Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal...

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Anomalous energy transport in FPU-β chain Sara Merino Aceituno Joint work with Antoine Mellet (University of Maryland) http://arxiv.org/abs/1411.5246 Imperial College London 9th November 2015. Kinetic theory with applications in physical sciences. University of Maryland, CSCAMM

Transcript of Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal...

Page 1: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Anomalous energy transport in FPU-β chain

Sara Merino AceitunoJoint work with Antoine Mellet (University of Maryland)

http://arxiv.org/abs/1411.5246

Imperial College London

9th November 2015.Kinetic theory with applications in physical sciences. University of

Maryland, CSCAMM

Page 2: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Anomalous

Energy transport in a solid

(1D-2D)

MACROSCOPIC MODEL:Fractional

Heat equation

∂tρ(t, x) = D∆xρ(t, x)

{∂tρ(t, x) = −∇x · ~(t, x) (mass cons.)

~(t, x) = − D∇xρ(t, x) (Fourier law)

MICROSCOPIC MODEL: FPU lattice/chain (Fermi Pasta Ulam)

H(q, p) =1

2

∑i∈Z

p2i + Vh(q) +

√λV (q)

(−∆x)γ/2ρ = F−1 (|k |γF(ρ)(k))

Page 3: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Anomalous

Energy transport in a solid

(1D-2D)

MACROSCOPIC MODEL:

Fractional

Heat equation

∂tρ(t, x) = D∆xρ(t, x)

{∂tρ(t, x) = −∇x · ~(t, x) (mass cons.)

~(t, x) = − D∇xρ(t, x) (Fourier law)

MICROSCOPIC MODEL: FPU lattice/chain (Fermi Pasta Ulam)

H(q, p) =1

2

∑i∈Z

p2i + Vh(q) +

√λV (q)

(−∆x)γ/2ρ = F−1 (|k |γF(ρ)(k))

Page 4: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Anomalous

Energy transport in a solid

(1D-2D)

MACROSCOPIC MODEL:

Fractional

Heat equation

∂tρ(t, x) = D∆xρ(t, x)

{∂tρ(t, x) = −∇x · ~(t, x) (mass cons.)

~(t, x) = − D∇xρ(t, x) (Fourier law)

MICROSCOPIC MODEL:

FPU lattice/chain (Fermi Pasta Ulam)

H(q, p) =1

2

∑i∈Z

p2i + Vh(q) +

√λV (q)

(−∆x)γ/2ρ = F−1 (|k |γF(ρ)(k))

Page 5: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Anomalous

Energy transport in a solid

(1D-2D)

MACROSCOPIC MODEL:

Fractional

Heat equation

∂tρ(t, x) = D∆xρ(t, x)

{∂tρ(t, x) = −∇x · ~(t, x) (mass cons.)

~(t, x) = − D∇xρ(t, x) (Fourier law)

MICROSCOPIC MODEL: FPU lattice/chain (Fermi Pasta Ulam)

H(q, p) =1

2

∑i∈Z

p2i + Vh(q) +

√λV (q)

(−∆x)γ/2ρ = F−1 (|k |γF(ρ)(k))

Page 6: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Anomalous

Energy transport in a solid

(1D-2D)

MACROSCOPIC MODEL:

Fractional

Heat equation

∂tρ(t, x) = D∆xρ(t, x)

{∂tρ(t, x) = −∇x · ~(t, x) (mass cons.)

~(t, x) = − D∇xρ(t, x) (Fourier law)

MICROSCOPIC MODEL: FPU lattice/chain (Fermi Pasta Ulam)

H(q, p) =1

2

∑i∈Z

p2i + Vh(q) +

√λV (q)

*Bonetto et al. [2000]

(−∆x)γ/2ρ = F−1 (|k |γF(ρ)(k))

Page 7: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Anomalous energy transport in a solid (1D-2D)

MACROSCOPIC MODEL:Fractional heat equation

∂tρ(t, x) = −D(−∆x)γ/2ρ(t, x)

{∂tρ(t, x) = −∇x · ~(t, x) (mass cons.)

~(t, x) = Anomalous Fourier Law

MICROSCOPIC MODEL: FPU-β lattice/chain (Fermi Pasta Ulam)

H(q, p) =1

2

∑i∈Z

p2i +

1

2

∑i∈Z

(qi+1 − qi )2 + β

∑i∈Z

(qi+1 − qi )4

(−∆x)γ/2ρ = F−1 (|k |γF(ρ)(k))

Page 8: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Previous results

MICROFPU chain/lattice

MACRO(Anomalous)

diffusion

Harmonic potentialNo relaxation to eq.

Cubic potential(FPU-alpha chain)

Quartic potential(FPU-beta chain)

Harmonic potential + noise conserving

momentum and energy

On site potential:- classic diffusion(Aoki, Lukkarinen, Spohn)

Next neighbour pot.:- 1d, 2d anomalous dif.- 3d or higher, classic dif.(Lepri, Livi, Politi)

(Olla, Basile, et.al)

Fractional heat equation

Heat equationNonlinear heat equation

Page 9: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Previous results

MICROFPU chain/lattice

KINETIC

MACRO(Anomalous)

diffusion

Harmonic potentialNo relaxation to eq.

Cubic potential(FPU-alpha chain)

Quartic potential(FPU-beta chain)

Harmonic potential + noise conserving

momentum and energy

On site potential:- classic diffusion(Aoki, Lukkarinen, Spohn)

Next neighbour pot.:- 1d, 2d anomalous dif.- 3d or higher, classic dif.(Lepri, Livi, Politi)

(Olla, Basile, et.al)

Fractional heat equation

Heat equationNonlinear heat equation

Heat – lattice vibrations – phonons

Spohn (formal)

Page 10: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Previous results

MICROFPU chain/lattice

KINETIC

MACRO(Anomalous)

diffusion

Harmonic potentialNo relaxation to eq.

Cubic potential(FPU-alpha chain)

Quartic potential(FPU-beta chain)

Harmonic potential + noise conserving

momentum and energy

On site potential:- classic diffusion(Aoki, Lukkarinen, Spohn)

Next neighbour pot.:- 1d, 2d anomalous dif.- 3d or higher, classic dif.(Lepri, Livi, Politi)

Free transport eq.C(W)=0

3-phonon Boltz. eq.C(W): bilinear op.

(we lose info in the kinetic limit; suggests weak mixing at micro

level)

next neighbour pot.

(Olla, Basile, et.al)

Fractional heat equation

Heat equationNonlinear heat equation

Heat – lattice vibrations – phonons

Spohn (formal)

Page 11: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Previous results

MICROFPU chain/lattice

KINETIC

MACRO(Anomalous)

diffusion

Harmonic potentialNo relaxation to eq.

Cubic potential(FPU-alpha chain)

Quartic potential(FPU-beta chain)

Harmonic potential + noise conserving

momentum and energy

On site potential:- classic diffusion(Aoki, Lukkarinen, Spohn)

Next neighbour pot.:- 1d, 2d anomalous dif.- 3d or higher, classic dif.(Lepri, Livi, Politi)

Free transport eq.C(W)=0

3-phonon Boltz. eq.C(W): bilinear op.

(we lose info in the kinetic limit; suggests weak mixing at micro

level)

next neighbour pot.

4-phonon Boltz. eq.C(W): bilinear op.

(Olla, Basile, et.al)

Linear Boltzmann Equation

Fractional heat equation

Heat equationNonlinear heat equation

(Bricmont -Kupianen)

(Olla, et. al.)(Mellet, Mouhot, Mischler, et. al.)

Linearised eq.(Lukkarinen, Spohn)

Heat – lattice vibrations – phonons

Spohn (formal)

(Basile, Olla, Spohn)

Page 12: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =

∑σ1,σ2,σ3=±1

∫T

∫T

∫T

F (k, k1, k2, k3)2

× δ(k + σ1k1 + σ2k2 + σ3k3)δ(ω + σ1ω1 + σ2ω2 + σ3ω3

× (W1W2W3 + W (σ1W2W3 + W1σ2W3 + W1W2σ3)) dk1 dk2 dk3

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 13: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =

∑σ1,σ2,σ3=±1

∫T

∫T

∫T

F (k, k1, k2, k3)2

× δ(k + σ1k1 + σ2k2 + σ3k3)δ(ω + σ1ω1 + σ2ω2 + σ3ω3

× (W1W2W3 + W (σ1W2W3 + W1σ2W3 + W1W2σ3))

dk1 dk2 dk3

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 14: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =∑

σ1,σ2,σ3=±1

∫T

∫T

∫T

F (k, k1, k2, k3)2

× δ(k + σ1k1 + σ2k2 + σ3k3)δ(ω + σ1ω1 + σ2ω2 + σ3ω3

× (W1W2W3 + W (σ1W2W3 + W1σ2W3 + W1W2σ3))

dk1 dk2 dk3

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 15: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =∑

σ1,σ2,σ3=±1

∫T

∫T

∫T

F (k , k1, k2, k3)2

× δ(k + σ1k1 + σ2k2 + σ3k3)δ(ω + σ1ω1 + σ2ω2 + σ3ω3

×(W1W2W3 + W (σ1W2W3 + W1σ2W3 + W1W2σ3)) dk1 dk2 dk3

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 16: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =∑

σ1,σ2,σ3=±1

∫T

∫T

∫T

F (k, k1, k2, k3)2

× δ(k + σ1k1 + σ2k2 + σ3k3)δ(ω + σ1ω1 + σ2ω2 + σ3ω3

× (W1W2W3 + W (σ1W2W3 + W1σ2W3 + W1W2σ3)) dk1 dk2 dk3

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 17: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =∑

σ1,σ2,σ3=±1

∫T

∫T

∫T

F (k, k1, k2, k3)2

×δ(k + σ1k1 + σ2k2 + σ3k3)δ(ω + σ1ω1 + σ2ω2 + σ3ω3)

× (W1W2W3 + W (σ1W2W3 + W1σ2W3 + W1W2σ3)) dk1 dk2 dk3

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 18: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =

∫ ∫ ∫F 2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[W1W2W3 + WW2W3 −WW1W3 −WW1W2] dk1 dk2 dk3.

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 19: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

∂tW (t, x , k) + ω′(k)∂xW (t, x , k) = C (W ) (t, x , k) ∈ (0,∞)× R× T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =

∫ ∫ ∫F 2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[W1W2W3 + WW2W3 −WW1W3 −WW1W2] dk1 dk2 dk3.

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 20: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

The kinetic model: the Boltzmann-phonon equation

The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation,properties and link to weakly anharmonic lattice dynamics’) - Peierls.

εα∂tWε(t, x , k)+εω′(k)∂xW ε(t, x , k) = C (W ε) (t, x , k) ∈ (0,∞)×R×T

k : wavenumber, ω(k): dispersion relation.

Four phonon-collision operator

C (W ) =

∫ ∫ ∫F 2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[W1W2W3 + WW2W3 −WW1W3 −WW1W2] dk1 dk2 dk3.

with

F := F (k, k1, k2, k3)2 =3∏

i=0

2 sin2(πki )

ω(ki ).

Equilibrium:1

βω + α; Cons.:

∫T

W dk = cte,

∫Tω(k)W dk = cte

Page 21: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Linearised 4-phonon Boltzmann equation

Around W ε = 1β0ω(k)

(1 + εf ε), L(f ) = W−10 DC (W0)(W0f ):

εγ

∂t f (t, x , k) +

ε

ω′(k)∂x f (t, x , k) = L(f ) (t, x , k) ∈ (0,∞)×R× [0, 1]

forω(k) = sin(πk) (dispersion relation)

and

L(f ) = ω(k)

∫ ∫ ∫δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3. (1)

’Regularise’ L (Lukkarinen and Spohn [2008])

L(f ) = K (f )−Vf Ker L = span {1, ω−1};∫T

f dk = cte,

∫Tω−1f dk = cte

Proposition (Lukkarinen and Spohn [2008])

c1| sin(πk)|5/3 ≤ V (k) ≤ c2| sin(πk)|5/3 c1, c2 > 0

Page 22: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Linearised 4-phonon Boltzmann equation

Around W ε = 1β0ω(k)

(1 + εf ε), L(f ) = W−10 DC (W0)(W0f ):

εγ

∂t f (t, x , k) +

ε

ω′(k)∂x f (t, x , k) = L(f ) (t, x , k) ∈ (0,∞)×R× [0, 1]

forω(k) = sin(πk) (dispersion relation)

and

L(f ) = ω(k)

∫ ∫ ∫δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3. (1)

’Regularise’ L (Lukkarinen and Spohn [2008])

L(f ) = K (f )−Vf Ker L = span {1, ω−1};∫T

f dk = cte,

∫Tω−1f dk = cte

Proposition (Lukkarinen and Spohn [2008])

c1| sin(πk)|5/3 ≤ V (k) ≤ c2| sin(πk)|5/3 c1, c2 > 0

Page 23: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Linearised 4-phonon Boltzmann equation

Around W ε = 1β0ω(k)

(1 + εf ε), L(f ) = W−10 DC (W0)(W0f ):

εγ

∂t f (t, x , k) +

ε

ω′(k)∂x f (t, x , k) = L(f ) (t, x , k) ∈ (0,∞)×R× [0, 1]

forω(k) = sin(πk) (dispersion relation)

and

L(f ) = ω(k)

∫ ∫ ∫δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3. (1)

’Regularise’ L (Lukkarinen and Spohn [2008])

L(f ) = K (f )−Vf Ker L = span {1, ω−1};∫T

f dk = cte,

∫Tω−1f dk = cte

Proposition (Lukkarinen and Spohn [2008])

c1| sin(πk)|5/3 ≤ V (k) ≤ c2| sin(πk)|5/3 c1, c2 > 0

Page 24: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Linearised 4-phonon Boltzmann equation

Around W ε = 1β0ω(k)

(1 + εf ε), L(f ) = W−10 DC (W0)(W0f ):

εγ

∂t f (t, x , k) +

ε

ω′(k)∂x f (t, x , k) = L(f ) (t, x , k) ∈ (0,∞)×R× [0, 1]

forω(k) = sin(πk) (dispersion relation)

and

L(f ) = ω(k)

∫ ∫ ∫δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3. (1)

’Regularise’ L (Lukkarinen and Spohn [2008])

L(f ) = K (f )−Vf Ker L = span {1, ω−1};∫T

f dk = cte,

∫Tω−1f dk = cte

Proposition (Lukkarinen and Spohn [2008])

c1| sin(πk)|5/3 ≤ V (k) ≤ c2| sin(πk)|5/3 c1, c2 > 0

Page 25: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Linearised 4-phonon Boltzmann equation

Around W ε = 1β0ω(k)

(1 + εf ε), L(f ) = W−10 DC (W0)(W0f ):

εγ∂t f (t, x , k) +εω′(k)∂x f (t, x , k) = L(f ) (t, x , k) ∈ (0,∞)×R× [0, 1]

forω(k) = sin(πk) (dispersion relation)

and

L(f ) = ω(k)

∫ ∫ ∫δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)

[ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3. (1)

’Regularise’ L (Lukkarinen and Spohn [2008])

L(f ) = K (f )−Vf Ker L = span {1, ω−1};∫T

f dk = cte,

∫Tω−1f dk = cte

Proposition (Lukkarinen and Spohn [2008])

c1| sin(πk)|5/3 ≤ V (k) ≤ c2| sin(πk)|5/3 c1, c2 > 0

Page 26: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

How does fractional phenomena occur?

ε2∂t fε + εω′(k)∂x f ε = L(f ε) (= K (f ε)− V (k)f ε)

As ε→ 0, f 0 ∈ ker L = span{1, ω−1} so f ε → T (t, x)

∂t

∫f ε dk︸ ︷︷ ︸T ε

+ ε−1∂x

∫ω′(k)f ε dk︸ ︷︷ ︸

=: I

= 0

I := ε−1∂x

∫L−1(ω′(k))L(f ε) dk

= ∂x

∫RN

L−1(ω′(k))ω′(k)∂x f ε dk + ε∂x

∫L−1(ω′(k))∂t f

ε dk

→ −κ∂2xT (t, x)

where

κ =

∫L−1(ω′(k))ω′(k)dk ≈ −

∫ 1

0

(ω′(k))2

Vdk

=∞, V ∼ sin(πk)5/3

Page 27: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

How does fractional phenomena occur?

ε2∂t fε + εω′(k)∂x f ε = L(f ε) (= K (f ε)− V (k)f ε)

As ε→ 0, f 0 ∈ ker L = span{1, ω−1} so f ε → T (t, x)

∂t

∫f ε dk︸ ︷︷ ︸T ε

+ ε−1∂x

∫ω′(k)f ε dk︸ ︷︷ ︸

=: I

= 0

I := ε−1∂x

∫L−1(ω′(k))L(f ε) dk

= ∂x

∫RN

L−1(ω′(k))ω′(k)∂x f ε dk + ε∂x

∫L−1(ω′(k))∂t f

ε dk

→ −κ∂2xT (t, x)

where

κ =

∫L−1(ω′(k))ω′(k)dk ≈ −

∫ 1

0

(ω′(k))2

Vdk

=∞, V ∼ sin(πk)5/3

Page 28: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

How does fractional phenomena occur?

ε2∂t fε + εω′(k)∂x f ε = L(f ε) (= K (f ε)− V (k)f ε)

As ε→ 0, f 0 ∈ ker L = span{1, ω−1} so f ε → T (t, x)

∂t

∫f ε dk︸ ︷︷ ︸T ε

+ ε−1∂x

∫ω′(k)f ε dk︸ ︷︷ ︸

=: I

= 0

I := ε−1∂x

∫L−1(ω′(k))L(f ε) dk

= ∂x

∫RN

L−1(ω′(k))ω′(k)∂x f ε dk + ε∂x

∫L−1(ω′(k))∂t f

ε dk

→ −κ∂2xT (t, x)

where

κ =

∫L−1(ω′(k))ω′(k)dk ≈ −

∫ 1

0

(ω′(k))2

Vdk

=∞, V ∼ sin(πk)5/3

Page 29: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

How does fractional phenomena occur?

ε2∂t fε + εω′(k)∂x f ε = L(f ε) (= K (f ε)− V (k)f ε)

As ε→ 0, f 0 ∈ ker L = span{1, ω−1} so f ε → T (t, x)

∂t

∫f ε dk︸ ︷︷ ︸T ε

+ ε−1∂x

∫ω′(k)f ε dk︸ ︷︷ ︸

=: I

= 0

I := ε−1∂x

∫L−1(ω′(k))L(f ε) dk

= ∂x

∫RN

L−1(ω′(k))ω′(k)∂x f ε dk + ε∂x

∫L−1(ω′(k))∂t f

ε dk

→ −κ∂2xT (t, x)

where

κ =

∫L−1(ω′(k))ω′(k)dk ≈ −

∫ 1

0

(ω′(k))2

Vdk

=∞, V ∼ sin(πk)5/3

Page 30: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

How does fractional phenomena occur?

ε2∂t fε + εω′(k)∂x f ε = L(f ε) (= K (f ε)− V (k)f ε)

As ε→ 0, f 0 ∈ ker L = span{1, ω−1} so f ε → T (t, x)

∂t

∫f ε dk︸ ︷︷ ︸T ε

+ ε−1∂x

∫ω′(k)f ε dk︸ ︷︷ ︸

=: I

= 0

I := ε−1∂x

∫L−1(ω′(k))L(f ε) dk

= ∂x

∫RN

L−1(ω′(k))ω′(k)∂x f ε dk + ε∂x

∫L−1(ω′(k))∂t f

ε dk

→ −κ∂2xT (t, x)

where

κ =

∫L−1(ω′(k))ω′(k)dk ≈ −

∫ 1

0

(ω′(k))2

Vdk

=∞, V ∼ sin(πk)5/3

Page 31: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

How does fractional phenomena occur?

ε2∂t fε + εω′(k)∂x f ε = L(f ε) (= K (f ε)− V (k)f ε)

As ε→ 0, f 0 ∈ ker L = span{1, ω−1} so f ε → T (t, x)

∂t

∫f ε dk︸ ︷︷ ︸T ε

+ ε−1∂x

∫ω′(k)f ε dk︸ ︷︷ ︸

=: I

= 0

I := ε−1∂x

∫L−1(ω′(k))L(f ε) dk

= ∂x

∫RN

L−1(ω′(k))ω′(k)∂x f ε dk + ε∂x

∫L−1(ω′(k))∂t f

ε dk

→ −κ∂2xT (t, x)

where

κ =

∫L−1(ω′(k))ω′(k)dk ≈ −

∫ 1

0

(ω′(k))2

Vdk =∞, V ∼ sin(πk)5/3

Page 32: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Note: formal restriction to the linearised case

εα∂tWε + εω′(k)∂xW ε = C (W ε),

Linearise

W ε = W0(1 + εf ε) where W0 =1

β0ω(k), β0 > 0.

εα∂t fε + εω′(k)∂x f ε = L(f ε) + ε

1

2Q(f ε, f ε) + ε2

1

6R(f ε, f ε, f ε). (2)

Proceeding as before

∂tTε + ∂x

1

ε

∫ω′(k)f ε dk = 0

ε−1∫ω′f εdk =

∫L−1(ω′)ω′∂x f εdk −

∫L−1(ω′)Q(f ε, f ε)dk +O(ε).

But Q(T ,T ) = 0.

Page 33: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Note: formal restriction to the linearised case

εα∂tWε + εω′(k)∂xW ε = C (W ε),

Linearise

W ε = W0(1 + εf ε) where W0 =1

β0ω(k), β0 > 0.

εα∂t fε + εω′(k)∂x f ε = L(f ε) + ε

1

2Q(f ε, f ε) + ε2

1

6R(f ε, f ε, f ε). (2)

Proceeding as before

∂tTε + ∂x

1

ε

∫ω′(k)f ε dk = 0

ε−1∫ω′f εdk =

∫L−1(ω′)ω′∂x f εdk −

∫L−1(ω′)Q(f ε, f ε)dk +O(ε).

But Q(T ,T ) = 0.

Page 34: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Note: formal restriction to the linearised case

εα∂tWε + εω′(k)∂xW ε = C (W ε),

Linearise

W ε = W0(1 + εf ε) where W0 =1

β0ω(k), β0 > 0.

εα∂t fε + εω′(k)∂x f ε = L(f ε) + ε

1

2Q(f ε, f ε) + ε2

1

6R(f ε, f ε, f ε). (2)

Proceeding as before

∂tTε + ∂x

1

ε

∫ω′(k)f ε dk = 0

ε−1∫ω′f εdk =

∫L−1(ω′)ω′∂x f εdk −

∫L−1(ω′)Q(f ε, f ε)dk +O(ε).

But Q(T ,T ) = 0.

Page 35: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Difficulties

It is not a linear Boltzmann equation (or radiative transfer equation);

∂t f + ω′(k)∂x f = L2(f )− L1(f ), Li Boltzmann operator

not a maximum principle.

No spectral gap.

The kernel is too big {1, ω−1}: spurious term.

T =

∫T

f (k) dk (energy), S =

∫T

f (k)

ω(k)dk (spurious term)

I S : fractional phenomena! Should reach equilibrium faster...

I Intuition: f ∈ L2 but ω−1 /∈ L2.

Page 36: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Difficulties

It is not a linear Boltzmann equation (or radiative transfer equation);

∂t f + ω′(k)∂x f = L2(f )− L1(f ), Li Boltzmann operator

not a maximum principle.

No spectral gap.

The kernel is too big {1, ω−1}: spurious term.

T =

∫T

f (k) dk (energy), S =

∫T

f (k)

ω(k)dk (spurious term)

I S : fractional phenomena! Should reach equilibrium faster...

I Intuition: f ∈ L2 but ω−1 /∈ L2.

Page 37: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Difficulties

It is not a linear Boltzmann equation (or radiative transfer equation);

∂t f + ω′(k)∂x f = L2(f )− L1(f ), Li Boltzmann operator

not a maximum principle.

No spectral gap.

The kernel is too big {1, ω−1}: spurious term.

T =

∫T

f (k) dk (energy), S =

∫T

f (k)

ω(k)dk (spurious term)

I S : fractional phenomena! Should reach equilibrium faster...

I Intuition: f ∈ L2 but ω−1 /∈ L2.

Page 38: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Difficulties

It is not a linear Boltzmann equation (or radiative transfer equation);

∂t f + ω′(k)∂x f = L2(f )− L1(f ), Li Boltzmann operator

not a maximum principle.

No spectral gap.

The kernel is too big {1, ω−1}: spurious term.

T =

∫T

f (k) dk (energy), S =

∫T

f (k)

ω(k)dk (spurious term)

I S : fractional phenomena! Should reach equilibrium faster...I Intuition: f ∈ L2 but ω−1 /∈ L2.

Page 39: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Theorem (Fractional diffusion limit for the linearised equation)

Let f ε be a solution of ε8/5∂t fε + εω′(k)∂x f ε = L(f ε) (BPE) with initial

data f0 ∈ L2(R× T). Then, up to a subsequence,

f ε(t, x , k) ⇀ T (t, x) L∞((0,∞); L2(x , k))-weak

where T is the weak limit of

T ε =1

〈V 〉

∫ 1

0V (k)f ε(k)dk

and solves the fractional diffusion equation

∂tT + κ(−∆x)4/5T = 0 in (0,∞)× R

κ ∈ (0,∞) with initial condition

T (0, x) =1

〈V 〉

∫ 1

0Vf0(t, x , k) dk.

Page 40: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (I)

f ε(t, x , k) = T ε(t, x)︸ ︷︷ ︸energy

+ Sε(t, x)ω(k)−1︸ ︷︷ ︸spurious term

+ ε45 hε(t, x , k)︸ ︷︷ ︸remainder

T ε ∈ L∞(0,∞; L2(R)),

hε ∈ L2V (T× R)

Proposition

The function S̃ε converges in distribution sense to

S̃(t, x) = −κ2κ3

(−∆)3/10T (t, x).

Page 41: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (I)

f ε(t, x , k) = T ε(t, x)︸ ︷︷ ︸energy

+ ε35 S̃ε(t, x)ω(k)−1︸ ︷︷ ︸spurious term

+ ε45 hε(t, x , k)︸ ︷︷ ︸remainder

T ε ∈ L∞(0,∞; L2(R)),

hε ∈ L2V (T× R)

Proposition

The function S̃ε converges in distribution sense to

S̃(t, x) = −κ2κ3

(−∆)3/10T (t, x).

Page 42: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (I)

f ε(t, x , k) = T ε(t, x)︸ ︷︷ ︸energy

+ ε35 S̃ε(t, x)ω(k)−1︸ ︷︷ ︸spurious term

+ ε45 hε(t, x , k)︸ ︷︷ ︸remainder

T ε ∈ L∞(0,∞; L2(R)),

hε ∈ L2V (T× R)

Proposition

The function S̃ε converges in distribution sense to

S̃(t, x) = −κ2κ3

(−∆)3/10T (t, x).

Page 43: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (II)

ε8/5∂t fε + εω′(k)∂x f ε = L(f ε)

• Project on the constant mode:

∂tT + κ1(−∆)4/5T + κ2(−∆)1/2S̃ = 0.

• Project on ω(k)−1:

κ2(−∆)1/2T + κ3(−∆)1/5S̃ = 0.

• ObtainS̃(t, x) = −κ2

κ3(−∆)3/10T (t, x).

and

∂tT +

(κ1 −

κ22κ3

)(−∆)4/5T = 0

Page 44: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (II)

ε8/5∂t fε + εω′(k)∂x f ε = L(f ε)

• Project on the constant mode:

∂tT + κ1(−∆)4/5T + κ2(−∆)1/2S̃ = 0.

• Project on ω(k)−1:

κ2(−∆)1/2T + κ3(−∆)1/5S̃ = 0.

• ObtainS̃(t, x) = −κ2

κ3(−∆)3/10T (t, x).

and

∂tT +

(κ1 −

κ22κ3

)(−∆)4/5T = 0

Page 45: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (II)

ε8/5∂t fε + εω′(k)∂x f ε = L(f ε)

• Project on the constant mode:

∂tT + κ1(−∆)4/5T + κ2(−∆)1/2S̃ = 0.

• Project on ω(k)−1:

κ2(−∆)1/2T + κ3(−∆)1/5S̃ = 0.

• ObtainS̃(t, x) = −κ2

κ3(−∆)3/10T (t, x).

and

∂tT +

(κ1 −

κ22κ3

)(−∆)4/5T = 0

Page 46: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (II)

ε8/5∂t fε + εω′(k)∂x f ε = L(f ε)

• Project on the constant mode:

∂tT + κ1(−∆)4/5T + κ2(−∆)1/2S̃ = 0.

• Project on ω(k)−1:

κ2(−∆)1/2T + κ3(−∆)1/5S̃ = 0.

• ObtainS̃(t, x) = −κ2

κ3(−∆)3/10T (t, x).

and

∂tT +

(κ1 −

κ22κ3

)(−∆)4/5T = 0

Page 47: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (II)

ε8/5∂t fε + εω′(k)∂x f ε = L(f ε)

• Project on the constant mode:

∂tT + κ1(−∆)4/5T + κ2(−∆)1/2S̃ = 0.

• Project on ω(k)−1:

κ2(−∆)1/2T + κ3(−∆)1/5S̃ = 0.

• ObtainS̃(t, x) = −κ2

κ3(−∆)3/10T (t, x).

and

∂tT +

(κ1 −

κ22κ3

)(−∆)4/5T = 0

Page 48: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (III)

Method: Laplace-Fourier Transform Method (like in Mellet, Mischler,Mouhot, ‘Fractional diffusion limit for collisional kinetic equations’).

f̂ ε(p, ξ, k) =

∫R

∫ ∞0

e−pte−iξx f ε(t, x , k) dt dx .

εαpf̂ ε − εαf̂0 + iεω′(k)ξ f̂ ε = K (f̂ ε)− V f̂ ε

We recall that L(f ) = K (f )− Vf with K (f ) =∫

K (k , k ′)f (k ′) dk ′. Thefact that

∫L(f ) dk = 0 and

∫1

ω(k)L(f ) dk = 0 for all f implies

V (k) =

∫K (k ′, k)dk ′,

V (k)

ω(k)=

∫K (k ′, k)

1

ω(k ′)dk ′

Multiplying the equation by K (k ′, k) and integrating with respect to k andk ′, we get...

Page 49: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Comments on the proof and results (III)

Method: Laplace-Fourier Transform Method (like in Mellet, Mischler,Mouhot, ‘Fractional diffusion limit for collisional kinetic equations’).

f̂ ε(p, ξ, k) =

∫R

∫ ∞0

e−pte−iξx f ε(t, x , k) dt dx .

εαpf̂ ε − εαf̂0 + iεω′(k)ξ f̂ ε = K (f̂ ε)− V f̂ ε

We recall that L(f ) = K (f )− Vf with K (f ) =∫

K (k , k ′)f (k ′) dk ′. Thefact that

∫L(f ) dk = 0 and

∫1

ω(k)L(f ) dk = 0 for all f implies

V (k) =

∫K (k ′, k)dk ′,

V (k)

ω(k)=

∫K (k ′, k)

1

ω(k ′)dk ′

Multiplying the equation by K (k ′, k) and integrating with respect to k andk ′, we get...

Page 50: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

• On the constant mode:

Fε1(f̂ 0) + aε1(p, ξ)T̂ ε(p, ξ) + aε2(p, ξ)̂̃Sε(p, ξ) + Rε1(p, ξ) = 0, (3)

in the limit

T̂0(ξ) + (−p − κ1|ξ|8/5)T̂ (p, ξ)− κ2|ξ|̂̃S = 0.

• ‘Projection’ on ω(k)−1:

Fε2(f̂ 0) + aε2(p, ξ)T̂ ε(p, ξ) + aε3(p, ξ)̂̃Sε(p, ξ) + Rε2(p, ξ) = 0, (4)

in the limit−κ2|ξ|T̂ (p, ξ)− κ3|ξ|2/5 ̂̃S = 0.

Page 51: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

• On the constant mode:

Fε1(f̂ 0) + aε1(p, ξ)T̂ ε(p, ξ) + aε2(p, ξ)̂̃Sε(p, ξ) + Rε1(p, ξ) = 0, (3)

in the limit

T̂0(ξ) + (−p − κ1|ξ|8/5)T̂ (p, ξ)− κ2|ξ|̂̃S = 0.

• ‘Projection’ on ω(k)−1:

Fε2(f̂ 0) + aε2(p, ξ)T̂ ε(p, ξ) + aε3(p, ξ)̂̃Sε(p, ξ) + Rε2(p, ξ) = 0, (4)

in the limit−κ2|ξ|T̂ (p, ξ)− κ3|ξ|2/5 ̂̃S = 0.

Page 52: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

• On the constant mode:

Fε1(f̂ 0) + aε1(p, ξ)T̂ ε(p, ξ) + aε2(p, ξ)̂̃Sε(p, ξ) + Rε1(p, ξ) = 0, (3)

in the limit

T̂0(ξ) + (−p − κ1|ξ|8/5)T̂ (p, ξ)− κ2|ξ|̂̃S = 0.

• ‘Projection’ on ω(k)−1:

Fε2(f̂ 0) + aε2(p, ξ)T̂ ε(p, ξ) + aε3(p, ξ)̂̃Sε(p, ξ) + Rε2(p, ξ) = 0, (4)

in the limit−κ2|ξ|T̂ (p, ξ)− κ3|ξ|2/5 ̂̃S = 0.

Page 53: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

• On the constant mode:

Fε1(f̂ 0) + aε1(p, ξ)T̂ ε(p, ξ) + aε2(p, ξ)̂̃Sε(p, ξ) + Rε1(p, ξ) = 0, (3)

in the limit

T̂0(ξ) + (−p − κ1|ξ|8/5)T̂ (p, ξ)− κ2|ξ|̂̃S = 0.

• ‘Projection’ on ω(k)−1:

Fε2(f̂ 0) + aε2(p, ξ)T̂ ε(p, ξ) + aε3(p, ξ)̂̃Sε(p, ξ) + Rε2(p, ξ) = 0, (4)

in the limit−κ2|ξ|T̂ (p, ξ)− κ3|ξ|2/5 ̂̃S = 0.

Page 54: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

MICROFPU chain/lattice

KINETIC

MACRO(Anomalous)

diffusion

Harmonic potentialNo relaxation to eq.

Cubic potential(FPU-alpha chain)

Quartic potential(FPU-beta chain)

Harmonic potential + noise conserving

momentum and energy

On site potential:- classic diffusion(Aoki, Lukkarinen, Spohn)

Next neighbour pot.:- 1d, 2d anomalous dif.- 3d or higher, classic dif.(Lepri, Livi, Politi)

Free transport eq.C(W)=0

3-phonon Boltz. eq.C(W): bilinear op.

(we lose info in the kinetic limit; suggests weak mixing at micro

level)

next neighbour pot.

4-phonon Boltz. eq.C(W): bilinear op.

(Olla, Basile, et.al)

Linear Boltzmann Equation

Fractional heat equation

Heat equationNonlinear heat equation

(Bricmont -Kupianen)

(Olla, et. al.)(Mellet, Mouhot, Mischler, et. al.)

Linearised eq.(Lukkarinen, Spohn)

Heat – lattice vibrations – phonons

Spohn (formal)

(Basile, Olla, Spohn)

Page 55: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

MICROFPU chain/lattice

KINETIC

MACRO(Anomalous)

diffusion

Harmonic potentialNo relaxation to eq.

Cubic potential(FPU-alpha chain)

Quartic potential(FPU-beta chain)

Harmonic potential + noise conserving

momentum and energy

On site potential:- classic diffusion(Aoki, Lukkarinen, Spohn)

Next neighbour pot.:- 1d, 2d anomalous dif.- 3d or higher, classic dif.(Lepri, Livi, Politi)

Free transport eq.C(W)=0

3-phonon Boltz. eq.C(W): bilinear op.

(we lose info in the kinetic limit; suggests weak mixing at micro

level)

next neighbour pot.

4-phonon Boltz. eq.C(W): bilinear op.

Spurious conservation of nb. of phonons

(Olla, Basile, et.al)

Linear Boltzmann Equation

Fractional heat equation

Heat equationNonlinear heat equation

(Mellet, Merino)

(Bricmont -Kupianen)

(Olla, et. al.)(Mellet, Mouhot, Mischler, et. al.)

Weaker mixing; smaller coeff.; slows down convergence to eq.

Linearised eq.(Lukkarinen, Spohn)

Heat – lattice vibrations – phonons

Spohn (formal)

(Basile, Olla, Spohn)

Page 56: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Final Remarks

Relation between different results for the FPU-β chain:

I MICRO:

at the level of the FPU (numerics),

Average energy current: je(N) =T− − T+

N1−α , α ∼ 0.4

I KINETIC:

the results of Lukkarinen-Spohn,

limt→∞

t3/5C (t) = constant

I MACRO:

the order of the fractional laplacian (−∆)4/5.

‘Similar’ equations exist in other context:

I Weak wave turbulence (4-wave kinetic equation).I Nordheim equation.

Page 57: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Final Remarks

Relation between different results for the FPU-β chain:

I MICRO: at the level of the FPU (numerics),

Average energy current: je(N) =T− − T+

N1−α , α ∼ 0.4

I KINETIC:

the results of Lukkarinen-Spohn,

limt→∞

t3/5C (t) = constant

I MACRO:

the order of the fractional laplacian (−∆)4/5.

‘Similar’ equations exist in other context:

I Weak wave turbulence (4-wave kinetic equation).I Nordheim equation.

Page 58: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Final Remarks

Relation between different results for the FPU-β chain:

I MICRO: at the level of the FPU (numerics),

Average energy current: je(N) =T− − T+

N1−α , α ∼ 0.4

I KINETIC: the results of Lukkarinen-Spohn,

limt→∞

t3/5C (t) = constant

I MACRO:

the order of the fractional laplacian (−∆)4/5.

‘Similar’ equations exist in other context:

I Weak wave turbulence (4-wave kinetic equation).I Nordheim equation.

Page 59: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Final Remarks

Relation between different results for the FPU-β chain:

I MICRO: at the level of the FPU (numerics),

Average energy current: je(N) =T− − T+

N1−α , α ∼ 0.4

I KINETIC: the results of Lukkarinen-Spohn,

limt→∞

t3/5C (t) = constant

I MACRO: the order of the fractional laplacian (−∆)4/5.

‘Similar’ equations exist in other context:

I Weak wave turbulence (4-wave kinetic equation).I Nordheim equation.

Page 60: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

Final Remarks

Relation between different results for the FPU-β chain:

I MICRO: at the level of the FPU (numerics),

Average energy current: je(N) =T− − T+

N1−α , α ∼ 0.4

I KINETIC: the results of Lukkarinen-Spohn,

limt→∞

t3/5C (t) = constant

I MACRO: the order of the fractional laplacian (−∆)4/5.

‘Similar’ equations exist in other context:

I Weak wave turbulence (4-wave kinetic equation).I Nordheim equation.

Page 61: Anomalous energy transport in FPU- chainThe kinetic model: the Boltzmann-phonon equation The formal kinetic limit (Spohn, ‘The phonon Boltzmann equation, properties and link to weakly

References I

Federico Bonetto, Joel L Lebowitz, and Luc Rey-Bellet. Fourier’s law: Achallenge for theorists. arXiv preprint math-ph/0002052, 2000.

Jani Lukkarinen and Herbert Spohn. Anomalous energy transport in thefpu-β chain. Communications on Pure and Applied Mathematics, 61(12):1753–1786, 2008.