Post on 02-Dec-2021
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) 79–188]
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 ≥ 2 , Λ = {0, 1, . . . , L− 1}d
Periodic BC: All arithmetic mod L
Dual lattice: Λ∗ = {0, 1L , . . . ,
L−1L }
d
“Integration” = finite sum:∫Λ∗
dk f (k) :=1
|Λ|∑k∈Λ∗
f (k)
“Dirac delta” = finite sum:
δΛ(k) := |Λ|1{k mod 1 = 0}
Fourier transform: (x ∈ Λ, k ∈ Λ∗)
f (k) =∑y∈Λ
f (y)e−i2πk·y ⇒ f (x) =
∫Λ∗
dk ′ f (k ′)ei2πk ′·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
dtψt(x) =
∑y∈Λ
α(x − y)ψt(y) + λ|ψt(x)|2ψt(x)
ψt : Λ→ C, t ∈ Rλ > 0 (defocusing)
Harmonic coupling determined by α : Zd → R.
α has finite range (for instance, nearest neighbour)
We assume also α(−x) = α(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
HΛ(ψ) =∑x ,y∈Λ
α(x − y)ψ(x)∗ψ(y) +1
2λ∑x∈Λ
|ψ(x)|4
Relate qx , px ∈ R to ψ by ψ(x) =1√2
(qx + ipx)
NLS equivalent to the Hamiltonian equations
qx = ∂pxHΛ , px = −∂qxHΛ
Thus HΛ(ψt) is conserved
By explicit differentiation, also∑
x |ψt(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 ψ = ψ0 (Grand canonical ensemble)
1
Zλβ,µe−β(HΛ(ψ)−µ‖ψ‖2)
∏x∈Λ
[d(Reψ(x)) d(Imψ(x))]
Define ω : Td → R by ω = Fx→kα.
We consider only β > 0 and µ < mink ω(k)⇒ Also the Gaussian measure at λ = 0 is well-defined
Zλβ,µ > 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 ψt exists and is unique for all t ∈ R with anyinitial data ψ0 ∈ CΛ. (conservation laws)
Initial state is stationary : E[F (ψt)] = E[F (ψ0)]
Also invariant under periodic translations:
E[F (τxψ)] = E[F (ψ)], (τxψ)(y) = ψ(y + x)
Translations commute with the time-evolution:
τxψt = ψt |ψ0=τxψ0
“Gauge invariance”: similar invariance properties hold fortranslations of total phase, ψ0(x) 7→ eiϕψ0(x), ϕ ∈ R.
Thus, for instance, E[ψt ] = 0, E[ψt′ψt ] = 0,
E[ψt′(x′)∗ψt(x)] = E[ψ0(0)∗ψt−t′(x − x ′)]
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 ∈ `2, and assume they have finite support.
Observable
QλΛ(τ) := E[〈f , ψ0〉∗〈e−iωλτλ−2
g , ψτλ−2〉]
Under additional assumptions on the decay of equilibriumcorrelations and on the dispersion relation:
Theorem
There is τ0 > 0 such that for all |τ | < τ0
limλ→0
limΛ→∞
QλΛ(τ) =
∫Td
dk g(k)∗f (k)W (k)e−Γ1(k)|τ |−iτΓ2(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(λ−2),
E[ψ0(k ′)∗ψt(k)] ≈ δΛ(k ′ − k)W (k)e−iωλren(k)te−|λ2t|Γ1(k)
W (k) = (β(ω(k)− µ))−1 = covariance function for λ = 0
ωλren(k) = ω(k) + λR0 + λ2Γ2(k)
Γ1(k) ≥ 0
⇒ k-space correlation decays exponentially in t,as dictated by e−|λ
2t|Γ1(k).
Nearest neighbour couplings (ωnn(k) = c −∑d
ν=1 cos(2πkν))satisfy all of our assumptions if d ≥ 4
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Setup Evol H Init Theorem Cumulants
Γj(k) are real, and Γ(k) = Γ1(k) + iΓ2(k) is given by
Γ(k1) = 2
∫ ∞0
dt
∫(Td )3
dk2dk3dk4δ(k1 + k2 − k3 − k4)
× eit(ω1+ω2−ω3−ω4) (W2W3 + W2W4 −W3W4)
with ωi = ω(ki ), Wi = W (ki ).
⇒ Γ1(k1) = 2π1
W (k1)2
∫(Td )3
dk2dk3dk4δ(k1 + k2 − k3 − k4)
× δ(ω1 + ω2 − ω3 − ω4)4∏
i=1
W (ki )
2Γ1(k) ≥ 0 coincides with the loss term of the linearisation ofCNL around W
Can be “derived” 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[∏i∈I
ψ0(ki , σi )]
=∑
S∈π(I )
∏A∈S
[δΛ
(∑i∈A
ki
)C|A|(kA, σA)
],
where the sum runs over all partitions S of the index set I .
Here truncated correlation (cumulant) functions are
Cn(k , σ) :=∑x∈Λn
1{x1=0}e−i2π
∑ni=1 xi ·kiE
[ n∏i=1
ψ0(xi , σi )]trunc
and for any random variables a1, . . . , an
E[ n∏i=1
ai
]trunc:= κ[a1, . . . , an] = ∂η1 · · · ∂ηn lnE[e
∑i ηiai ]
∣∣∣η=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 λ and for all n ≥ 4 the truncatedcorrelation functions (cumulants) should satisfy
supΛ,σ∈{±1}n
∑x∈Λn
1{x1=0}
∣∣∣E[ n∏i=1
ψ0(xi , σi )]trunc∣∣∣ ≤ λcn0n!
For n = 2 should have∑‖x‖∞≤L/2
∣∣∣E[ψ0(0)∗ψ0(x)]− E[ψ0(0)∗ψ0(x)]λ=0L=∞
∣∣∣ ≤ λ2c20
Proven in [Abdesselam, Procacci, and Scoppola, 2009]
Estimates imply that ‖Cn‖∞ <∞⇒ cumulant expansion encodes all singularities in ki
The rest is “just” 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,“gauge invariant” and with fast decay of correlations
Then there always is wt(x) such that
E[ψt(x′)∗ψt(x)] = wt(x
′ − x)
Kinetic conjecture: Wτ = limλ→0 limΛ→∞(Fwτλ−2) solves ahomogeneous non-linear Boltzmann–Peierls equation
∂τWτ (k) = CNL[Wτ (·)],
CNL[h](k1) = 4π
∫(Td )3
dk2dk3dk4 δ(k1 + k2 − k3 − k4)
× δ(ω1 + ω2 − ω3 − ω4) [h2h3h4 − h1(h2h3 + h2h4 − 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 ψ(x), x ∈ Zd , which is (very)strongly mixing under lattice translations:
Assume that the fields in well separated regions becomeasymptotically independent as the separation grows.
Then κ[ψ(x), ψ(x + y1), . . . , ψ(x + yn−1)]→ 0 as |yi | → ∞.How fast? `1- or `2-summably?
Not true for corresponding moments: E[|ψ(x)|2|ψ(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 ψ(x), x ∈ Zd , which is (very)strongly mixing under lattice translations:
Assume that the fields in well separated regions becomeasymptotically independent as the separation grows.
Then κ[ψ(x), ψ(x + y1), . . . , ψ(x + yn−1)]→ 0 as |yi | → ∞.How fast? `1- or `2-summably?
Not true for corresponding moments: E[|ψ(x)|2|ψ(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
supx∈Zd
∑y∈(Zd )n−1
|κ[ψ(x), ψ(x + y1), . . .]|p <∞, ∀n.
If 1 ≤ p ≤ 2, can take Fourier-transform in y⇒ functions F (n)(x , k), L∞ in x ∈ Zd and L2-integrable fork ∈ (Td)n−1.
`1-clustering implies that F (n)(x , k) is continuous anduniformly bounded (⇒ 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 ψt , with randominitial data for ψ0.
Cumulants are multilinear and permutation invariant
⇒ ∂tκ[ψt(x`)n`=1] =
n∑`=1
κ[∂tψt(x`), ψt(x`′)`′ 6=`]
Solution? How to iterate into a closed hierarchy?
Computations often simplified by using Wick polynomialrepresentation of ∂tψt
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 ψ0 which is
`1-clustering
Gauge invariant: ψ0(x) ∼ eiθψ0(x) for any θ ∈ R⇒ also ψt will then be gauge invariant.
Slowly varying in space: the cumulants vary only slowly underspatial translations
Then, for instance, (x , y) 7→ E[ψ0(x)∗ψ0(x + y)] isslowly varying in x and `1-summable in y .
Denote
Wt(x , k) =∑y∈Zd
e−i2πk·yE[ψt(x)∗ψt(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
ψt(x ,+1) = ψt(x) and ψt(x ,−1) = ψt(x)∗
IF `p-clustering is preserved by the time-evolution, should study
Order-n “Wigner functions”: x ∈ Zd , k ∈ (Td)n−1, σ ∈ {±1}n
F(n)t (x , k , σ)
=∑
y∈(Zd )n−1
e−i2πk·yκ[ψt(x , σ1), ψt(x + y1, σ2), . . . , ψt(x + yn−1, σn)]
1 Now Wt(x , k) = F(2)t (x , k , (−1, 1))
2 Its derivative involves only W and F (4)
3 Compute also the derivative of F (4) and “solve” 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 t−1, λ� 1
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
With ω(k) = α(k) and ρt(x) = E[|ψt(x)|2] =∫dk Wt(x , k),
∂tWt(x , k) = −i∑z
α(z)ei2πk·z (Wt(x , k)−Wt(x − z , k))
−i2λ∑z
(ρt(x + z)− ρt(x))
∫dk ′ ei2πz·(k′−k)Wt(x , k
′)
−iλ
∫dk ′1dk ′2
(F
(4)t (x , k ′1, k
′2, k − k ′1 − k ′2)− F
(4)t (x , k ′1, k
′2, k)
)
≈ − 1
2π∇kω(k) · ∇xWt(x , k) + 2λ∇xρt(x) · 1
2π∇kWt(x , k)
+ λ2CNL[Wt(x , ·)](k)
O(λ) term is of Vlasov–Poisson type
The first two terms vanish for spatially homogeneous states
Using the same recipe in Part I yields Γ(k)
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras `p -clustering DNLS
With ω(k) = α(k) and ρt(x) = E[|ψt(x)|2] =∫dk Wt(x , k),
∂tWt(x , k) = −i∑z
α(z)ei2πk·z (Wt(x , k)−Wt(x − z , k))
−i2λ∑z
(ρt(x + z)− ρt(x))
∫dk ′ ei2πz·(k′−k)Wt(x , k
′)
−iλ
∫dk ′1dk ′2
(F
(4)t (x , k ′1, k
′2, k − k ′1 − k ′2)− F
(4)t (x , k ′1, k
′2, k)
)≈ − 1
2π∇kω(k) · ∇xWt(x , k) + 2λ∇xρt(x) · 1
2π∇kWt(x , k)
+ λ2CNL[Wt(x , ·)](k)
O(λ) term is of Vlasov–Poisson type
The first two terms vanish for spatially homogeneous states
Using the same recipe in Part I yields Γ(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 “wrong”equilibrium 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 “true”conservation laws:
For instance, if diffusive, L2 � eL for system L� 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 =N−1∑j=0
[12p
2j + 1
2q2j − 1
2δ(qj−1qj + qjqj+1) + 14λq
4j
]
λ ≥ 0 is the coupling constant for the onsite anharmonicity
If λ = 0, the evolution is explicitly solvable using normalmodes whose dispersion relations are ±ω(k) with
ω(k) =(1− 2δ cos(2πk)
)1/2
The parameter 0 < δ ≤ 12 controls the pinning onsite potential
This model is expected to have (diffusive) normal heatconduction for λ, δ > 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 , σ) =1√
2ω(k)[ω(k)q(k, t) + iσp(k , t)] , σ ∈ {±1}
⇒ d
dtat(k, σ) = −iσω(k)at(k , σ)
−iσλ∑
σ′∈{±1}3
∫(Λ∗)3
d3k ′δΛ
(k −
3∑j=1
k ′j
) 3∏`=0
1√2ω(k ′`)
3∏j=1
at(k′j , σ′j )
Here k ∈ Λ∗ := {− 12 + 1
N , . . . ,12−
1N ,
12}
for a finite periodic chain of length N (= |Λ∗|)
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) :=
∫Λ∗
dk ′〈at(k ′)∗at(k)〉 =∑y∈Λ
e−i2πy ·k〈at(0)∗at(y)〉
We expect (“kinetic conjecture”) that then the following limit exists
Wτ (k) = limλ→0
limL→∞
wλ−2τ (k; L)
Describes evolution of covariances for large lattices (L→∞)at kinetic time-scales (t = λ−2τ = O(λ−2))
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Boltzmann–Peierls equation 26
In addition, the limiting Wigner functions should satisfy thefollowing phonon Boltzmann equation
∂
∂tW (k0, t) = 12πλ2
∑σ∈{±1}3
∫T3
d3k3∏`=0
1
2ω`
×δ(k0 +3∑
j=1
σjkj) δ(ω0 +
3∑j=1
σjωj
)×[W1W2W3 + W0(σ1W2W3 + σ2W1W3 + σ3W1W2)
]Here Wi = W (ki , t), ωi = ω(ki )
For chosen nearest neighbour interactions, the collisionδ-functions have solutions only if
∑3j=1σj = −1
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Boltzmann–Peierls equation 27
Kinetic equation (spatially homogeneous initial data)
∂
∂tW (k0, t) =
9π
4λ2
∫T3
d3k1
ω0ω1ω2ω3
× δ(ω0 + ω1 − ω2 − ω3)δ(k0 + k1 − k2 − k3)
×[W1W2W3 + W0W2W3 −W0W1W3 −W0W1W2
]
Stationary solutions are
W (k) =1
β′(ω(k)− µ′)
µ′ results from number conservation which is broken by theoriginal evolution (then expect µ′ = 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 + λ × perturbation
wt(k) =∫
dk ′〈at(k ′)∗at(k)〉 Initial state: translation invariant & “chaotic”
↓ ↓ (weak coupling)
∂τWτ (k) = C[Wτ ](k) Boltzmann equation for Wτ = limλ→0
wλ−2τ
↓ ↓∂τS [Wτ ] = σ[Wτ ] S = kinetic entropy (H-function)
σ = entropy production ≥ 0
↓ ↓σ[W eql] = 0 ⇐ 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
N〈|a(k , t)|2〉
Computing the covariance from simulated equilibrium states(one parameter, β) and fitting numerically to the kineticformula (two parameters, β′, µ′) yields
β 1 10 100 1000
β′ 0.912 8.98 97.1 986.4µ′ -0.488 -0.229 -0.0426 -0.0120
As expected, β′ ≈ β and µ′ ≈ 0 for large β
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
Set N = 64 (periodic BC), δ = 14 (pinning)
Consider two sets of non-equilibrium initial data:
A) Bimodal momentum distribution (λ = 1):Choose an initial ”temperature” β0 and sample positions qjfrom the corresponding equilibrium distribution and themomenta pj from the bimodal distribution
Z−1 exp[−β0
(4p4
j − 12p
2j
)]B) Random phase, with given initial Wigner function (λ = 1
2 , 10):Take a function W0(k) and compute initial qj and pj from
a(k) =√
NW0(k) eiϕ(k)
where each ϕ(k) is i.i.d. randomly distributed, uniformly on[0, 2π]
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
A) Bimodal initial data
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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
Intro DNLS Kinetic FPU Comments Extras Pre-therm Phonons Wigner BPeq Numerical
B) Random phase initial data (λ = 12)
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(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? (λ = 10)
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Figure : Time evolution of the density and energy differences usingλ = 10 (⇒ kinetic time-scale (β/λ)2 ≈ 10) and longer simulation timetmax = 106
Will this trend continue until true equilibrium values have beenreached?
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Open
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 Vlasov–Poisson 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(λ) times to expand at into aperturbation sum (we choose N0! ≈ λ−p, 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 the“moments to cumulants formula”.
Error terms These will involve also as for some s > 0.The expectation is estimated by a Schwarzbound and stationarity of the equilibriummeasure⇒ The bound involves again only
finite moments of a0.
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
4 Each cumulant induces linear dependencies between the wavevectors. These can be encoded in “Feynman graphs”.
5 This results in a sum with roughly (N0!)2 non-zero terms.However, most of these vanish in the limit λ→ 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(λ) := ln gmom,t(λ), gmom,t(λ) := E[eλ·ψt ] .
Then with ∂Jλ :=∏
i∈J ∂λi , yJ =
∏i∈J yi ,
κ[ψt(x)J ] = ∂Jλgt(0) , E[ψt(x)J ] = ∂Jλgmom,t(0)
Define
Gw(ψt , λ) =eλ·ψt
E[eλ·ψt ]
⇒ ∂tκ[ψt(x)J ] = ∂Jλ∂tgt(λ)∣∣λ=0
= ∂JλE[λ · ∂tψt Gw(ψt , λ)]∣∣λ=0
=∑`∈J
E[∂tψt(x`) ∂J\`λ Gw(ψt , 0)]
∂JλGw(ψt , 0) = :ψt(x)J : are called Wick polynomials
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
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 ′j = κ[yj ′ , yj ] denotes the covariance matrix ,
Gw(y , λ) = exp[λ · (y − 〈y〉)− λ · Cλ/2] .
⇒ 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[yJ′:yJ :
]=
∑π∈P(J′∪J)
∏A∈π
(κ[yA]1{A 6⊂J}) (1)
:yJ : are µ-a.s. unique polynomials of order |J| such that (1)holds for every J ′
Multi-truncated moments-to-cumulants formula
Suppose L ≥ 1 is given and consider a collection of L + 1 indexsequences J ′, J`, ` = 1, . . . , L. Then with I = J ′ ∪ (∪L`=1J`)
E[yJ′
L∏`=1
:yJ` :
]=
∑π∈P(I )
∏A∈π
(κ[yA]1{A 6⊂J`, ∀`}
).
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
∂tyj(t) =∑I
M Ij (t) :y(t)I :
Then the cumulants satisfy
∂tκ[y(t)I ′ ] =∑`∈I ′
∑I
M I` (t)E
[:y(t)I : :y(t)I
′\`:]
where the truncated moments-to-cumulants formula implies
E[
:y(t)I : :y(t)I′\`:]
=∑π∈P(I∪(I ′\`))
∏A∈π
(κ[y(t)A]1{A∩I 6=∅, A∩(I ′\`) 6=∅}
)⇒ 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 functionsψ : R× Zd → C which satisfy
i∂tψt(x) =∑y∈Zd
α(x − y)ψt(y) + λ|ψt(x)|2ψt(x)
Assuming that E[ψt(x)] = 0 and using the WP one gets
i∂tψt(x) =∑y∈Zd
α(x − y) :ψt(y): +2λρt(x) :ψt(x):
+λ :ψt(x)∗ψt(x)ψt(x):
ρt(x) = E[ψt(x)∗ψt(x)] = E[|ψt(x)|2]
This splitting was called “pair truncation” 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 − ∆T2 T + ∆T
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 = 〈p2i 〉, J =
1
N
∑j〈Jj ,j+1〉 ≈ 〈Ji ,i+1〉
3 Fourier’s Law predicts that when ∆T → 0,
− N
∆TJ → κ(T ,N) .
Repeat for several ∆T , and estimate κ(T ,N) from the slope.
4 Increase N to estimate κ(T ) = limN→∞
κ(T ,N).
Jani Lukkarinen Kinetic theory of lattice waves
Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy
Comparison to kinetic prediction 44
Simulations yield good agreement with the kinetic prediction ofT 2κ(T ) ≈ 0.28 δ−3/2 for T → 0, δ small (black solid line)
� T = 0.1
© 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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● microscopic
● kinetic
500 1000 1500 2000 2500t
-3.458
-3.46
-3.462
-3.464
S(t)
(a) Bimodal initial data
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● microscopic
● 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) =
∫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