a microscopic derivation of gibbs measures for nonlinear … · 2019. 8. 22. · vedran sohinger...

A microscopic derivation of Gibbs measures fornonlinear Schrödinger equations with unbounded

interaction potentials

Vedran Sohinger (University of Warwick)

partly joint work withJürg Fröhlich (ETH Zürich)

Antti Knowles (University of Geneva)Benjamin Schlein (University of Zürich)

Quantissima in the Serenissima III, VeniceAugust 22, 2019.

V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 1 / 25

The nonlinear Schrödinger equationConsider 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 orw = 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 theprobability measure on the space of fields φ : Λ→ C

µ(dφ) ..=1

Ze−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-almostsurely.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, ...


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

∑16i<j6n

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⊗n0 implies Ψn,t ∼ φ⊗nt .

(Hepp (1974), Ginibre-Velo (1979), Spohn (1980), Erdos-Schlein-Yau(2006, 2007), Lieb-Seiringer (2006), Klainerman-Machedon (2008),T.Chen-Pavlovic (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 .


Outline of strategy and goals

The main strategy1 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 = limτ→∞

γτ,p .

GoalsPart 1: Consider bounded interaction potentials w.Part 2: Consider more singular (optimal) w.


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

Z0e−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)1/2φ(k) has a Gaussian distribution.

φ ≡ φω =∑k∈Zd

gk(ω)

(|k|2 + κ)1/2e2πik·x , (gk) = i.i.d. complex Gaussians.

→ Classical free field .Series converges almost surely in H1− d2−ε(Λ).


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 pointwisenonnegative.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 .

Ww ..=1

2

∫dxdy

(|φω(x)|2 −∞

)w(x− y)

(|φω(y)|2 −∞

).

→ Rigorously defined by frequency truncation.Ww > 0 if w > 0.


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) ≡ L2sym(Λp) define the classical p-particle correlation function

γp by its operator kernel

γp(x1, . . . , xp; y1, . . . , yp)..= ρ

(φω(y1) · · ·φω(yp)φ

ω(x1) · · ·φω(xp)).

The γp encode ρ, and hence dµ.


The quantum problem

Equilibrium of H(n) is governed by the Gibbs state1

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

τ2

∑16i<j6n

w(xi − xj)

]≡ Hτ,0 +Wτ .

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

Pτ..= e−Hτ .


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 thatsatisfy

[φτ (x), φ∗τ (y)] =1

τδ(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 functionγτ,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 .


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 haveγτ,p

Tr−→ γp as τ →∞ .

The convergence is in the trace class.(Here, ‖A‖Tr

..= Tr |A|.)(ii) Let d = 2, 3 and w ∈ L∞(Td) be of positive type (w > 0). The

convergence holds in the Hilbert-Schmidt class after a renormalisationprocedure and with a slight modification of the grand canonical ensemblePτ = e−Hτ (needed for technical reasons).


Derivation of Gibbs measures: unbounded interaction.

Theorem 2: S. (Preprint 2019).(i) Let d = 1 and w ∈ Lq(T1), 1 6 q 6∞ be pointwise nonnegative. We have

γτ,pTr−→ 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,pHS−→ 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 byLewin-Nam-Rougerie (J. Éc. Polytech. Math., 2015).Higher dimensions: non local, non translation-invariant interactions.Lewin-Nam-Rougerie (JMP, 2018) : 1D non-periodic problem withsubharmonic trapping.Fröhlich, Knowles, Schlein, S. (Preprint 2017): time-dependent problemin 1D. → Corresponds to the invariance of the measure.Lewin-Nam-Rougerie (Preprint 2018) : 2D problem without modifiedgrand canonical ensemble.


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

limτ→∞

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.


Idea of proof: Borel summation

The Borel transform of a formal power series:

A(z) =∑m>0

αmzm 7−→ B(z) ..=

∑m>0

αmm!

zm .

Formally, we have

A(z) =

∫ ∞0

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| 6 Cmm! , |RM (z)| 6 CMM !|z|M .


Estimating aτ,m

Use Duhamel’s formula and write

aτ,m =1

Tr(e−Hτ,0

) Tr

((−1)m

∫ 1

0

dt1

∫ t1

0

dt2 · · ·∫ tm−1

0

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) Derivation of Gibbs measures for NLS Quantissima, August 2019 16 / 25

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

0

dt1

∫ t1

0

dt2 · · ·∫ tm−1

0

dtm =1

m!.

Main work: For fixed t1, . . . , tm, each graph contributes O(Cm).Conclude that |aτ,m| 6 Cmm!.


The graph structure: Algorithm for w ∈ L∞

In general, we work with Gτ,t ..= e−tτ

(−∆+κ)

τ(e(−∆+κ)/τ−1)for t > −1.

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

Tddx1

∫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)


The graph structure

We have∣∣∣∣ ∫Td

dx1

∫Td

dx2 w(x1 − y1)Gτ,t(x1;x2)w(x2 − y2)Gτ,−t(x1;x2)

∣∣∣∣6 ‖w‖2L∞(Td)

∫Td

dx1

∫Td

dx2Gτ,t(x1;x2)Gτ,−t(x2;x1) ,

which is

= ‖w‖2L∞(Td)

∫Td

dx1

∫Td

dx2Gτ,0(x1;x2)Gτ,0(x2;x1)

= ‖w‖2L∞(Td) ‖Gτ,0‖2HS ∼

∑k∈Zd

1

(|k|2 + κ)2.

→ Bounded uniformly in t > 0, τ > 1.


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 etτ (∆−κ) for t ∈ (0, 1)

Gτ,t for t ∈ (−1, 0] .

Difficulties:Gτ,t is singular for t ∼ −1.etτ

(∆−κ) is singular for t ∼ 0.

Decompose Qτ,t in another way.


Proof of Theorem 2: Splitting of Qτ,t

We writeQτ,t = Q

(1)τ,t +

1

τQ

(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)

0 for t = 0 .

Here {t} = t− btc.Q

(1)τ,t is better behaved than Gτ,t for negative t.

Q(2)τ,t retains some good boundedness properties, e.g.

0 6∫

dy Q(2)τ,t (x; y) =

∫dy Q

(2)τ,t (y;x) 6 1 .


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 ‖wτ‖L∞(T2) 6 τβ , for β < 1.(ii) wτ > 0.(iii) wτ → w in Lq(T2) as τ →∞.

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


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) =

∫dxw(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).


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) 6L

(1 + |k|)ε

for all k ∈ Z2.With setup as in Theorem 1, we have

γτ,pHS−→ γp as τ →∞ .

→ Classical variant of this endpoint case: Bourgain (JMPA, 1997).


Thank you for your attention!


a microscopic derivation of gibbs measures for nonlinear … · 2019. 8. 22. · vedran sohinger...

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