a microscopic derivation of gibbs measures for nonlinear … · 2019. 8. 22. · vedran sohinger...
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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, ...
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
∑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 .
V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 4 / 25
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.
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
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).
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 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.
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) ≡ 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.
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 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 .
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 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 .
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 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).
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 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.
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
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.
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) =∑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 .
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
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!.
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−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)
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)
∣∣∣∣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.
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 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.
V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 20 / 25
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 .
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 ‖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.
V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 22 / 25
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).
V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 23 / 25
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).
V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 24 / 25
Thank you for your attention!
V. Sohinger (University of Warwick) Derivation of Gibbs measures for NLS Quantissima, August 2019 25 / 25
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