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Probabilistic methods for discrete nonlinearSchrodinger equations
Sourav Chatterjee(NYU)
Kay Kirkpatrick(Universite Paris IX)
Sourav Chatterjee Probability with discrete NLS
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
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
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
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
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
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
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
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
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
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
Fraction of mass at the heaviest site
β
cθ
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
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
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
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
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
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
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
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
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
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
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