A microscopic derivation of Gibbs measures for nonlinear Schrödinger equations with unbounded interaction potentials Vedran Sohinger (University of Warwick) partly joint work with Jürg Fröhlich (ETH Zürich) Antti Knowles (University of Geneva) Benjamin Schlein (University of Zürich) Quantissima in the Serenissima III, Venice August 22, 2019. V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 1 / 25
The nonlinear Schrödinger equation Consider the spatial domain Λ = T d for d = 1 , 2 , 3 . Study the nonlinear Schrödinger equation (NLS) . � � � � d y w ( x − y ) | φ t ( y ) | 2 φ t ( x ) i ∂ t φ t ( x ) = − ∆ + κ φ t ( x ) + φ 0 ( x ) = Φ( x ) ∈ H s (Λ) . Chemical potential κ > 0 ; Interaction potential w ∈ L q (Λ) is positive or w = δ . Conserved energy (Hamiltonian) � � � |∇ φ ( x ) | 2 + κ | φ ( x ) | 2 � + 1 d x d y | φ ( x ) | 2 w ( x − y ) | φ ( y ) | 2 . H ( φ ) = d x 2 The Gibbs measure d µ associated with Hamiltonian flow is the probability measure on the space of fields φ : Λ → C � 1 µ (d φ ) . Z . Z e − H ( φ ) d φ , e − H ( φ ) d φ . . = . = → formally invariant under flow of NLS. V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 2 / 25
Gibbs measures for the NLS: known results Rigorous construction: CQFT literature in the 1970 -s (Nelson, Glimm-Jaffe, Simon), also Lebowitz-Rose-Speer (1988). Proof of invariance: Bourgain and Zhidkov (1990s). → Measure supported on low-regularity Sobolev spaces. Application to PDE: Obtain low-regularity solutions of NLS µ -almost surely . Recent advances: Bourgain-Bulut, Burq-Tzvetkov, Burq-Thomann-Tzvetkov, Cacciafesta- de Suzzoni, Deng, Genovese-Lucá-Valeri, Nahmod-Oh-Rey-Bellet-Staffilani, Nahmod-Rey-Bellet-Sheffield-Staffilani, Oh-Pocovnicu, Oh-Quastel, Oh-Tzvetkov, Oh-Tzvetkov-Wang, Thomann-Tzvetkov, Tzvetkov, ... V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 3 / 25
Derivation of Gibbs measures: informal statement Formally, NLS is a classical limit of many-body quantum theory . On H ( n ) ≡ L 2 sym (Λ n ) we consider the n -body Hamiltonian n � � � � H ( n ) . . = − ∆ x i + κ + λ w ( x i − x j ) . i =1 1 � i<j � n Solve n-body Schrödinger equation i ∂ t Ψ n,t = H ( n ) Ψ n,t . Obtain that, for λ = 1 /n as n → ∞ Ψ n, 0 ∼ φ ⊗ n implies Ψ n,t ∼ φ ⊗ n . 0 t (Hepp (1974), Ginibre-Velo (1979), Spohn (1980), Erd˝ os-Schlein-Yau (2006, 2007), Lieb-Seiringer (2006), Klainerman-Machedon (2008), T.Chen-Pavlovi´ c (2010), Ammari-Nier (2011), X.Chen-Holmer (2012), Lewin-Nam-Rougerie (2014), S. (2014), Lewin-Nam-Schlein (2015), Bossmann-Teufel (2018,2019), . . . ). Problem: Obtain Gibbs measure d µ as many-body quantum limit . V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 4 / 25
Outline of strategy and goals The main strategy Give rigorous definition of classical Gibbs measure d µ . 1 ‘Encode’ d µ in terms of a sequence of operators ( γ p ) p . 2 Define many-body quantum Gibbs states and ‘encode’ them in terms of a 3 sequence of operators ( γ τ,p ) p . Show that 4 γ p = τ →∞ γ τ,p . lim Goals Part 1: Consider bounded interaction potentials w . Part 2: Consider more singular (optimal) w . V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 5 / 25
The Wiener measure and classical free field � Let H 0 ( φ ) . d x ( |∇ φ ( x ) | 2 + κ | φ ( x ) | 2 ) . . = Define the Wiener measure d µ 0 � 1 µ 0 (d φ ) . Z 0 . e − H 0 ( φ ) d φ , e − H 0 ( φ ) d φ . . = . = Z 0 Write a k . φ ( k ) and d 2 a k . . = � . = d Im a k d Re a k . � e − c ( | k | 2 + κ ) | a k | 2 d 2 a k � µ 0 (d φ ) = . e − c ( | k | 2 + κ ) | a k | 2 d 2 a k k ∈ Z d For φ ∈ supp d µ 0 , ( | k | 2 + κ ) 1 / 2 � φ ( k ) has a Gaussian distribution . � g k ( ω ) φ ≡ φ ω = ( | k | 2 + κ ) 1 / 2 e 2 πik · x , ( g k ) = i.i.d. complex Gaussians . k ∈ Z d → Classical free field . Series converges almost surely in H 1 − d 2 − ε (Λ) . V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 6 / 25
The classical system and Gibbs measures The classical interaction is � . = 1 W . d x d y | φ ω ( x ) | 2 w ( x − y ) | φ ω ( y ) | 2 . 2 In [0 , + ∞ ) almost surely if d = 1 and w ∈ L ∞ ( T 1 ) is pointwise nonnegative . In this case d µ ≪ d µ 0 . For d = 2 , 3 , W is infinite almost surely even if w ∈ L ∞ ( T d ) . Perform a renormalisation in the form of Wick ordering . � � � � � . = 1 W w . | φ ω ( x ) | 2 − ∞ | φ ω ( y ) | 2 − ∞ d x d y w ( x − y ) . 2 → Rigorously defined by frequency truncation. W w � 0 if ˆ w � 0 . V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 7 / 25
The classical system and Gibbs measures Classical Gibbs state ρ ( · ) : Given X ≡ X ( ω ) a random variable, let � X e − W d µ 0 ρ ( X ) . . = E µ ( X ) = � . e − W d µ 0 On H ( p ) ≡ L 2 sym (Λ p ) define the classical p -particle correlation function γ p by its operator kernel � � γ p ( x 1 , . . . , x p ; y 1 , . . . , y p ) . . = ρ φ ω ( y 1 ) · · · φ ω ( y p ) φ ω ( x 1 ) · · · φ ω ( x p ) . The γ p encode ρ , and hence d µ . V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 8 / 25
The quantum problem Equilibrium of H ( n ) is governed by the Gibbs state Z ( n ) e − H ( n ) , 1 . = Tr e − H ( n ) . Z ( n ) . Work on the Bosonic Fock space � F . H ( n ) . . = n ∈ N Consider a large parameter τ > 1 . (Here 1 /τ plays role of Planck’s constant). For d = 1 , consider the quantum Hamiltonian on F � � � � n � � � 1 + 1 H τ . . = − ∆ x i + κ w ( x i − x j ) ≡ H τ, 0 + W τ . τ 2 τ n ∈ N i =1 1 � i<j � n W τ should be properly renormalised when d = 2 , 3 . The grand canonical ensemble is: P τ . . = e − H τ . V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 9 / 25
The quantum Gibbs state Quantum Gibbs state ρ τ ( · ) : Given A ∈ L ( F ) we define its expectation . = Tr F ( A P τ ) ρ τ ( A ) . Tr F ( P τ ) . Work with quantum fields (operator-valued distributions) φ τ , φ ∗ τ on F that satisfy τ ( y )] = 1 [ φ τ ( x ) , φ ∗ [ φ τ ( x ) , φ τ ( y )] = [ φ ∗ τ ( x ) , φ ∗ τ δ ( x − y ) , τ ( y )] = 0 . → φ ω , φ ∗ Heuristic: φ τ ← → φ ω . τ ← On H ( p ) ≡ L 2 sym (Λ p ) define the quantum p -particle correlation function γ τ,p by its kernel � � γ τ,p ( x 1 , . . . , x p ; y 1 , . . . , y p ) . . = ρ τ φ ∗ τ ( y 1 ) · · · φ ∗ τ ( y p ) φ τ ( x 1 ) · · · φ τ ( x p ) . The γ τ,p encode ρ τ , and hence P τ . V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 10 / 25
Derivation of Gibbs measures: w ∈ L ∞ . Theorem 1: Fröhlich, Knowles, Schlein, S. (CMP , 2017). (i) Let d = 1 and w ∈ L ∞ ( T 1 ) be pointwise nonnegative or w = δ . Then for all p ∈ N we have Tr as γ τ,p − → γ p τ → ∞ . The convergence is in the trace class. (Here, �A� Tr . . = Tr |A| .) (ii) Let d = 2 , 3 and w ∈ L ∞ ( T d ) be of positive type ( ˆ w � 0 ). The convergence holds in the Hilbert-Schmidt class after a renormalisation procedure and with a slight modification of the grand canonical ensemble P τ = e − H τ (needed for technical reasons). V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 11 / 25
Derivation of Gibbs measures: unbounded interaction. Theorem 2: S. (Preprint 2019). (i) Let d = 1 and w ∈ L q ( T 1 ) , 1 � q � ∞ be pointwise nonnegative. We have Tr as γ τ,p − → γ p τ → ∞ . (ii) Let d = 2 , 3 and w ∈ L q ( T d ) be of positive type, where � (1 , ∞ ] , d = 2 q ∈ (3 , ∞ ] , d = 3 . With renormalisation and modification of P τ as in Theorem 1, we have HS as γ τ,p − → γ p τ → ∞ . → Optimal range of w for NLS: Bourgain (JMPA, 1997). V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 12 / 25
Related results 1 D results: previously shown using variational techniques by Lewin-Nam-Rougerie (J. Éc. Polytech. Math., 2015). Higher dimensions: non local, non translation-invariant interactions. Lewin-Nam-Rougerie (JMP , 2018) : 1 D non-periodic problem with subharmonic trapping. Fröhlich, Knowles, Schlein, S. (Preprint 2017): time-dependent problem in 1 D . → Corresponds to the invariance of the measure . Lewin-Nam-Rougerie (Preprint 2018) : 2 D problem without modified grand canonical ensemble. V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 13 / 25
Recommend
More recommend
