Anomalous energy transport in FPU-β chain

Transcription

1 Anomalous energy transport in FPU-β chain Sara Merino Aceituno Joint work with Antoine Mellet (University of Maryland) Imperial College London 9th November Kinetic theory with applications in physical sciences. University of Maryland, CSCAMM

2 Energy transport in a solid Heat equation t ρ(t, x) = D x ρ(t, x) { t ρ(t, x) = x j(t, x) (mass cons.) j(t, x) = D x ρ(t, x) (Fourier law)

3 Energy transport in a solid MACROSCOPIC MODEL: Heat equation t ρ(t, x) = D x ρ(t, x) { t ρ(t, x) = x j(t, x) (mass cons.) j(t, x) = D x ρ(t, x) (Fourier law)

4 Energy transport in a solid MACROSCOPIC MODEL: Heat equation t ρ(t, x) = D x ρ(t, x) MICROSCOPIC MODEL: { t ρ(t, x) = x j(t, x) (mass cons.) j(t, x) = D x ρ(t, x) (Fourier law)

5 Energy transport in a solid MACROSCOPIC MODEL: Heat equation t ρ(t, x) = D x ρ(t, x) { t ρ(t, x) = x j(t, x) (mass cons.) j(t, x) = D x ρ(t, x) (Fourier law) MICROSCOPIC MODEL: FPU lattice/chain (Fermi Pasta Ulam)

6 Energy transport in a solid MACROSCOPIC MODEL: Heat equation t ρ(t, x) = D x ρ(t, x) { t ρ(t, x) = x j(t, x) (mass cons.) j(t, x) = D x ρ(t, x) (Fourier law) MICROSCOPIC MODEL: FPU lattice/chain (Fermi Pasta Ulam) H(q, p) = 1 2 *Bonetto et al. [2000] pi 2 + V h (q) + λv (q) i Z

7 Anomalous energy transport in a solid (1D-2D) MACROSCOPIC MODEL: Fractional heat equation t ρ(t, x) = D( x ) γ/2 ρ(t, x) { t ρ(t, x) = x j(t, x) (mass cons.) j(t, x) = Anomalous Fourier Law MICROSCOPIC MODEL: FPU-β lattice/chain (Fermi Pasta Ulam) H(q, p) = 1 pi i+1 q i ) 2 2 i Z i Z(q 2 + β (q i+1 q i ) 4 i Z ( x ) γ/2 ρ = F 1 ( k γ F(ρ)(k))

8 Previous results MICRO FPU chain/lattice Harmonic potential No relaxation to eq. Cubic potential (FPU-alpha chain) On site potential: - classic diffusion (Aoki, Lukkarinen, Spohn) Quartic potential (FPU-beta chain) Next neighbour pot.: - 1d, 2d anomalous dif. - 3d or higher, classic dif. (Lepri, Livi, Politi) Harmonic potential + noise conserving momentum and energy (Olla, Basile, et.al) MACRO (Anomalous) diffusion Nonlinear heat equation Fractional heat equation Heat equation

9 Previous results MICRO FPU chain/lattice Spohn (formal) Heat lattice vibrations phonons Harmonic potential No relaxation to eq. Cubic potential (FPU-alpha chain) On site potential: - classic diffusion (Aoki, Lukkarinen, Spohn) Quartic potential (FPU-beta chain) Next neighbour pot.: - 1d, 2d anomalous dif. - 3d or higher, classic dif. (Lepri, Livi, Politi) Harmonic potential + noise conserving momentum and energy (Olla, Basile, et.al) KINETIC MACRO (Anomalous) diffusion Nonlinear heat equation Fractional heat equation Heat equation

10 Previous results MICRO FPU chain/lattice Spohn (formal) Heat lattice vibrations phonons KINETIC Harmonic potential No relaxation to eq. Free transport eq. C(W)=0 next neighbour pot. Cubic potential (FPU-alpha chain) 3-phonon Boltz. eq. C(W): bilinear op. (we lose info in the kinetic limit; suggests weak mixing at micro level) On site potential: - classic diffusion (Aoki, Lukkarinen, Spohn) Quartic potential (FPU-beta chain) Next neighbour pot.: - 1d, 2d anomalous dif. - 3d or higher, classic dif. (Lepri, Livi, Politi) Harmonic potential + noise conserving momentum and energy (Olla, Basile, et.al) MACRO (Anomalous) diffusion Nonlinear heat equation Fractional heat equation Heat equation

11 Previous results MICRO FPU chain/lattice Spohn (formal) Heat lattice vibrations phonons KINETIC Harmonic potential No relaxation to eq. Free transport eq. C(W)=0 next neighbour pot. Cubic potential (FPU-alpha chain) 3-phonon Boltz. eq. C(W): bilinear op. On site potential: - classic diffusion (Aoki, Lukkarinen, Spohn) Quartic potential (FPU-beta chain) 4-phonon Boltz. eq. C(W): bilinear op. Next neighbour pot.: - 1d, 2d anomalous dif. - 3d or higher, classic dif. (Lepri, Livi, Politi) Harmonic potential + noise conserving momentum and energy Linearised eq. (Lukkarinen, Spohn) (Olla, Basile, et.al) (Basile, Olla, Spohn) MACRO (Anomalous) diffusion (we lose info in the kinetic limit; suggests weak mixing at micro level) Nonlinear heat equation (Bricmont - Kupianen) Fractional heat equation (Olla, et. al.) (Mellet, Mouhot, Mischler, et. al.) Linear Boltzmann Equation Heat equation

12 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) k: wavenumber, ω(k): dispersion relation. (t, x, k) (0, ) R T

13 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) k: wavenumber, ω(k): dispersion relation. Four phonon-collision operator C(W ) = T T T (t, x, k) (0, ) R T dk 1 dk 2 dk 3

14 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) k: wavenumber, ω(k): dispersion relation. Four phonon-collision operator C(W ) = σ 1,σ 2,σ 3 =±1 T T T (t, x, k) (0, ) R T dk 1 dk 2 dk 3

15 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) k: wavenumber, ω(k): dispersion relation. Four phonon-collision operator C(W ) = σ 1,σ 2,σ 3 =±1 T T T (t, x, k) (0, ) R T (W 1 W 2 W 3 + W (σ 1 W 2 W 3 + W 1 σ 2 W 3 + W 1 W 2 σ 3 )) dk 1 dk 2 dk 3

16 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) k: wavenumber, ω(k): dispersion relation. Four phonon-collision operator C(W ) = σ 1,σ 2,σ 3 =±1 T T T F (k, k 1, k 2, k 3 ) 2 (t, x, k) (0, ) R T with (W 1 W 2 W 3 + W (σ 1 W 2 W 3 + W 1 σ 2 W 3 + W 1 W 2 σ 3 )) dk 1 dk 2 dk 3 F := F (k, k 1, k 2, k 3 ) 2 = 3 i=0 2 sin 2 (πk i ). ω(k i )

17 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) k: wavenumber, ω(k): dispersion relation. Four phonon-collision operator C(W ) = σ 1,σ 2,σ 3 =±1 T T T F (k, k 1, k 2, k 3 ) 2 (t, x, k) (0, ) R T δ(k + σ 1 k 1 + σ 2 k 2 + σ 3 k 3 )δ(ω + σ 1 ω 1 + σ 2 ω 2 + σ 3 ω 3 ) with (W 1 W 2 W 3 + W (σ 1 W 2 W 3 + W 1 σ 2 W 3 + W 1 W 2 σ 3 )) dk 1 dk 2 dk 3 F := F (k, k 1, k 2, k 3 ) 2 = 3 i=0 2 sin 2 (πk i ). ω(k i )

18 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) (t, x, k) (0, ) R T k: wavenumber, ω(k): dispersion relation. Four phonon-collision operator C(W ) = F 2 δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 ) [W 1 W 2 W 3 + WW 2 W 3 WW 1 W 3 WW 1 W 2 ] dk 1 dk 2 dk 3. with F := F (k, k 1, k 2, k 3 ) 2 = 3 i=0 2 sin 2 (πk i ). ω(k i )

19 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. t W (t, x, k) + ω (k) x W (t, x, k) = C(W ) (t, x, k) (0, ) R T k: wavenumber, ω(k): dispersion relation. Four phonon-collision operator C(W ) = F 2 δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 ) [W 1 W 2 W 3 + WW 2 W 3 WW 1 W 3 WW 1 W 2 ] dk 1 dk 2 dk 3. with Equilibrium: 3 F := F (k, k 1, k 2, k 3 ) 2 2 sin 2 (πk i ) =. ω(k i ) i=0 1 βω + α ; Cons.: W dk = cte, ω(k)w dk = cte T T

20 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. ε α t W ε (t, x, k)+εω (k) x W 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 2 δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 ) [W 1 W 2 W 3 + WW 2 W 3 WW 1 W 3 WW 1 W 2 ] dk 1 dk 2 dk 3. with Equilibrium: 3 F := F (k, k 1, k 2, k 3 ) 2 2 sin 2 (πk i ) =. ω(k i ) i=0 1 βω + α ; Cons.: W dk = cte, ω(k)w dk = cte T T

21 Linearised 4-phonon Boltzmann equation Around W ε = 1 β 0 ω(k) (1 + εf ε ), L(f ) = W 1 0 DC(W 0 )(W 0 f ): t f (t, x, k)+ ω (k) x f (t, x, k) = L(f ) (t, x, k) (0, ) R [0, 1] for ω(k) = sin(πk) (dispersion relation)

22 Linearised 4-phonon Boltzmann equation Around W ε = 1 β 0 ω(k) (1 + εf ε ), L(f ) = W 1 0 DC(W 0 )(W 0 f ): t f (t, x, k)+ ω (k) x f (t, x, k) = L(f ) (t, x, k) (0, ) R [0, 1] for and ω(k) = sin(πk) L(f ) = ω(k) (dispersion relation) δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 ) [ω 3 f 3 + ω 2 f 2 ω 1 f 1 ωf ] dk 1 dk 2 dk 3. (1)

23 Linearised 4-phonon Boltzmann equation Around W ε = 1 β 0 ω(k) (1 + εf ε ), L(f ) = W 1 0 DC(W 0 )(W 0 f ): t f (t, x, k)+ ω (k) x f (t, x, k) = L(f ) (t, x, k) (0, ) R [0, 1] for and ω(k) = sin(πk) L(f ) = ω(k) (dispersion relation) δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 ) [ω 3 f 3 + ω 2 f 2 ω 1 f 1 ωf ] dk 1 dk 2 dk 3. (1) Regularise L (Lukkarinen and Spohn [2008]) L(f ) = K(f ) Vf Ker L = span {1, ω 1 }; f dk = cte, T T ω 1 f dk = cte

24 Linearised 4-phonon Boltzmann equation Around W ε = 1 β 0 ω(k) (1 + εf ε ), L(f ) = W 1 0 DC(W 0 )(W 0 f ): t f (t, x, k)+ ω (k) x f (t, x, k) = L(f ) (t, x, k) (0, ) R [0, 1] for and ω(k) = sin(πk) L(f ) = ω(k) (dispersion relation) δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 ) [ω 3 f 3 + ω 2 f 2 ω 1 f 1 ωf ] dk 1 dk 2 dk 3. (1) Regularise L (Lukkarinen and Spohn [2008]) L(f ) = K(f ) Vf Ker L = span {1, ω 1 }; f dk = cte, Proposition (Lukkarinen and Spohn [2008]) T T ω 1 f dk = cte c 1 sin(πk) 5/3 V (k) c 2 sin(πk) 5/3 c 1, c 2 > 0

25 Linearised 4-phonon Boltzmann equation Around W ε = 1 β 0 ω(k) (1 + εf ε ), L(f ) = W 1 0 DC(W 0 )(W 0 f ): ε γ t f (t, x, k)+εω (k) x f (t, x, k) = L(f ) (t, x, k) (0, ) R [0, 1] for and ω(k) = sin(πk) L(f ) = ω(k) (dispersion relation) δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 ) [ω 3 f 3 + ω 2 f 2 ω 1 f 1 ωf ] dk 1 dk 2 dk 3. (1) Regularise L (Lukkarinen and Spohn [2008]) L(f ) = K(f ) Vf Ker L = span {1, ω 1 }; f dk = cte, Proposition (Lukkarinen and Spohn [2008]) T T ω 1 f dk = cte c 1 sin(πk) 5/3 V (k) c 2 sin(πk) 5/3 c 1, c 2 > 0

26 How does fractional phenomena occur? ε 2 t f ε + εω (k) x f ε = L(f ε ) (= K(f ε ) V (k)f ε )

27 How does fractional phenomena occur? ε 2 t f ε + 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)

28 How does fractional phenomena occur? ε 2 t f ε + 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 + ε 1 x ω (k)f ε dk = 0 } {{ } } {{ } T ε =: I

29 How does fractional phenomena occur? ε 2 t f ε + 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 + ε 1 x ω (k)f ε dk = 0 } {{ } } {{ } T ε =: I I := ε 1 x L 1 (ω (k))l(f ε ) dk = x R N L 1 (ω (k))ω (k) x f ε dk + ε x L 1 (ω (k)) t f ε dk

30 How does fractional phenomena occur? ε 2 t f ε + 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 + ε 1 x ω (k)f ε dk = 0 } {{ } } {{ } T ε =: I I := ε 1 x L 1 (ω (k))l(f ε ) dk = x R N L 1 (ω (k))ω (k) x f ε dk + ε x κ 2 x T (t, x) L 1 (ω (k)) t f ε dk where κ = 1 L 1 (ω (k))ω (k)dk 0 (ω (k)) 2 dk V

31 How does fractional phenomena occur? ε 2 t f ε + 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 + ε 1 x ω (k)f ε dk = 0 } {{ } } {{ } T ε =: I I := ε 1 x L 1 (ω (k))l(f ε ) dk = x R N L 1 (ω (k))ω (k) x f ε dk + ε x κ 2 x T (t, x) L 1 (ω (k)) t f ε dk where κ = 1 L 1 (ω (k))ω (k)dk 0 (ω (k)) 2 dk =, V sin(πk) 5/3 V

32 Note: formal restriction to the linearised case ε α t W ε + εω (k) x W ε = C(W ε ), Linearise W ε = W 0 (1 + εf ε ) where W 0 = 1 β 0 ω(k), β 0 > 0.

33 Note: formal restriction to the linearised case ε α t W ε + εω (k) x W ε = C(W ε ), Linearise W ε = W 0 (1 + εf ε ) where W 0 = 1 β 0 ω(k), β 0 > 0. ε α t f ε + εω (k) x f ε = L(f ε ) + ε 1 2 Q(f ε, f ε ) + ε R(f ε, f ε, f ε ). (2)

34 Note: formal restriction to the linearised case ε α t W ε + εω (k) x W ε = C(W ε ), Linearise W ε = W 0 (1 + εf ε ) where W 0 = 1 β 0 ω(k), β 0 > 0. ε α t f ε + εω (k) x f ε = L(f ε ) + ε 1 2 Q(f ε, f ε ) + ε R(f ε, f ε, f ε ). (2) Proceeding as before t T ε 1 + x ω (k)f ε dk = 0 ε ε 1 ω f ε dk = L 1 (ω )ω x f ε dk L 1 (ω )Q(f ε, f ε )dk + O(ε). But Q(T, T ) = 0.

35 Difficulties It is not a linear Boltzmann equation (or radiative transfer equation); t f + ω (k) x f = L 2 (f ) L 1 (f ), L i Boltzmann operator not a maximum principle.

36 Difficulties It is not a linear Boltzmann equation (or radiative transfer equation); t f + ω (k) x f = L 2 (f ) L 1 (f ), L i Boltzmann operator not a maximum principle. No spectral gap.

37 Difficulties It is not a linear Boltzmann equation (or radiative transfer equation); t f + ω (k) x f = L 2 (f ) L 1 (f ), L i Boltzmann operator not a maximum principle. No spectral gap. The kernel is too big {1, ω 1 }: spurious term. f (k) T = f (k) dk (energy), S = dk (spurious term) ω(k) T T S: fractional phenomena! Should reach equilibrium faster...

38 Difficulties It is not a linear Boltzmann equation (or radiative transfer equation); t f + ω (k) x f = L 2 (f ) L 1 (f ), L i Boltzmann operator not a maximum principle. No spectral gap. The kernel is too big {1, ω 1 }: spurious term. f (k) T = f (k) dk (energy), S = dk (spurious term) ω(k) T T S: fractional phenomena! Should reach equilibrium faster... Intuition: f L 2 but ω 1 / L 2.

39 Theorem (Fractional diffusion limit for the linearised equation) Let f ε be a solution of ε 8/5 t f ε + εω (k) x f ε = L(f ε ) (BPE) with initial data f 0 L 2 (R T). Then, up to a subsequence, f ε (t, x, k) T (t, x) L ((0, ); L 2 (x, k))-weak where T is the weak limit of T ε = 1 V 1 and solves the fractional diffusion equation 0 V (k)f ε (k)dk t T + κ( x ) 4/5 T = 0 in (0, ) R κ (0, ) with initial condition T (0, x) = 1 V 1 0 Vf 0 (t, x, k) dk.

40 Comments on the proof and results (I) f ε (t, x, k) = T ε (t, x) + S ε (t, x)ω(k) 1 + ε 4 5 h ε (t, x, k) }{{}}{{}}{{} energy spurious term remainder T ε L (0, ; L 2 (R)), h ε L 2 V (T R)

41 Comments on the proof and results (I) f ε (t, x, k) = T ε (t, x) + ε 3 5 S ε (t, x)ω(k) 1 + ε 4 5 h ε (t, x, k) }{{}}{{}}{{} energy spurious term remainder T ε L (0, ; L 2 (R)), h ε L 2 V (T R)

42 Comments on the proof and results (I) f ε (t, x, k) = T ε (t, x) + ε 3 5 S ε (t, x)ω(k) 1 + ε 4 5 h ε (t, x, k) }{{}}{{}}{{} energy spurious term remainder T ε L (0, ; L 2 (R)), h ε L 2 V (T R) Proposition The function S ε converges in distribution sense to S(t, x) = κ 2 κ 3 ( ) 3/10 T (t, x).

43 Comments on the proof and results (II) ε 8/5 t f ε + εω (k) x f ε = L(f ε )

44 Comments on the proof and results (II) Project on the constant mode: ε 8/5 t f ε + εω (k) x f ε = L(f ε ) t T + κ 1 ( ) 4/5 T + κ 2 ( ) 1/2 S = 0.

45 Comments on the proof and results (II) ε 8/5 t f ε + εω (k) x f ε = L(f ε ) Project on the constant mode: t T + κ 1 ( ) 4/5 T + κ 2 ( ) 1/2 S = 0. Project on ω(k) 1 : κ 2 ( ) 1/2 T + κ 3 ( ) 1/5 S = 0.

46 Comments on the proof and results (II) ε 8/5 t f ε + εω (k) x f ε = L(f ε ) Project on the constant mode: t T + κ 1 ( ) 4/5 T + κ 2 ( ) 1/2 S = 0. Project on ω(k) 1 : κ 2 ( ) 1/2 T + κ 3 ( ) 1/5 S = 0. Obtain S(t, x) = κ 2 κ 3 ( ) 3/10 T (t, x).

47 Comments on the proof and results (II) ε 8/5 t f ε + εω (k) x f ε = L(f ε ) Project on the constant mode: t T + κ 1 ( ) 4/5 T + κ 2 ( ) 1/2 S = 0. Project on ω(k) 1 : κ 2 ( ) 1/2 T + κ 3 ( ) 1/5 S = 0. Obtain and S(t, x) = κ 2 κ 3 ( ) 3/10 T (t, x). ( ) t T + κ 1 κ2 2 ( ) 4/5 T = 0 κ 3

48 Comments on the proof and results (III) Method: Laplace-Fourier Transform Method (like in Mellet, Mischler, Mouhot, Fractional diffusion limit for collisional kinetic equations ). f ε (p, ξ, k) = R 0 e pt e iξx f ε (t, x, k) dt dx.

49 Comments on the proof and results (III) Method: Laplace-Fourier Transform Method (like in Mellet, Mischler, Mouhot, Fractional diffusion limit for collisional kinetic equations ). f ε (p, ξ, k) = R 0 e pt e iξx f ε (t, x, k) dt dx. ε α pf ε ε α f0 + iεω (k)ξf ε = K( f ε ) V f ε We recall that L(f ) = K(f ) Vf with K(f ) = K(k, k )f (k ) dk. The fact 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 1, k) ω(k ) dk Multiplying the equation by K(k, k) and integrating with respect to k and k, we get...

50 On the constant mode: F ε 1( f 0 ) + a ε 1(p, ξ) T ε (p, ξ) + a ε 2(p, ξ) S ε (p, ξ) + R ε 1(p, ξ) = 0, (3)

51 On the constant mode: F1( f ε 0 ) + a1(p, ε ξ) T ε (p, ξ) + a2(p, ε ξ) S ε (p, ξ) + R1(p, ε ξ) = 0, (3) in the limit T 0 (ξ) + ( p κ 1 ξ 8/5 ) T (p, ξ) κ 2 ξ S = 0.

52 On the constant mode: F1( f ε 0 ) + a1(p, ε ξ) T ε (p, ξ) + a2(p, ε ξ) S ε (p, ξ) + R1(p, ε ξ) = 0, (3) in the limit T 0 (ξ) + ( p κ 1 ξ 8/5 ) T (p, ξ) κ 2 ξ S = 0. Projection on ω(k) 1 : F2( f ε 0 ) + a2(p, ε ξ) T ε (p, ξ) + a3(p, ε ξ) S ε (p, ξ) + R2(p, ε ξ) = 0, (4)

53 On the constant mode: F1( f ε 0 ) + a1(p, ε ξ) T ε (p, ξ) + a2(p, ε ξ) S ε (p, ξ) + R1(p, ε ξ) = 0, (3) in the limit T 0 (ξ) + ( p κ 1 ξ 8/5 ) T (p, ξ) κ 2 ξ S = 0. Projection on ω(k) 1 : F2( f ε 0 ) + a2(p, ε ξ) T ε (p, ξ) + a3(p, ε ξ) S ε (p, ξ) + R2(p, ε ξ) = 0, (4) in the limit κ 2 ξ T (p, ξ) κ 3 ξ 2/5 S = 0.

54 MICRO FPU chain/lattice Spohn (formal) Heat lattice vibrations phonons KINETIC Harmonic potential No relaxation to eq. Free transport eq. C(W)=0 next neighbour pot. Cubic potential (FPU-alpha chain) 3-phonon Boltz. eq. C(W): bilinear op. On site potential: - classic diffusion (Aoki, Lukkarinen, Spohn) Quartic potential (FPU-beta chain) 4-phonon Boltz. eq. C(W): bilinear op. Next neighbour pot.: - 1d, 2d anomalous dif. - 3d or higher, classic dif. (Lepri, Livi, Politi) Harmonic potential + noise conserving momentum and energy Linearised eq. (Lukkarinen, Spohn) (Olla, Basile, et.al) (Basile, Olla, Spohn) MACRO (Anomalous) diffusion (we lose info in the kinetic limit; suggests weak mixing at micro level) Nonlinear heat equation (Bricmont - Kupianen) Fractional heat equation (Olla, et. al.) (Mellet, Mouhot, Mischler, et. al.) Linear Boltzmann Equation Heat equation

55 MICRO FPU chain/lattice Spohn (formal) Heat lattice vibrations phonons KINETIC MACRO (Anomalous) diffusion Harmonic potential No relaxation to eq. Free transport eq. C(W)=0 next neighbour pot. Cubic potential (FPU-alpha chain) 3-phonon Boltz. eq. C(W): bilinear op. (we lose info in the kinetic limit; suggests weak mixing at micro level) Nonlinear heat equation (Bricmont - Kupianen) On site potential: - classic diffusion (Aoki, Lukkarinen, Spohn) Weaker mixing; smaller coeff.; slows down convergence to eq. Quartic potential (FPU-beta chain) 4-phonon Boltz. eq. C(W): bilinear op. Spurious conservation of nb. of phonons Fractional heat equation Next neighbour pot.: - 1d, 2d anomalous dif. - 3d or higher, classic dif. (Lepri, Livi, Politi) (Mellet, Merino) Harmonic potential + noise conserving momentum and energy Linearised eq. (Lukkarinen, Spohn) (Olla, et. al.) (Mellet, Mouhot, Mischler, et. al.) (Olla, Basile, et.al) (Basile, Olla, Spohn) Linear Boltzmann Equation Heat equation

56 Final Remarks Relation between different results for the FPU-β chain: MICRO: KINETIC: MACRO:

57 Final Remarks Relation between different results for the FPU-β chain: MICRO: at the level of the FPU (numerics), Average energy current: j e (N) = T T +, α 0.4 N1 α KINETIC: MACRO:

58 Final Remarks Relation between different results for the FPU-β chain: MICRO: at the level of the FPU (numerics), Average energy current: j e (N) = T T +, α 0.4 N1 α KINETIC: the results of Lukkarinen-Spohn, lim t t3/5 C(t) = constant MACRO:

59 Final Remarks Relation between different results for the FPU-β chain: MICRO: at the level of the FPU (numerics), Average energy current: j e (N) = T T +, α 0.4 N1 α KINETIC: the results of Lukkarinen-Spohn, lim t t3/5 C(t) = constant MACRO: the order of the fractional laplacian ( ) 4/5.

60 Final Remarks Relation between different results for the FPU-β chain: MICRO: at the level of the FPU (numerics), Average energy current: j e (N) = T T +, α 0.4 N1 α KINETIC: the results of Lukkarinen-Spohn, lim t t3/5 C(t) = constant MACRO: the order of the fractional laplacian ( ) 4/5. Similar equations exist in other context: Weak wave turbulence (4-wave kinetic equation). Nordheim equation.

61 References I Federico Bonetto, Joel L Lebowitz, and Luc Rey-Bellet. Fourier s law: A challenge for theorists. arxiv preprint math-ph/ , Jani Lukkarinen and Herbert Spohn. Anomalous energy transport in the fpu-β chain. Communications on Pure and Applied Mathematics, 61 (12): , 2008.