Kinetic theory of (lattice) waves

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]



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



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)



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



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



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 ′)]



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)



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



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. . . )



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



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. . .


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)]



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)]



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]



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]



`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]



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



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)



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



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)



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)


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)]



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



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



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∗|)



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))



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



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)



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)



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



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]



A) Bimodal initial data

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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)



B) Random phase initial data (λ = 12)

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-0.4 -0.2 0 0.2 0.4k

0.02

0.04

0.06W(k,t)

(c) t = 500

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0.06W(k,t)

(d) t = 1000

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-0.4 -0.2 0 0.2 0.4k

0.02

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0.06W(k,t)

(e) t = 5000

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-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)



Eventual relaxation towards equilibrium? (λ = 10)

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0.2 0.4 0.6 0.8 1×106t

-5×10-4

0

5×10-4Δρ(t)

(a) ρsim(t)− ρsim(tmax)

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0.2 0.4 0.6 0.8 1×106t

-5×10-4

0

5×10-4Δe(t)

(b) esim(t)− esim(tmax)

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?


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?


Intro DNLS Kinetic FPU Comments Extras Proof Wick Simulations Entropy

Appendix



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.



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.



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



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



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`, ∀`}

).



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



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)



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).



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



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


Documents

Kinetic theory of (lattice) waves