Nonlinear Gibbs measure and equilibrium Bose gases Phan Th` anh Nam - PowerPoint PPT Presentation

Nonlinear Gibbs measure and equilibrium Bose gases Phan Th` anh Nam (LMU Munich) Joint work with Mathieu Lewin and Nicolas Rougerie ICMP Montreal, July 27, 2018 1 / 10

Goal Nonlinear Gibbs measure � �� �� d µ ( u ) = “ z − 1 exp |∇ u | 2 + κ | u | 2 + | u | 2 ( w ∗ | u | 2 ) − β du ” Ω invariant under NLS flow u = ( − ∆ + κ ) u + ( w ∗ | u | 2 ) u i ˙ used in Euclidean Quantum Field Theory (Glimm-Jaffe, Simon ’70s, ...) NLS equation with rough initial data (Lebowitz-Rose-Speer ’88, Bourgain ’90s, Burq-Thomann-Tzvetkov ’00s, ...) Stochastic PDE (da Prato-Debbussche ’03, Hairer ’14, ...) Goal: µ arised from many-body quantum mechanics, in a mean-field limit Difficulty: µ is singular, energy functional is + ∞ almost everywhere 2 / 10

Gibbs measure Free (Gaussian) measuse h = − ∆ + κ > 0 on bounded domain Ω ⊂ R d , hu j = λ j u j . Then � λ j � � π e − λ j | α j | 2 d α j d µ 0 ( u ) = “ z − 1 0 e −� u , hu � du ” := , α j = � u j , u � ∈ C j ≥ 1 is well defined on Sobolev space H s if and only if s < 1 − d / 2 Interacting measure d µ ( u ) = “ z − 1 e −� u , hu �−D ( u ) du ” := z − 1 e −D ( u ) d µ 0 ( u ) r is well-defined when d = 1, 0 ≤ w ∈ a δ 0 + L ∞ and �� D ( u ) = 1 w ( x − y ) | u ( x ) | 2 | u ( y ) | 2 dxdy 2 w ∈ L 1 and d = 2 , 3, 0 ≤ � �� � �� � D ( u ) = 1 | u ( x ) | 2 − �| u ( x ) | 2 � µ 0 | u ( y ) | 2 − �| u ( y ) | 2 � µ 0 w ( x − y ) 2 3 / 10 1

Many-body quantum model Bosonic Gibbs state Γ λ, T = Z − 1 λ, T e − H λ / T with grand-canonical Hamiltonian   ∞ n ∞ � � � �   L 2 sym (Ω n ) H λ = ( − ∆ x j + κ ) + λ w ( x i − x j ) on n =0 j =1 1 ≤ i < j ≤ n n =0 � �� x ( − ∆ + κ ) a x dx + λ a ∗ w ( x − y ) a ∗ x a ∗ = y a x a y dxdy 2 Mean-field limit λ = T − 1 → 0 formally leads to semiclassical approximation � � �� � − 1 1 a ∗ w ( x − y ) a ∗ x a ∗ Z T − 1 , T = Tr exp x ( − ∆ + κ ) a x dx − y a x a y dxdy 2 T 2 T � ( T /π ) dim L 2 (Ω) w ( x − y ) | u ( x ) | 2 | u ( y ) | 2 dxdy du � u ( x )( − ∆+ κ ) u ( x ) dx − 1 �� e − ∼ 2 L 2 (Ω) 4 / 10

1D result � � �� � − 1 x ( − ∆ + κ ) a x dx − λ Γ λ, T = Z − 1 a ∗ w ( x − y ) a ∗ x a ∗ λ, T exp y a x a y dxdy T 2 T Theorem (Lewin-N-Rougerie ’15) Assume d = 1, 0 ≤ w ∈ a δ 0 + L ∞ and λ = T − 1 → 0. Then � Z λ, T e −D ( u ) d µ 0 ( u ) → z r = Z 0 , T L 2 (Ω) and � k ! T k Γ ( k ) | u ⊗ k �� u ⊗ k | d µ ( u ) , λ, T → ∀ k ≥ 1 L 2 (Ω) strongly in trace class Remarks Reduced density matrices Γ ( k ) λ, T ( x 1 , ..., x k ; y 1 , ..., y k ) = Tr[ a ∗ x 1 ... a ∗ x k a y 1 ... a y k Γ λ, T ] Fragmentation of Bose-Einstein condensates µ determined completely by all moments 5 / 10

Renormalized Hamiltonian d ≥ 2 �� � �� � D ( u ) = 1 | u ( x ) | 2 − �| u ( x ) | 2 � µ 0 | u ( y ) | 2 − �| u ( y ) | 2 � µ 0 w ( x − y ) 2 � � � � � � � 2 � � = 1 � � | u ( x ) | 2 e ik · x dx − | u ( x ) | 2 e ik · x dx w ( k ) � dk � � 2 µ 0 Ω Ω � � � � x ( − ∆ + κ ) a x dx + λ � 2 dk a ∗ � d Γ( e ik · x ) − � d Γ( e ik · x ) � Γ 0 , T � H λ = w ( k ) � 2 � �� x ( − ∆ + V T ( x )) a x dx + λ y a x a y dxdy + λ a ∗ w ( x − y ) a ∗ x a ∗ = 2 � ρ 0 , T , w ∗ ρ 0 , T � 2 � � 1 with V T ( x ) = κ + λ w (0) / 2 − λ w ∗ ρ 0 , T ( x ) , ρ 0 , T ( x ) = ( x ; x ) − ∆+ κ e − 1 T In homogeneous case ( − ∆ periodic on unit torus)   T in d = 1  � 1 ρ 0 , T ( x ) = ∼ T log T in d = 2 | k | 2+ κ   − 1 e T T 3 / 2 k ∈ (2 π Z ) d in d = 3 and V T is simply a (modified) chemical potential 6 / 10

2D result � � � �� a ∗ x ( − ∆ + V T ( x )) a x dx + λ w ( x − y ) a ∗ x a ∗ y a x a y dxdy + E T Γ λ, T = Z − 1 2 λ, T exp − T Theorem (Lewin-N-Rougerie ’18) w ( k )(1 + | k | ) ∈ L 1 and λ = T − 1 → 0. Then Assume d = 2, 0 ≤ � � Z λ, T e −D ( u ) d µ 0 ( u ) → z r = Z 0 , T and � k ! T k Γ ( k ) | u ⊗ k �� u ⊗ k | d µ ( u ) , λ, T → ∀ k ≥ 1 strongly in Schatten space S p for all p > 1. Moreover, � Γ (1) λ, T − Γ (1) 0 , T | u ⊗ k �� u ⊗ k | ( d µ ( u ) − d µ 0 ( u )) → in trace class T Similar result expected in d = 3 (work in progress) 7 / 10

Remarks Fr¨ ohlich-Knowles-Schlein-Sohinger 2017: µ arised in d ≤ 3 for e − ε H 0 / T e − ( H λ − 2 ε H 0 ) / T e − ε H 0 / T Rescaling T �→ 1 and Ω �→ [0 , L ] d with L → ∞ : free density � � 1 1 + ∞ in d = 1 , 2 � → ρ c = 1 k 2+ κ L d e | 2 π k | 2 − 1 dk in d = 3 e − 1 L 2 R 3 k ∈ 2 π Z d Thus the Gibbs measure tells us the behavior just below the critical density , or equivalently just above the critical temperature for BEC Deuchert-Seiringer-Yngvason 2018: BEC transition in thermodynamic and Gross-Pitaevskii limit 8 / 10

Ideas of proofs Variational approach : � � − log Z λ, T + λ Γ λ, T minimizes = inf H (Γ , Γ 0 , T ) T Tr( W Γ) � �� � Z 0 , T Γ ≥ 0 , Tr Γ=1 Tr(Γ(log Γ − log Γ 0 , T )) � � � µ minimizes − log z r = inf H cl ( ν, µ 0 ) + D ( u ) d ν ( u ) � �� � ν prob. measure d ν d ν � d µ 0 log d µ 0 d µ 0 Quantum to classical by quantum de Finetti theorem � k ! T k Γ ( k ) | u ⊗ k �� u ⊗ k | d ν ( u ) , λ, T ⇀ ∀ k ≥ 1 In d = 1 the result essentially follows from lim inf H (Γ λ, T , Γ 0 , T ) ≥ H cl ( ν, µ 0 ) (Berezin-Lieb) � lim inf λ T Tr( W Γ λ, T ) = lim inf 1 T 2 Tr( w Γ (2) λ, T ) ≥ D ( u ) d ν ( u ) (Fatou) 9 / 10

Ideas of proofs For d ≥ 2, renormalized interaction has no sign � Fatou’s lemma fails to apply! Localization method Γ λ, T ≈ (Γ λ, T ) P ⊗ (Γ 0 , T ) Q , P = 1 ( − ∆ + κ ≤ Λ) , Q = 1 − P Use quantitative de Finetti for P modes, and error estimate for Q modes Lemma (Variance estimate: d = 2 , Λ ≥ T δ ) �� � 2 � 1 d Γ( Q ) − � d Γ( Q ) � Γ λ, T → 0 T 2 Γ λ, T Proof. Reduce two-body to one-body problem � � H λ − ε d Γ( Q ) d Γ( Q ) e − �� � 2 � Tr T � � d Γ( Q ) − � d Γ( Q ) � λ, T ≈ T ∂ ε =0 , H λ − ε d Γ( Q ) Γ λ, T e − Tr T then control g ′ (0) by g ( ε ) − g (0) and g ′′ , thanks to Taylor’s expansion g ( ε ) = g (0) + g ′ (0) ε + ε 2 2 g ′′ ( θ ε ) 10 / 10