Intro DNLS Kinetic FPU Comments Extras
Kinetic theory of (lattice) waves
Jani Lukkarinen
based on joint works with
Herbert Spohn (TU Munchen),
Matteo Marcozzi (U Geneva), Alessia Nota (U Bonn),
Christian Mendl (Stanford U), Jianfeng Lu (Duke U)
Bristol 2018
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Part I
Time-correlations in stationary statesof the discrete NLS
[JL and H. Spohn, Invent. Math. 183 (2011) 79β188]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Setup for the rigorous result 3
Finite lattice: L β₯ 2 , Ξ = {0, 1, . . . , Lβ 1}d
Periodic BC: All arithmetic mod L
Dual lattice: Ξβ = {0, 1L , . . . ,
Lβ1L }
d
βIntegrationβ = finite sum:β«Ξβ
dk f (k) :=1
|Ξ|βkβΞβ
f (k)
βDirac deltaβ = finite sum:
Ξ΄Ξ(k) := |Ξ|1{k mod 1 = 0}
Fourier transform: (x β Ξ, k β Ξβ)
f (k) =βyβΞ
f (y)eβi2ΟkΒ·y β f (x) =
β«Ξβ
dk β² f (k β²)ei2Οk β²Β·x
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Evolution equations 4
Discrete nonlinear Schrodinger equation
id
dtΟt(x) =
βyβΞ
Ξ±(x β y)Οt(y) + Ξ»|Οt(x)|2Οt(x)
Οt : Ξβ C, t β RΞ» > 0 (defocusing)
Harmonic coupling determined by Ξ± : Zd β R.
Ξ± has finite range (for instance, nearest neighbour)
We assume also Ξ±(βx) = Ξ±(x)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Conservation laws 5
Hamiltonian function
HΞ(Ο) =βx ,yβΞ
Ξ±(x β y)Ο(x)βΟ(y) +1
2Ξ»βxβΞ
|Ο(x)|4
Relate qx , px β R to Ο by Ο(x) =1β2
(qx + ipx)
NLS equivalent to the Hamiltonian equations
qx = βpxHΞ , px = ββqxHΞ
Thus HΞ(Οt) is conserved
By explicit differentiation, alsoβ
x |Οt(x)|2 is conserved
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Initial state 6
Probability distribution of Ο = Ο0 (Grand canonical ensemble)
1
Zλβ,Β΅eβΞ²(HΞ(Ο)βΒ΅βΟβ2)
βxβΞ
[d(ReΟ(x)) d(ImΟ(x))]
Define Ο : Td β R by Ο = FxβkΞ±.
We consider only Ξ² > 0 and Β΅ < mink Ο(k)β Also the Gaussian measure at Ξ» = 0 is well-defined
Zλβ,¡ > 0 is the normalization constant
Let E denote expectation over the initial data
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Properties of the system 7
The solution Οt exists and is unique for all t β R with anyinitial data Ο0 β CΞ. (conservation laws)
Initial state is stationary : E[F (Οt)] = E[F (Ο0)]
Also invariant under periodic translations:
E[F (ΟxΟ)] = E[F (Ο)], (ΟxΟ)(y) = Ο(y + x)
Translations commute with the time-evolution:
ΟxΟt = Οt |Ο0=ΟxΟ0
βGauge invarianceβ: similar invariance properties hold fortranslations of total phase, Ο0(x) 7β eiΟΟ0(x), Ο β R.
Thus, for instance, E[Οt ] = 0, E[Οtβ²Οt ] = 0,
E[Οtβ²(xβ²)βΟt(x)] = E[Ο0(0)βΟtβtβ²(x β x β²)]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Field-field correlation function 8
Fix test-functions f , g β `2, and assume they have finite support.
Observable
QΞ»Ξ(Ο) := E[γf , Ο0γβγeβiΟΞ»ΟΞ»β2
g , ΟΟΞ»β2γ]
Under additional assumptions on the decay of equilibriumcorrelations and on the dispersion relation:
Theorem
There is Ο0 > 0 such that for all |Ο | < Ο0
limΞ»β0
limΞββ
QΞ»Ξ(Ο) =
β«Td
dk g(k)βf (k)W (k)eβΞ1(k)|Ο |βiΟΞ2(k)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Summary of the main result 9
Loosely : for all not too large t = O(Ξ»β2),
E[Ο0(k β²)βΟt(k)] β Ξ΄Ξ(k β² β k)W (k)eβiΟΞ»ren(k)teβ|Ξ»2t|Ξ1(k)
W (k) = (Ξ²(Ο(k)β Β΅))β1 = covariance function for Ξ» = 0
ΟΞ»ren(k) = Ο(k) + Ξ»R0 + Ξ»2Ξ2(k)
Ξ1(k) β₯ 0
β k-space correlation decays exponentially in t,as dictated by eβ|Ξ»
2t|Ξ1(k).
Nearest neighbour couplings (Οnn(k) = c ββd
Ξ½=1 cos(2ΟkΞ½))satisfy all of our assumptions if d β₯ 4
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Ξj(k) are real, and Ξ(k) = Ξ1(k) + iΞ2(k) is given by
Ξ(k1) = 2
β« β0
dt
β«(Td )3
dk2dk3dk4Ξ΄(k1 + k2 β k3 β k4)
Γ eit(Ο1+Ο2βΟ3βΟ4) (W2W3 + W2W4 βW3W4)
with Οi = Ο(ki ), Wi = W (ki ).
β Ξ1(k1) = 2Ο1
W (k1)2
β«(Td )3
dk2dk3dk4Ξ΄(k1 + k2 β k3 β k4)
Γ Ξ΄(Ο1 + Ο2 β Ο3 β Ο4)4β
i=1
W (ki )
2Ξ1(k) β₯ 0 coincides with the loss term of the linearisation ofCNL around W
Can be βderivedβ following the same recipe as for kineticequations (more later. . . )
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Main tool to handle non-Gaussian initial data 11Moments to cumulants formula
Cumulant expansion
For any index set I ,
E[βiβI
Ο0(ki , Οi )]
=β
SβΟ(I )
βAβS
[Ξ΄Ξ
(βiβA
ki
)C|A|(kA, ΟA)
],
where the sum runs over all partitions S of the index set I .
Here truncated correlation (cumulant) functions are
Cn(k , Ο) :=βxβΞn
1{x1=0}eβi2Ο
βni=1 xi Β·kiE
[ nβi=1
Ο0(xi , Οi )]trunc
and for any random variables a1, . . . , an
E[ nβi=1
ai
]trunc:= ΞΊ[a1, . . . , an] = βΞ·1 Β· Β· Β· βΞ·n lnE[e
βi Ξ·iai ]
β£β£β£Ξ·=0
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Assumption: decay of initial correlations 12`1-clustering of the equilibrium measure
For sufficiently small Ξ» and for all n β₯ 4 the truncatedcorrelation functions (cumulants) should satisfy
supΞ,Οβ{Β±1}n
βxβΞn
1{x1=0}
β£β£β£E[ nβi=1
Ο0(xi , Οi )]truncβ£β£β£ β€ Ξ»cn0n!
For n = 2 should haveββxβββ€L/2
β£β£β£E[Ο0(0)βΟ0(x)]β E[Ο0(0)βΟ0(x)]Ξ»=0L=β
β£β£β£ β€ Ξ»2c20
Proven in [Abdesselam, Procacci, and Scoppola, 2009]
Estimates imply that βCnββ <ββ cumulant expansion encodes all singularities in ki
The rest is βjustβ analysis of oscillatory integrals. . .
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
Part II
Asymptotic independence,evolution of cumulants,
and kinetic theory
[JL and M. Marcozzi, J. Math. Phys. 57 (2016) 083301 (27pp)]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
Goal: Kinetic theory of homogeneous DNLS 14
Assume that the initial state is translation invariant,βgauge invariantβ and with fast decay of correlations
Then there always is wt(x) such that
E[Οt(xβ²)βΟt(x)] = wt(x
β² β x)
Kinetic conjecture: WΟ = limΞ»β0 limΞββ(FwΟΞ»β2) solves ahomogeneous non-linear BoltzmannβPeierls equation
βΟWΟ (k) = CNL[WΟ (Β·)],
CNL[h](k1) = 4Ο
β«(Td )3
dk2dk3dk4 Ξ΄(k1 + k2 β k3 β k4)
Γ Ξ΄(Ο1 + Ο2 β Ο3 β Ο4) [h2h3h4 β h1(h2h3 + h2h4 β h3h4)]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
Why study cumulants? 15
Observation: If y , z are independent random variables we have
E[ynzm] = E[yn]E[zm] 6= 0
whereas the corresponding cumulant is zero if n,m 6= 0.
Consider a random lattice field Ο(x), x β Zd , which is (very)strongly mixing under lattice translations:
Assume that the fields in well separated regions becomeasymptotically independent as the separation grows.
Then ΞΊ[Ο(x), Ο(x + y1), . . . , Ο(x + ynβ1)]β 0 as |yi | β β.How fast? `1- or `2-summably?
Not true for corresponding moments: E[|Ο(x)|2|Ο(x + y)|2]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
Why study cumulants? 15
Observation: If y , z are independent random variables we have
E[ynzm] = E[yn]E[zm] 6= 0
whereas the corresponding cumulant is zero if n,m 6= 0.
Consider a random lattice field Ο(x), x β Zd , which is (very)strongly mixing under lattice translations:
Assume that the fields in well separated regions becomeasymptotically independent as the separation grows.
Then ΞΊ[Ο(x), Ο(x + y1), . . . , Ο(x + ynβ1)]β 0 as |yi | β β.How fast? `1- or `2-summably?
Not true for corresponding moments: E[|Ο(x)|2|Ο(x + y)|2]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
`p-clustering states 16
Call the field `p-clustering if
supxβZd
βyβ(Zd )nβ1
|ΞΊ[Ο(x), Ο(x + y1), . . .]|p <β, βn.
If 1 β€ p β€ 2, can take Fourier-transform in yβ functions F (n)(x , k), Lβ in x β Zd and L2-integrable fork β (Td)nβ1.
`1-clustering implies that F (n)(x , k) is continuous anduniformly bounded (β helps in nonlinearities)
Many examples of `1-clustering thermal Gibbs states,e.g., discrete NLS [Abdesselam, Procacci, Scoppola]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
Time-evolution of cumulants? 17
As before, consider deterministic evolution of Οt , with randominitial data for Ο0.
Cumulants are multilinear and permutation invariant
β βtΞΊ[Οt(x`)n`=1] =
nβ`=1
ΞΊ[βtΟt(x`), Οt(x`β²)`β² 6=`]
Solution? How to iterate into a closed hierarchy?
Computations often simplified by using Wick polynomialrepresentation of βtΟt
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
An example: Discrete NLS 18
Consider the DNLS with initial data for Ο0 which is
`1-clustering
Gauge invariant: Ο0(x) βΌ eiΞΈΟ0(x) for any ΞΈ β Rβ also Οt will then be gauge invariant.
Slowly varying in space: the cumulants vary only slowly underspatial translations
Then, for instance, (x , y) 7β E[Ο0(x)βΟ0(x + y)] isslowly varying in x and `1-summable in y .
Denote
Wt(x , k) =βyβZd
eβi2ΟkΒ·yE[Οt(x)βΟt(x + y)]
In the spatially homogeneous case, Wt(x , k) = Wt(k)= Wigner function (as defined in earlier works)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
Higher order Wigner functions 19
Οt(x ,+1) = Οt(x) and Οt(x ,β1) = Οt(x)β
IF `p-clustering is preserved by the time-evolution, should study
Order-n βWigner functionsβ: x β Zd , k β (Td)nβ1, Ο β {Β±1}n
F(n)t (x , k , Ο)
=β
yβ(Zd )nβ1
eβi2ΟkΒ·yΞΊ[Οt(x , Ο1), Οt(x + y1, Ο2), . . . , Οt(x + ynβ1, Οn)]
1 Now Wt(x , k) = F(2)t (x , k , (β1, 1))
2 Its derivative involves only W and F (4)
3 Compute also the derivative of F (4) and βsolveβ both byintegrating out the free evolution (in Duhamel form)
4 Insert F (4) result into W , and check/argue that the remainingF (4) and F (6) can be ignored for tβ1, Ξ»οΏ½ 1
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
With Ο(k) = Ξ±(k) and Οt(x) = E[|Οt(x)|2] =β«dk Wt(x , k),
βtWt(x , k) = βiβz
Ξ±(z)ei2ΟkΒ·z (Wt(x , k)βWt(x β z , k))
βi2Ξ»βz
(Οt(x + z)β Οt(x))
β«dk β² ei2ΟzΒ·(kβ²βk)Wt(x , k
β²)
βiΞ»
β«dk β²1dk β²2
(F
(4)t (x , k β²1, k
β²2, k β k β²1 β k β²2)β F
(4)t (x , k β²1, k
β²2, k)
)
β β 1
2ΟβkΟ(k) Β· βxWt(x , k) + 2Ξ»βxΟt(x) Β· 1
2ΟβkWt(x , k)
+ Ξ»2CNL[Wt(x , Β·)](k)
O(Ξ») term is of VlasovβPoisson type
The first two terms vanish for spatially homogeneous states
Using the same recipe in Part I yields Ξ(k)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
With Ο(k) = Ξ±(k) and Οt(x) = E[|Οt(x)|2] =β«dk Wt(x , k),
βtWt(x , k) = βiβz
Ξ±(z)ei2ΟkΒ·z (Wt(x , k)βWt(x β z , k))
βi2Ξ»βz
(Οt(x + z)β Οt(x))
β«dk β² ei2ΟzΒ·(kβ²βk)Wt(x , k
β²)
βiΞ»
β«dk β²1dk β²2
(F
(4)t (x , k β²1, k
β²2, k β k β²1 β k β²2)β F
(4)t (x , k β²1, k
β²2, k)
)β β 1
2ΟβkΟ(k) Β· βxWt(x , k) + 2Ξ»βxΟt(x) Β· 1
2ΟβkWt(x , k)
+ Ξ»2CNL[Wt(x , Β·)](k)
O(Ξ») term is of VlasovβPoisson type
The first two terms vanish for spatially homogeneous states
Using the same recipe in Part I yields Ξ(k)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Part III
Kinetic theory of the onsiteFPU-chain (DNKG) with
pre-thermalization
[C.B. Mendl, J. Lu, and JL,Phys. Rev. E 94 (2016) 062104 (9 pp)]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Pre-thermalization 22
Pre-thermalization = quasi-thermalizationThe system is thermalized but with βwrongβequilibrium states (e.g. extra conservation laws)
Happens if there are quasi-conserved observables with verylong equilibration times (e.g. with relaxation times of order eL
for system size L)
Problematic, since may interfere with relaxation of the βtrueβconservation laws:
For instance, if diffusive, L2 οΏ½ eL for system LοΏ½ 1
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
FPU-type chains 23
We consider a chain of classical particles with nearest neighbourinteractions, dynamics defined by
The Hamiltonian
H =Nβ1βj=0
[12p
2j + 1
2q2j β 1
2Ξ΄(qjβ1qj + qjqj+1) + 14Ξ»q
4j
]
Ξ» β₯ 0 is the coupling constant for the onsite anharmonicity
If Ξ» = 0, the evolution is explicitly solvable using normalmodes whose dispersion relations are Β±Ο(k) with
Ο(k) =(1β 2Ξ΄ cos(2Οk)
)1/2
The parameter 0 < Ξ΄ β€ 12 controls the pinning onsite potential
This model is expected to have (diffusive) normal heatconduction for Ξ», Ξ΄ > 0
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Normal modes 24
From a solution (qi (t), pi (t)) define
Phonon fields
at(k , Ο) =1β
2Ο(k)[Ο(k)q(k, t) + iΟp(k , t)] , Ο β {Β±1}
β d
dtat(k, Ο) = βiΟΟ(k)at(k , Ο)
βiΟΞ»β
Οβ²β{Β±1}3
β«(Ξβ)3
d3k β²Ξ΄Ξ
(k β
3βj=1
k β²j
) 3β`=0
1β2Ο(k β²`)
3βj=1
at(kβ²j , Οβ²j )
Here k β Ξβ := {β 12 + 1
N , . . . ,12β
1N ,
12}
for a finite periodic chain of length N (= |Ξβ|)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Wigner function 25
Assume a spatially homogeneous state and consider thecorresponding Wigner function
wt(k ; L) :=
β«Ξβ
dk β²γat(k β²)βat(k)γ =βyβΞ
eβi2Οy Β·kγat(0)βat(y)γ
We expect (βkinetic conjectureβ) that then the following limit exists
WΟ (k) = limΞ»β0
limLββ
wΞ»β2Ο (k; L)
Describes evolution of covariances for large lattices (Lββ)at kinetic time-scales (t = Ξ»β2Ο = O(Ξ»β2))
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
BoltzmannβPeierls equation 26
In addition, the limiting Wigner functions should satisfy thefollowing phonon Boltzmann equation
β
βtW (k0, t) = 12ΟΞ»2
βΟβ{Β±1}3
β«T3
d3k3β`=0
1
2Ο`
ΓΞ΄(k0 +3β
j=1
Οjkj) Ξ΄(Ο0 +
3βj=1
ΟjΟj
)Γ[W1W2W3 + W0(Ο1W2W3 + Ο2W1W3 + Ο3W1W2)
]Here Wi = W (ki , t), Οi = Ο(ki )
For chosen nearest neighbour interactions, the collisionΞ΄-functions have solutions only if
β3j=1Οj = β1
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
BoltzmannβPeierls equation 27
Kinetic equation (spatially homogeneous initial data)
β
βtW (k0, t) =
9Ο
4Ξ»2
β«T3
d3k1
Ο0Ο1Ο2Ο3
Γ Ξ΄(Ο0 + Ο1 β Ο2 β Ο3)Ξ΄(k0 + k1 β k2 β k3)
Γ[W1W2W3 + W0W2W3 βW0W1W3 βW0W1W2
]
Stationary solutions are
W (k) =1
Ξ²β²(Ο(k)β Β΅β²)
Β΅β² results from number conservation which is broken by theoriginal evolution (then expect Β΅β² = 0)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Spatially homogeneous kinetic theory 28
Microsystem Dynamics: free evolution + Ξ» Γ perturbation
wt(k) =β«
dk β²γat(k β²)βat(k)γ Initial state: translation invariant & βchaoticβ
β β (weak coupling)
βΟWΟ (k) = C[WΟ ](k) Boltzmann equation for WΟ = limΞ»β0
wΞ»β2Ο
β ββΟS [WΟ ] = Ο[WΟ ] S = kinetic entropy (H-function)
Ο = entropy production β₯ 0
β βΟ[W eql] = 0 β W eql from an equilibrium state
(classifies stationary solutions)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Numerical simulations (Christian Mendl) 29
To compare in more detail to kinetic theory, we considerseveral stochastic, periodic and translation invariant initialdata: Then
Wsim(k, t) =1
Nγ|a(k , t)|2γ
Computing the covariance from simulated equilibrium states(one parameter, Ξ²) and fitting numerically to the kineticformula (two parameters, Ξ²β², Β΅β²) yields
Ξ² 1 10 100 1000
Ξ²β² 0.912 8.98 97.1 986.4Β΅β² -0.488 -0.229 -0.0426 -0.0120
As expected, Ξ²β² β Ξ² and Β΅β² β 0 for large Ξ²
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Set N = 64 (periodic BC), Ξ΄ = 14 (pinning)
Consider two sets of non-equilibrium initial data:
A) Bimodal momentum distribution (Ξ» = 1):Choose an initial βtemperatureβ Ξ²0 and sample positions qjfrom the corresponding equilibrium distribution and themomenta pj from the bimodal distribution
Zβ1 exp[βΞ²0
(4p4
j β 12p
2j
)]B) Random phase, with given initial Wigner function (Ξ» = 1
2 , 10):Take a function W0(k) and compute initial qj and pj from
a(k) =β
NW0(k) eiΟ(k)
where each Ο(k) is i.i.d. randomly distributed, uniformly on[0, 2Ο]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
A) Bimodal initial data
βββββββββ
βββββ
ββββββ
ββββββ
βββ
ββββββ
ββββββ
ββββββ
ββββββββββββββββββ
-0.4 -0.2 0 0.2 0.4k
0.03
0.04
0.05
W(k,t)
(a) t = 0
βββββββββ
βββββββ
βββ
β
β
β
β
β
β
ββ
ββ
β
β
βββ
β
β
ββ
ββ
β
β
β
β
β
βββββββββββββββββ
βββ
-0.4 -0.2 0 0.2 0.4k
0.03
0.04
0.05
W(k,t)
(b) t = 150
ββββββββββββ
ββββ
βββ
β
β
β
β
βββββ
β
β
β
βββ
β
β
βββ
β
ββ
β
β
β
ββ
ββββββββββββββ
ββββ
-0.4 -0.2 0 0.2 0.4k
0.03
0.04
0.05
W(k,t)
(c) t = 250
βββββββββββββ
ββββββββ
β
β
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ββββββ
βββββββ
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βββββββββββββββββββ
ββββ²β²β²β²β²β²β²β²β²β²β²
β²β²β²β²
β²β²β²β²β²β²
β²
β²
β²β²β²β²β²
β²β²β²β²β²β²
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β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²
β²β²β²
-0.4 -0.2 0 0.2 0.4k
0.03
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W(k,t)
(d) t = 500
βββββββββββββ
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β²β²β²β²
β²β²β²β²β²β²β²β²β²β²β²β²
β²β²β²β²β²β²β²β²β²β²β²β²β²
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-0.4 -0.2 0 0.2 0.4k
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(e) t = 1000
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β²β²β²β²β²
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β²β²β²β²β²β²β²β²β²β²β²β²β²β²
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-0.4 -0.2 0 0.2 0.4k
0.03
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0.05
W(k,t)
(f) t = 2500
Wigner function from simulations (blue dots) vs.solving the kinetic equation (yellow triangles) starting at t = 500.(black dashed line) Kinetic equilibrium profile fitted to (f)
Jani Lukkarinen Kinetic theory of lattice waves
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B) Random phase initial data (Ξ» = 12)
βββββββββ
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(a) t = 0
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β‘β‘β‘
-0.4 -0.2 0 0.2 0.4k
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(b) t = 250
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-0.4 -0.2 0 0.2 0.4k
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(c) t = 500
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β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²
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β²β²β²β²
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-0.4 -0.2 0 0.2 0.4k
0.02
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0.06W(k,t)
(d) t = 1000
βββββββββββ
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ββββββββββββββββββββββββββββββββββββββ²β²β²β²β²β²
β²β²β²β²β²β²β²
β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²
β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²
-0.4 -0.2 0 0.2 0.4k
0.02
0.04
0.06W(k,t)
(e) t = 5000
ββββββββββββββ
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βββββββββββββββββββββββββββββββββββββ²β²β²β²β²β²β²β²β²β²β²β²β²β²
β²β²β²β²β²β²β²β²β²β²β²β²β²β²
β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²β²
-0.4 -0.2 0 0.2 0.4k
0.02
0.04
0.06W(k,t)
(f) t = 10000
Wigner function from simulations (blue dots) vs.solving the kinetic equation (yellow triangles) starting at t = 500.(f) Expected equilibrium distribution (red dot-dashed line)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Eventual relaxation towards equilibrium? (Ξ» = 10)
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
0.2 0.4 0.6 0.8 1Γ106t
-5Γ10-4
0
5Γ10-4ΞΟ(t)
(a) Οsim(t)β Οsim(tmax)
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
0.2 0.4 0.6 0.8 1Γ106t
-5Γ10-4
0
5Γ10-4Ξe(t)
(b) esim(t)β esim(tmax)
Figure : Time evolution of the density and energy differences usingΞ» = 10 (β kinetic time-scale (Ξ²/Ξ»)2 β 10) and longer simulation timetmax = 106
Will this trend continue until true equilibrium values have beenreached?
Jani Lukkarinen Kinetic theory of lattice waves
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Open problems 34
Pre-thermalization: What is going on here?
Is it 1D effect only?
. . . or finite size?
. . . or just for some initial data?
Inhomogeneous initial data:Does kinetic theory perform as well?. . . with the VlasovβPoisson term?Proofs and proper assumptions?
For rigorous proofs of time-correlations:a priori estimates for propagation of clustering forequilibrium time-correlations derived in[JL, M. Marcozzi and A. Nota, arXiv:1601.08163 ]
Anything similar for non-stationary states?
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Appendix
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Outline of the proof 36
1 Show that it is enough to prove the result assuming t > 0
2 Iterate a Duhamel formula N0(Ξ») times to expand at into aperturbation sum (we choose N0! β Ξ»βp, for a small p)
3 There are two types of terms in the expansion:
Main terms These will contain a finite monomial of a0 whoseexpectation can be evaluated using theβmoments to cumulants formulaβ.
Error terms These will involve also as for some s > 0.The expectation is estimated by a Schwarzbound and stationarity of the equilibriummeasureβ The bound involves again only
finite moments of a0.
Jani Lukkarinen Kinetic theory of lattice waves
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4 Each cumulant induces linear dependencies between the wavevectors. These can be encoded in βFeynman graphsβ.
5 This results in a sum with roughly (N0!)2 non-zero terms.However, most of these vanish in the limit Ξ»β 0, due tooscillating phase factors.
6 Careful classification of graphs: we use a special resolution ofthe wave vector constraints which allows an estimation basedon identifying, and iteratively estimating, certain graphmotives.
7 Only a small fraction of the graphs (leading graphs) willremain. These consist of graphs obtained by iterative additionof one of the 20 leading motives.
8 The limit of the leading graphs is explicitly computable, andtheir sum yields the result in the main theorem.
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Wick polynomials 38
Generating functions
gt(Ξ») := ln gmom,t(Ξ»), gmom,t(Ξ») := E[eλ·Οt ] .
Then with βJΞ» :=β
iβJ βΞ»i , yJ =
βiβJ yi ,
ΞΊ[Οt(x)J ] = βJΞ»gt(0) , E[Οt(x)J ] = βJΞ»gmom,t(0)
Define
Gw(Οt , Ξ») =eλ·Οt
E[eλ·Οt ]
β βtΞΊ[Οt(x)J ] = βJΞ»βtgt(Ξ»)β£β£Ξ»=0
= βJΞ»E[Ξ» Β· βtΟt Gw(Οt , Ξ»)]β£β£Ξ»=0
=β`βJ
E[βtΟt(x`) βJ\`Ξ» Gw(Οt , 0)]
βJΞ»Gw(Οt , 0) = :Οt(x)J : are called Wick polynomials
Jani Lukkarinen Kinetic theory of lattice waves
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WP have been mainly used for Gaussian fields.They were introduced in quantum field theory where theunperturbed measure concerns Gaussian (free) fields
Gaussian case has significant simplifications:If Cj β²j = ΞΊ[yj β² , yj ] denotes the covariance matrix ,
Gw(y , Ξ») = exp[Ξ» Β· (y β γyγ)β Ξ» Β· CΞ»/2] .
β Wick polynomials are Hermite polynomials
The resulting orthogonality properties are used in the Wienerchaos expansion and Malliavin calculus
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Truncated moments-to-cumulants formula
E[yJβ²:yJ :
]=
βΟβP(Jβ²βͺJ)
βAβΟ
(ΞΊ[yA]1{A 6βJ}) (1)
:yJ : are Β΅-a.s. unique polynomials of order |J| such that (1)holds for every J β²
Multi-truncated moments-to-cumulants formula
Suppose L β₯ 1 is given and consider a collection of L + 1 indexsequences J β², J`, ` = 1, . . . , L. Then with I = J β² βͺ (βͺL`=1J`)
E[yJβ²
Lβ`=1
:yJ` :
]=
βΟβP(I )
βAβΟ
(ΞΊ[yA]1{A 6βJ`, β`}
).
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Suppose that the evolution equation of the random variables yj(t)can be written in a form
βtyj(t) =βI
M Ij (t) :y(t)I :
Then the cumulants satisfy
βtΞΊ[y(t)I β² ] =β`βI β²
βI
M I` (t)E
[:y(t)I : :y(t)I
β²\`:]
where the truncated moments-to-cumulants formula implies
E[
:y(t)I : :y(t)Iβ²\`:]
=βΟβP(Iβͺ(I β²\`))
βAβΟ
(ΞΊ[y(t)A]1{Aβ©I 6=β , Aβ©(I β²\`) 6=β }
)β evolution hierarchy for cumulants
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
The discrete NLS equation on the lattice Zd deals with functionsΟ : RΓ Zd β C which satisfy
iβtΟt(x) =βyβZd
Ξ±(x β y)Οt(y) + Ξ»|Οt(x)|2Οt(x)
Assuming that E[Οt(x)] = 0 and using the WP one gets
iβtΟt(x) =βyβZd
Ξ±(x β y) :Οt(y): +2Ξ»Οt(x) :Οt(x):
+Ξ» :Οt(x)βΟt(x)Οt(x):
Οt(x) = E[Οt(x)βΟt(x)] = E[|Οt(x)|2]
This splitting was called βpair truncationβ in [JL, Spohn](Part I)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Molecular dynamics simulations (Kenichiro Aoki, 2006) 43
T β βT2 T + βT
2N sites
1 Simulate a chain of N particles with two heat baths(Nose-Hoover) at ends, waiting until a steady state reached
2 Measure temperature and current profiles:
Ti = γp2i γ, J =
1
N
βjγJj ,j+1γ β γJi ,i+1γ
3 Fourierβs Law predicts that when βT β 0,
β N
βTJ β ΞΊ(T ,N) .
Repeat for several βT , and estimate ΞΊ(T ,N) from the slope.
4 Increase N to estimate ΞΊ(T ) = limNββ
ΞΊ(T ,N).
Jani Lukkarinen Kinetic theory of lattice waves
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Comparison to kinetic prediction 44
Simulations yield good agreement with the kinetic prediction ofT 2ΞΊ(T ) β 0.28 Ξ΄β3/2 for T β 0, Ξ΄ small (black solid line)
οΏ½ T = 0.1
Β© T = 0.4
4 T = 4
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Evolution of entropy 45
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ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β microscopic
β kinetic
500 1000 1500 2000 2500t
-3.458
-3.46
-3.462
-3.464
S(t)
(a) Bimodal initial data
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β microscopic
β kinetic
2000 4000 6000 8000 10000t
-3.58
-3.57
-3.56
-3.55S(t)
(b) Random phase initial data
Time evolution of entropy S(t) =
β«dk logW (k , t)
(blue dots) W = Wigner function measured from simulations
(orange dots) W = solution to the kinetic equation, initial data fromt = 500 simulation results
Jani Lukkarinen Kinetic theory of lattice waves