Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and...

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Probabilistic methods for discrete nonlinear Schr¨ odinger equations Sourav Chatterjee (NYU) Kay Kirkpatrick (Universit´ e Paris IX) Sourav Chatterjee Probability with discrete NLS

Transcript of Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and...

Page 1: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Probabilistic methods for discrete nonlinearSchrodinger equations

Sourav Chatterjee(NYU)

Kay Kirkpatrick(Universite Paris IX)

Sourav Chatterjee Probability with discrete NLS

Page 2: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

The cubic nonlinear Schrodinger equation

I Linear Schrodinger equation:

i∂tψ = −∆ψ,

where ψ : Rd × R→ C.

I Cubic nonlinear Schrodinger equation:

i∂tψ = −∆ψ + κ|ψ|2ψ.

I κ can be +1 (defocusing equation) or −1 (focusing equation).

I In this talk, we will concentrate on the focusing equation, thatis, κ = −1.

I Crucial to understanding a variety of physical phenomena:I Bose-Einstein condensationI Langmuir waves in plasmasI Nonlinear opticsI Envelopes of water wavesI etc...

Sourav Chatterjee Probability with discrete NLS

Page 3: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

The Gibbs measure

I The cubic NLS is a Hamiltonian system, with Hamiltonian

H(ψ) = 2

∫|∇ψ|2dx + κ

∫|ψ|4dx .

I One method of studying the behavior of Hamiltonian systemsis through invariant measures for the dynamics.

I Gibbs measures, i.e. measures that have density of the form

e−βH(ψ)

are (formally) invariant by Liouville’s theorem.

I Here β is a parameter, sometimes called the ‘inversetemperature’.

Sourav Chatterjee Probability with discrete NLS

Page 4: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

The mass cut-off

I The Gibbs measure for the focusing NLS has infinite mass(even in 1D) and hence cannot be normalized to give aprobability measure.

I This necessitates a mass cut-off: consider densities of the form

e−βH(ψ)1{N(ψ)≤B}

where B is another parameter and N(ψ) =∫|ψ|2dx .

I Even after mass cut-off, it is not obvious that everythingmakes sense.

I Following the pioneering work of Lebowitz-Rose-Speer, theapproach was given a rigorous foundation through the worksof McKean-Vaninsky, Bourgain and Rider.

I Most of the work till now in 1D. Bourgain and Tzvetkov didthe 2D defocusing case. Brydges-Slade found some problemsin defining the Gibbs measure in the 2D focusing case.

I Analysis of Gibbs measures can lead to conclusions aboutwell-posedness (Bourgain).

Sourav Chatterjee Probability with discrete NLS

Page 5: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

3D and higher

I It is not clear (and probably not true) that Gibbs measures forthe focusing NLS can be defined analogously in dimensions 3and higher.

I Our approach: work with a discretized system withoutworrying about taking continuum limits. Gibbs measures canalways be defined for discrete systems.

I Try to understand the fine properties of the system and makeinferences about the dynamics.

I One long-term motivation: the soliton resolution conjecture.

Sourav Chatterjee Probability with discrete NLS

Page 6: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

The soliton resolution conjecture

I The linear Schrodinger is a dispersive system: while∫Rd |ψ|2dx is conserved over time, for any compact set K ,

limt→∞

∫K|ψ|2dx = 0.

I In its simplest form, the soliton conjecture for the nonlinearNLS says that there is a compact set K such that the aboveequation does not hold for K , but [see below].

I One can find initial data that disperses even for the nonlinearNLS. But this is the worst-case scenario.

I The conjecture says that complete dispersion does not happenin the typical case, although it is not clear what ‘typical’means. [Readable reference: Terry Tao’s blog on this.]

I The conjecture has been proved in 1D (cubic), where thesystem is completely integrable. All other cases are unknown.

Sourav Chatterjee Probability with discrete NLS

Page 7: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Formulation via the invariant measure approach

I One way to quantify ‘typical’: a set of probability 1 under aninvariant probability measure whose support is the space of allinitial data with a given mass and energy (say).

I Characteristics of a random pick from the invariant measuredo not change as it evolves over time according to the NLSflow.

I Implication: If the random initial data has positive mass in aneighborhood of the origin with probability 1, then it willcontinue to have this characteristic for all time, satisfying thesoliton resolution conjecture.

I Difficulty: We do not know such invariant measures for thecontinuum system (focusing cubic) in 3D and higher.

I Note: The existence of soliton solutions, e.g. ground statesolitons, is known. The difficulty is in proving that suchbehavior is typical.

Sourav Chatterjee Probability with discrete NLS

Page 8: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

The discrete cubic NLS

I Since we cannot handle the continuum system, one alternativeis to discretize space.

I Let G = (V ,E ) be the lattice approximation of thed-dimensional unit torus [0, 1]d , whereV = {0, 1/L, 2/L, . . . , (L− 1)/L}d and E consists ofnearest-neighbor edges.

I Let h = 1/L denote the lattice spacing. Let n = Ld be thesize of the graph.

I The discrete nearest-neighbor Laplacian on G is defined as

∆f (x) :=1

h2

∑y : (x ,y)∈E

(f (y)− f (x)).

I The discrete cubic NLS on G is a family of coupled ODEswith fx(t) := f (x , t), satisfying for all x ∈ V

id

dtfx = −∆fx + κ|fx |2fx .

Sourav Chatterjee Probability with discrete NLS

Page 9: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

The discrete Gibbs measure

I The discrete Hamiltonian associated with the focusing NLS is:

H(f ) :=2

n

∑(x ,y)∈E

∣∣∣∣ fx − fyh

∣∣∣∣2 − 1

n

∑x∈V

|fx |4.

I Note analogy with continuum Hamiltonian:

H(f ) = 2

∫|∇f |2dx −

∫|f |4dx .

I Let N(f ) = 1n

∑x∈V |fx |2 denote the mass.

I Given β,B > 0, the probability measure µ on Cn defined as

d µ :=1

Ze−β

eH(f )1{eN(f )≤B}

∏x∈V

dfx

is an invariant probability measure for the discrete focusingcubic NLS. Here Z = Z (β,B) is the normalizing constant.

Sourav Chatterjee Probability with discrete NLS

Page 10: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

The limiting free energy

I Let m : [2,∞)→ R be the function

m(θ) :=θ

2− 1

2+θ

2

√1− 2

θ+ log

(1

2− 1

2

√1− 2

θ

).

I Let θc be the unique real zero of m. Numerically,θc ≈ 2.455407.

Theorem (Chatterjee & Kirkpatrick, 2010)

Suppose d ≥ 3. As the lattice size n→∞ (and hence the latticespacing h→ 0),

log Z (β,B)

n→

{log(Bπe) if βB2 ≤ θc ,log(Bπe) + m(βB2) if βB2 > θc .

Sourav Chatterjee Probability with discrete NLS

Page 11: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Features of the Gibbs measure

I Let ψ be a random initial data from the Gibbs measure.I Let M1(ψ) and M2(ψ) denote the largest and second largest

components of the vector (|ψx |2)x∈V .I When βB2 > θc , there is high probability that M1(ψ) ≈ an

and M2(ψ) = o(n), where

a = a(β,B) :=B

2+

B

2

√1− 2

βB2.

I Moreover,∑

x |ψx |2 ≈ Bn with high probability.I Largest component carries more than half of the total mass:

maxx

|ψx |2∑y |ψy |2

≈ a

B>

1

2.

I On the other hand, when βB2 < θc , then M1(ψ) = o(n) andmaxx |ψx |2/

∑y |ψy |2 ≈ 0.

Sourav Chatterjee Probability with discrete NLS

Page 12: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Fraction of mass at the heaviest site

β

1

a

Figure: The fraction of mass at the heaviest site jumps from roughly zerofor small inverse temperature, to roughly .71 at the critical threshold.(Here B = 1.)

Sourav Chatterjee Probability with discrete NLS

Page 13: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Typicality of soliton-like solutions

Theorem (Chatterjee & Kirkpatrick, 2010)

Suppose βB2 > θc and d ≥ 3. Suppose ψ(x , t) is a solution of thediscrete cubic NLS on lattice of size n with random initial dataψ(·, 0) from the Gibbs measure µ. Then for any q ∈ (0, 1) that issufficiently close to 1, with probability at least 1− exp(−nq) thereis a point x0 such that for all t ∈ [0, exp(nq)],∣∣∣∣ |ψ(x0, t)|2

n− a(β,B)

∣∣∣∣ ≤ n−(1−q)/4

andmaxx 6=x0 |ψ(x , t)|2

n≤ n−(1−q).

Sourav Chatterjee Probability with discrete NLS

Page 14: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Stability of the soliton: proof sketch

I If βB2 > θc , we show that with high probability, there is asingle point with abnormally large value of the wave functionat t = 0.

I As time progresses, the invariance of the Gibbs measureimplies that the above picture must hold for all but a verysmall proportion of times.

I We use estimates related to the NLS flow to show that if thesoliton has to jump from one site to another, that cannothappen in a time period as small as allowed by the precedingestimate. Hence the soliton cannot jump.

I The above argument is a simple combination of probabilisticand PDE methods.

Sourav Chatterjee Probability with discrete NLS

Page 15: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Analysis of the Gibbs measure

I Simplifying step: in d ≥ 3, the interaction term in theHamiltonian has almost no effect if n is large. In other words,the problem attains a mean-field character. Not obvious apriori. Not true in d = 1 or 2.

I Upon ignoring the interaction term, the problem boils down toanalyzing the uniform probability distribution on subsets of Cn

like

Γa,b =

{f ∈ Cn :

∑x∈V

|fx |2 ≈ bn,∑x∈V

|fx |4 ≈ a2n2

}.

Here ≈ means equal up to error of order smaller than themain term.

I To show: A uniformly chosen point from Γa,b has exactly oneabnormally large component with high probability.

I Also, need to compute the volume of Γa,b to connect uniformdistributions with Gibbs measures.

Sourav Chatterjee Probability with discrete NLS

Page 16: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

A simple problem

I Suppose Z1,Z2 are i.i.d. complex Gaussian random variables,and we are given that |Z1|4 + |Z2|4 ≈ n, where n is a largenumber.

I What can you say about the pair (Z1,Z2), conditional on thisevent?

I Conditionally, (Z1,Z2) is distributed on the L4 annulus{(z1, z2) : |z1|4 + |z2|4 ≈ n} with probability density functionproportional to exp(−|z1|2 − |z2|2).

I Thus, the conditional distribution must concentrate on thepart of the annulus where |z1|2 + |z2|2 is minimized. Thishappens where the L4 and L2 balls touch.

I This implies that with high probability, either |Z1|4 ≈ n or|Z2|4 ≈ n.

Sourav Chatterjee Probability with discrete NLS

Page 17: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

A slightly more complex problem

I The above logic continues to hold if we replace (Z1,Z2) by(Z1, . . . ,Zk), as long as k is sufficiently small compared to n.

I Precisely, if k � n/ log n, then the same argument shows thatconditional on |Z1|4 + · · ·+ |Zk |4 ≈ n, we can say that withhigh probability, there is exactly one i such that |Zi |4 ≈ n.

I However, we are interested in the case where k is of the sameorder as n. The previous argument breaks down becausevolumes compete with densities.

I But the result is still true, with a small modification.

Sourav Chatterjee Probability with discrete NLS

Page 18: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

An even more complex problem

I Suppose Z1, . . . ,Zn are i.i.d. standard complex Gaussianrandom variables.

I Let A be the event |Z1|4 + · · · |Zn|4 ≈ bn where b is a constantgreater than E|Z1|4 = 8 and ≈ has some appropriate meaning.

I What does the vector (Z1, . . . ,Zn) look like, conditional on A?

Sourav Chatterjee Probability with discrete NLS

Page 19: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Analysis of the complex problem

I Note that

P(A) ≥ P(|Z1|4 ≈ (b − 8)n)P(|Z2|4 + · · ·+ |Zn|4 ≈ 8n)

≈ e−c√

n,

where c denotes a constant depending on b.I On the other hand, if B denotes the event that the histogram

of the data {|Z1|2, . . . , |Zn|2} deviates significantly from itsexpected shape, then P(B) ≤ e−dn, where d is a constantdepending on the deviation.

I Thus, P(B|A) ≤ P(B)/P(A) = very small.I Together, this implies that given A, there must exist a

relatively small set S ⊆ {1, . . . , n} such that∑i∈S |Zi |4 ≈ (b − 8)n.

I The argument for the simpler problem can be now applied toshow that there is exactly one i such that |Zi |2 ≈ (b − 8)n.

Sourav Chatterjee Probability with discrete NLS

Page 20: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

From Gaussian measures to Gibbs measures

I When the standard Gaussian distribution is restricted to asubset of a thin L2 annulus (e.g. intersection of L2 and L4

annuli), it becomes the uniform distribution on this set.

I The Gibbs measures are approximately convex combinationsof uniform distributions on an infinite collection ofintersections of L2 and L4 annuli.

I Thus, our analysis proceeds as: Gaussian→ Uniform→ Gibbs.

Sourav Chatterjee Probability with discrete NLS

Page 21: Probabilistic methods for discrete nonlinear Schr odinger …souravc/beam-nls-trans.pdf · 3D and higher I It is not clear (and probably not true) that Gibbs measures for the focusing

Continuum limit

I Unfortunately, the Gibbs measures do not tend to acontinuum limit at the grid size goes to zero.

I The reason is the energy levels attained by the Gibbs measureare too low to admit good behavior in the limit.

I In fact, we have a theorem proving convergence to white noiseplus one exceptional point.

I Future goal: to develop a continuum version. Requires newideas in addition to the existing ones.

Sourav Chatterjee Probability with discrete NLS