main result Proof strategy and Uniform estimates What’s next? ∀ k ≥ 1, the sequence of k particle Husimi measures, m ( k ) N , t , is weakly compact in L 1 ( R 6 k ) All the remainder terms, R k , ˜ R k and � R k , converge to 0 in weak sense. Li CHEN Joint work with Jinyeop Lee and Matthew Liew Combined mean-field and semiclassical limits of large fermionic sy