A microscopic derivation of Gibbs measures for nonlinear Schrdinger - - PowerPoint PPT Presentation

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 = Td for d = 1, 2, 3. Study the nonlinear Schrödinger equation (NLS).

• i∂tφt(x) =
• − ∆ + κ
• φt(x) +
• dy w(x − y) |φt(y)|2 φt(x)

φ0(x) = Φ(x) ∈ Hs(Λ) . Chemical potential κ > 0 ; Interaction potential w ∈ Lq(Λ) is positive or w = δ. Conserved energy (Hamiltonian) H(φ) =

• dx
• |∇φ(x)|2 + κ|φ(x)|2

+ 1 2

• dx dy |φ(x)|2 w(x − y) |φ(y)|2 .

The Gibbs measure dµ associated with Hamiltonian flow is the probability measure on the space of fields φ : Λ → C µ(dφ) . .= 1 Z e−H(φ) dφ , Z . .=

• e−H(φ) dφ .

→ 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) ≡ L2

sym(Λn) we consider the n-body Hamiltonian

H(n) . .=

n

• i=1
• −∆xi + κ
• + λ
• 1i<jn

w(xi − xj) . Solve n-body Schrödinger equation i∂tΨn,t = H(n)Ψn,t . Obtain that, for λ = 1/n as n → ∞ Ψn,0 ∼ φ⊗n implies Ψn,t ∼ φ⊗n

t

. (Hepp (1974), Ginibre-Velo (1979), Spohn (1980), Erd˝

• s-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 dµ 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

1

Give rigorous definition of classical Gibbs measure dµ.

2

‘Encode’ dµ in terms of a sequence of operators (γp)p.

3

Define many-body quantum Gibbs states and ‘encode’ them in terms of a sequence of operators (γτ,p)p.

4

Show that γp = lim

τ→∞ γτ,p .

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 H0(φ) . .=

• dx (|∇φ(x)|2 + κ|φ(x)|2).

Define the Wiener measure dµ0 µ0(dφ) . .= 1 Z0 e−H0(φ) dφ , Z0 . .=

• e−H0(φ) dφ .

Write ak . .= φ(k) and d2ak . .= d Imak d Reak. µ0(dφ) =

• k∈Zd

e−c(|k|2+κ)|ak|2d2ak

• e−c(|k|2+κ)|ak|2d2ak

. For φ ∈ supp dµ0, (|k|2 + κ)1/2 φ(k) has a Gaussian distribution. φ ≡ φω =

• k∈Zd

gk(ω) (|k|2 + κ)1/2 e2πik·x , (gk) = i.i.d. complex Gaussians. → Classical free field . Series converges almost surely in H1− 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 W . .= 1 2

• dx dy |φω(x)|2 w(x − y) |φω(y)|2 .

In [0, +∞) almost surely if d = 1 and w ∈ L∞(T1) is pointwise nonnegative. In this case dµ ≪ dµ0. For d = 2, 3, W is infinite almost surely even if w ∈ L∞(Td). Perform a renormalisation in the form of Wick ordering . W w . .= 1 2

• dx dy
• |φω(x)|2 − ∞
• w(x − y)
• |φω(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 ρ(X) . .= Eµ(X) =

• X e−W dµ0
• e−W dµ0

. On H(p) ≡ L2

sym(Λp) define the classical p-particle correlation function

γp by its operator kernel γp(x1, . . . , xp; y1, . . . , yp) . .= ρ

• φω(y1) · · · φω(yp)φω(x1) · · · φω(xp)
• .

The γp encode ρ, and hence dµ.

• 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 1 Z(n) e−H(n) , Z(n) . .= Tr e−H(n) . Work on the Bosonic Fock space F . .=

• n∈N

H(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 Hτ . .=

• n∈N
• 1

τ

n

• i=1
• −∆xi + κ
• + 1

τ 2

• 1i<jn

w(xi − xj)

• ≡ Hτ,0 + Wτ .

Wτ should be properly renormalised when d = 2, 3. The grand canonical ensemble is: Pτ . .= e−Hτ .

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 9 / 25

The quantum Gibbs state

Quantum Gibbs state ρτ(·): Given A ∈ L(F) we define its expectation ρτ(A) . .= TrF(APτ) TrF(Pτ) . Work with quantum fields (operator-valued distributions) φτ, φ∗

τ on F that

satisfy [φτ(x), φ∗

τ(y)] = 1

τ δ(x − y) , [φτ(x), φτ(y)] = [φ∗

τ(x), φ∗ τ(y)] = 0 .

Heuristic: φτ ← → φω , φ∗

τ ←

→ φω. On H(p) ≡ L2

sym(Λp) define the quantum p-particle correlation function

γτ,p by its kernel γτ,p(x1, . . . , xp; y1, . . . , yp) . .= ρτ

• φ∗

τ(y1) · · · φ∗ τ(yp)φτ(x1) · · · φτ(xp)

• .

The γτ,p encode ρτ, and hence Pτ.

• 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∞(T1) be pointwise nonnegative or w = δ. Then for all p ∈ N we have γτ,p

Tr

− → γp as τ → ∞ . The convergence is in the trace class. (Here, ATr . .= Tr |A|.) (ii) Let d = 2, 3 and w ∈ L∞(Td) 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 Pτ = e−Hτ (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 ∈ Lq(T1), 1 q ∞ be pointwise nonnegative. We have γτ,p

Tr

− → γp as τ → ∞ . (ii) Let d = 2, 3 and w ∈ Lq(Td) be of positive type, where q ∈

• (1, ∞] , d = 2

(3, ∞] , d = 3 . With renormalisation and modification of Pτ as in Theorem 1, we have γτ,p

HS

− → γp as 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

1D 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) : 1D non-periodic problem with subharmonic trapping. Fröhlich, Knowles, Schlein, S. (Preprint 2017): time-dependent problem in 1D. → Corresponds to the invariance of the measure. Lewin-Nam-Rougerie (Preprint 2018) : 2D problem without modified grand canonical ensemble.

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 13 / 25

Idea of the proof: perturbative expansion in interaction

Example: Consider the Classical relative partition function A(z) . .=

• e−zW dµ0

Quantum relative partition function Aτ(z) = Tr

• e−Hτ,0−zWτ

Tr(e−Hτ,0) . A(z) and Aτ(z) are analytic in Re z > 0. We want to show that lim

τ→∞ Aτ(z) = A(z)

for Re z > 0 . Problem: The series expansions A(z) =

∞

• m=0

amzm , Aτ(z) =

∞

• m=0

aτ,mzm have radius of convergence zero. Solution: Use Borel summation.

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 14 / 25

Idea of proof: Borel summation

The Borel transform of a formal power series: A(z) =

• m0

αmzm − → B(z) . .=

• m0

αm m! zm . Formally, we have A(z) = ∞ dt e−t B(tz) . For M ∈ N, write A(z) =

M−1

• m=0

αmzm + RM(z) . By Sokal (1980), it suffices to prove |αm| Cmm! , |RM(z)| CMM!|z|M .

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 15 / 25

Estimating aτ,m

Use Duhamel’s formula and write aτ,m = 1 Tr

• e−Hτ,0 Tr
• (−1)m

1 dt1 t1 dt2 · · · tm−1 dtm e−(1−t1)Hτ,0 Wτ e−(t1−t2)Hτ,0 Wτ · · · e−(tm−1−tm)Hτ,0 Wτ e−tmHτ,0

• .

→ A normalised trace of an iterated time integral. Rewrite aτ,m using the quantum Wick theorem 1 Tr(e−Hτ,0) Tr

• φ∗

τ(x1) · · · φ∗ τ(xk)φτ(y1) · · · φτ(yk) e−Hτ,0

=

• π∈Sk

k

• j=1

1 Tr(e−Hτ,0) Tr

• φ∗

τ(xj)φτ(yπ(j)) e−Hτ,0

. Factors are the quantum Green function Gτ(x; y) . .= 1 Tr(e−Hτ,0) Tr

• φ∗

τ(x)φτ(y) e−Hτ,0

= 1 τ

• e(−∆+κ)/τ − 1

.

• V. Sohinger (University of Warwick)

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The graph structure

The pairing of φ∗

τ, φτ gives rise to a graph structure .

2m copies of φ∗

τ, 2m copies of φτ.

→ Total number of graphs is at most (2m)! = O(Cmm!2). One gains

1 m! from the time integral

1 dt1 t1 dt2 · · · tm−1 dtm = 1 m! . Main work: For fixed t1, . . . , tm, each graph contributes O(Cm). Conclude that |aτ,m| Cmm!.

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 17 / 25

The graph structure: Algorithm for w ∈ L∞

In general, we work with Gτ,t . .=

e− t

τ (−∆+κ)

τ(e(−∆+κ)/τ −1) for t > −1.

One has Gτ,t(x; y) 0. Example: Consider, for t ∈ (0, 1)

• Td dx1
• Td dx2 w(x1 − y1) Gτ,t(x1; x2) w(x2 − y2) Gτ,−t(x1; x2) .

w(x1 − y1) x1 x1 y1 y1 Time t1 Time t2 = t1 − t. w(x2 − y2) x2 x2 y2 y2 Gτ,t(x2; x1) Gτ,−t(x1; x2)

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 18 / 25

The graph structure

We have

• Td dx1
• Td dx2 w(x1 − y1) Gτ,t(x1; x2) w(x2 − y2) Gτ,−t(x1; x2)
• w2

L∞(Td)

• Td dx1
• Td dx2 Gτ,t(x1; x2) Gτ,−t(x2; x1) ,

which is = w2

L∞(Td)

• Td dx1
• Td dx2 Gτ,0(x1; x2) Gτ,0(x2; x1)

= w2

L∞(Td) Gτ,02 HS ∼

• k∈Zd

1 (|k|2 + κ)2 . → Bounded uniformly in t 0, τ 1.

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 19 / 25

Theorem 2: Unbounded interactions

Earlier approach relies crucially on w ∈ L∞. Analyse weights corresponding to each edge for fixed time. (∼ time-evolved Green function+time-evolved delta function). For t ∈ (−1, 1) we consider the quantity Qτ,t . .=

• Gτ,t + 1

τ e

t τ (∆−κ)

for t ∈ (0, 1) Gτ,t for t ∈ (−1, 0] . Difficulties:

Gτ,t is singular for t ∼ −1. e

t τ (∆−κ) is singular for t ∼ 0.

Decompose Qτ,t in another way.

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Derivation of Gibbs measures for NLS Quantissima, August 2019 20 / 25

Proof of Theorem 2: Splitting of Qτ,t

We write Qτ,t = Q(1)

τ,t + 1

τ Q(2)

τ,t ,

where for t ∈ (−1, 1), we define Q(1)

τ,t .

.= e−{t}h/τ τ(eh/τ − 1) = Gτ,{t} Q(2)

τ,t .

.=

• e−{t}h/τ

for t ∈ (−1, 0) ∪ (0, 1) for t = 0 . Here {t} = t − ⌊t⌋. Q(1)

τ,t is better behaved than Gτ,t for negative t.

Q(2)

τ,t retains some good boundedness properties, e.g.

• dy Q(2)

τ,t(x; y) =

• dy Q(2)

τ,t(y; x) 1 .

• V. Sohinger (University of Warwick)

Derivation of Gibbs measures for NLS Quantissima, August 2019 21 / 25

The interaction potential

Set d = 2 . Consider w ∈ Lq(T2), q > 1 of positive type. On the quantum level, work with τ-dependent interaction potential wτ.

(i) wτ ∈ L∞(T2) and wτL∞(T2) τ β, for β < 1. (ii) ˆ wτ 0. (iii) wτ → w in Lq(T2) as τ → ∞.

Constructed by appropriate truncation in Fourier space. Decompose graphs into connected components. Since wτL∞(T2) is not bounded uniformly in τ, need to carefully distribute the interactions over the connected components.

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Optimality of q

Recall the expansion of the classical relative partition function A(z) =

∞

• m=0

amzm By the classical Wick theorem, we have a1

• dx
• dy w(x − y) G2(x; y) =
• dx w(x) G2(x; 0) ,

where G = (−∆ + κ)−1. We have G ∈ Lr(Td × Td) where r ∈      [1, ∞] , d = 1 [1, ∞) , d = 2 [1, 3) , d = 3 . By duality, we thus obtain the optimal range of q; Bourgain (JMPA, 1997).

• V. Sohinger (University of Warwick)

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L1 interactions in two dimensions.

Theorem 3: S. (Preprint 2019).

Let d = 2 and w ∈ L1(Td) satisfy the following assumptions. w is of positive type. w is pointwise nonnegative. There exist ε > 0 and L > 0 such that

• w(k)

L (1 + |k|)ε for all k ∈ Z2. With setup as in Theorem 1, we have γτ,p

HS

− → γp as τ → ∞ . → Classical variant of this endpoint case: Bourgain (JMPA, 1997).

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Thank you for your attention!

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