Thermalization and conduction in one-dimensional chains: a wave - - PowerPoint PPT Presentation

Thermalization and conduction in one-dimensional chains: a wave turbulence approach

Miguel Onorato

Universit` a di Torino, Dipartimento di Fisica miguel.onorato@gmail.com in collaboration with

• Y. L’vov (Rensselaer Polytechnic Institute - New York)
• L. Pistone (Universit`

a di Torino - Torino)

• D. Proment (University of East Anglia - Norwich)
• S. Chibbaro ( Institut Jean Le Rond d’Alembert - Paris)
• M. Bustamante (University College Dublin - Dublin)
• L. Rondoni (Politecnico di Torino- Torino)
• G. Dematteis (Universit`

a di Torino - Torino)

November 4, 2019

The weakly nonlinear one-dimensional chain model

N equal masses connected by a weakly nonlinear spring The Hamiltonian H =

N

• j=1

1 2mp2

j + κ

2(qj − qj+1)2

• +α

3

N

• j=1

(qj−qj+1)3+β 4

N

• j=1

(qj−qj+1)4+...

Enrico Fermi (1901-1954) John Pasta (1909-1984) Stanislaw Ulam (1918-1984) Mary Tsingou-Menzel (1928- ) MANIAC I (1952-1957)

The result expected by Fermi and collaborators

Equipartition of linear energy in Fourier space for large times Qk = 1 N

N−1

• j=0

qje−i 2πkj

N , Pk = 1

N

N−1

• j=0

pje−i 2πkj

N ,

then Ek = |Pk|2 + ω2

k|Qk|2 = const

with ωk = 2

• sin

πk N

The Los Alamos report

Following up on the “little discovery”

Soliton theory Theory of integrable PDEs Hamiltonian Chaos

Some years after FPUT: solitons and integrability in physics

In the limit of long waves (continuum limit) the α-FPUT system reduces to the Korteweg-de Vries (KdV) equation: ∂η ∂t + η ∂η ∂x + ∂3η ∂x3 = 0

Numerical simulations of the KdV

ZK showed, besides recurrence, the formation of train of solitons

Experimental demonstration of the ZK solitons

The wave tank in Berlin (5 m × 90 m × 15 cm)

Trillo et. al PRL 2016

FPUT recurrence in shallow water (Trillo et. al PRL 2016)

5 15 25 35 45 55 65 75 0.5 1

distance z Fourier amplitudes (a.u.)

recurrence

(a)

1st harmonic 2nd harmonic 3rd harmonic 65 75 85 95 5 10

elevation [cm]

(b) 15m 90 100 110 120 5 10

elevation [cm]

(c) 65m 100 110 120 130 5 10

elevation [cm] time [s]

(d) 75m

Literature and reviews

Some reviews: Ford, J. ”The Fermi-Pasta-Ulam problem: paradox turns discovery.” Physics Reports 213.5 (1992): 271-310. Berman, G. P., and F. M. Izrailev. ”The Fermi-Pasta-Ulam problem: fifty years of progress.” Chaos (Woodbury, NY) 15.1 (2005): 15104 Carati, A., L. Galgani, and A. Giorgilli. ”The Fermi-Pasta-Ulam problem as a challenge for the foundations of physics.” Chaos: An Interdisciplinary Journal of Nonlinear Science 15.1 (2005): 015105-015105. Weissert, Thomas P. ”The genesis of simulation in dynamics: pursuing the Fermi-Pasta-Ulam problem.” Springer-Verlag New York, Inc., 1999. Gallavotti, G., ed. ”The Fermi-Pasta-Ulam problem: a status report.”

• Vol. 728. Springer, 2008.

Open questions

... but FPU is not an integrable system... Does the system thermalize for arbitrary small nonlinearity? If yes, what is the time scale of thermalization for finite N? What is the thermalization time scale in the thermodynamic limit? How does thermalization time scale depend on the number of particles?

The models

α-FPUT ¨ qj = (qj+1 + qj−1 − 2qj) + α

• (qj+1 − qj)2 − (qj−1 − qj)2

β-FPUT ¨ qj = (qj+1 + qj−1 − 2qj) + β

• (qj+1 − qj)3 − (qj−1 − qj)3

Discrete Nonlinear Klein Gordon (DNKG) ¨ qj = (qj+1 + qj−1 − 2qj) − qj − gq3

j ,

Toda Lattice ¨ qj = 1 2α (exp[2α(qj+1 − qj)] − exp[2α(qj − qj−1)])

The linear and the weakly nonlinear regime

Linear regime For α-FPUT, β-FPUT, Toda ωk = 2| sin (kπ/N) | For DNKG ωk =

• 1 + 4 sin (kπ/N)2

Weakly nonlinear regime β ∼ g ∼ α2 ∼ ǫ

Normal modes

Assuming periodic boundary conditions, we introduce the wave action variable ak = 1 √2ωk (ωkQk + iPk), with Pk = ˙ Qk and ωk = 2| sin(πk/N)| Because of the absence of three wave interactions, i.e.: k1 ± k2 ± k3 = 0 ω1 ± ω2 ± ω3 = 0 quadratic nonlinearity can be removed from α-FPUT and Toda.

Same (approximate) Hamiltonian for all 4 models

H N =

N−1

• k=0

ωk|ak|2 + ǫ

• k1,k2,k3,k4
• T (1)

1,2,3,4(a∗ 1a2a3a4 + c.c.)δ1−2−3−4+

+ 1 2T (2)

1,2,3,4a∗ 1a∗ 2a3a4δ1+2−3−4 + 1

4T (3)

1,2,3,4(a1a2a3a4 + c.c.)δ1+2+3+4

• with

δ1±2±3±4 = δ(k1±k2±k3±k4), ai = a(ki, t), T1,2,3,4 = T(k1, k2, k3, k4) ǫ ∼ β ∼ g ∼ α2 Starting point for statistical theory (see Nazarenko 2011)

The thermodynamic limit

N → ∞, L → ∞ with L N = ∆x = const Then the dispersion relations become: ωκ = 2| sin(κ/2)|, ωκ =

• 1 + 4 sin(κ/2)2

with κ ∈ R. The following 4-wave resonant interactions are satisfied: κ1 + κ2 − κ3 − κ4 = 0 ω1 + ω2 − ω3 − ω4 = 0 Standard Wave Turbulence can be developed

The Wave Kinetic Equation

Look for an evolution equation for the correlator < a(κi, t)a(κj, t)∗ >= niδ(κi − κj) with ni = n(κi, t) Assume random initial phases and amplitudes ∂n(κ1, t) ∂t = J(κ1, t) J(κ1, t) = ǫ2 2π T 2

1,2,3,4n1n2n3n4

1 n1 + 1 n2 − 1 n3 − 1 n4

• δ(∆κ)δ(∆ω)dκ2,3

δ(∆κ) = κ1 + κ2 − κ3 − κ4 δ(∆ω) = ω(κ1) + ω(κ2) − ω(κ3) − ω(κ4)

The Wave Kinetic Equation

Conserved quantities: E = 2π ω(κ)n(κ, t)dκ, N = 2π n(κ, t)dκ, Existence of an H-theorem: H = 2π ln(n(κ, t))dκ, with dH dt ≤ 0 The Rayleigh-Jeans distribution dH/dt = 0 → n(k, t) = T ω(κ) + µ Thermalization time scale: 1/ǫ2

Small N regime

ωk = 2| sin(πk/N)| with k ∈ Z k1 ± k2 ± k3 ± k4 = 0 (mod N) ω1 ± ω2 ± ω3 ± ω4 = 0 It can be shown that only the following interactions are possible (of Umklapp type): k1 + k2 − k3 − k4 = 0 (mod N) ω1 + ω2 − ω3 − ω4 = 0

Umklapp (flip-over) scattering

Normal process (N-process) and Umklapp process (U-process). Example of an Umklapp scattering with N = 32 (kmax = 16), k1 = 7, k2 = 9, k3 = −7, k4 = 23 → outside the Brillouin zone, therefore the wave-number is flip-over k′

4 = k4 − N = −9

Small N regime

For N power of 2, the above system has solutions for integer values of k: Trivial solutions: all wave numbers are equal or k1 = k3, k2 = k4,

• r

k1 = k4, k2 = k3 Nontrivial solutions: {k1, k2; k3, k4} =

• k1, N

2 − k1; N − k1, N 2 + k1

• with k1 = 1, 2, . . . , ⌊N/4⌋

However.... Four-waves resonant interactions are isolated No efficient mixing (and thermalization) can be achieved via a four-wave resonant process (for weak nonlinearity)

Removing non resonant interactions

H N =

N−1

• k=0

ωk|ak|2 + ǫ

• k1,k2,k3,k4
• T (1)

1,2,3,4(a∗ 1a2a3a4 + c.c.)δ1−2−3−4+

+ 1 2T (2)

1,2,3,4a∗ 1a∗ 2a3a4δ1+2−3−4 + 1

4T (3)

1,2,3,4(a1a2a3a4 + c.c.)δ1+2+3+4

• Eliminate the non-resonant terms from the Hamiltonian using a

near-identity (canonical) transformation from {ia, a∗} to {ib, b∗} a1 = b1 + ǫ

• k2,k3,k4

(B(1)

1,2,3,4b2b3b4δ1−2−3−4 + B(2) 1,2,3,4b∗ 2b3b4δ1+2−3−4+

+ B(3)

1,2,3,4b∗ 2b∗ 3b4δ1+2+3−4 + B(4) 1,2,3,4b∗ 2b∗ 3b∗ 4δ1+2+3+4) + O(ǫ2)

with B1,2,3,4 ≃ T1,2,3,4/(ω1 ± ω2 ± ω3 ± ω4).

Removing non-resonant four-wave interactions: the appearance six-wave interactions in the β-FPUT

check for exact resonances at higher order idb1 dt = ω1b1 + ǫ

• k2,k3,k4

T1,2,3,4b∗

2b3b4δ1+2−3−4+

+ ǫ2 W1,2,3,4,5,6b∗

2b∗ 3b4b5b6δ1+2+3−4−5−6

Resonant conditions: k1 + k2 + k3 − k4 − k5 − k6 = 0 (mod N) ω1 + ω2 + ω3 − ω4 − ω5 − ω6 = 0 Non-isolated solutions exist for integer values of k with arbitrary N. dn1 dt ∼ ǫ4....

Estimation of the equipartition time scale for incoherent waves

Look for the evolution equation of < b(ki, t)b(kj, t)∗ >= n(ki, t)δi−j dn1 dt ∼ ǫ2 < b∗

1b∗ 2b∗ 3b4b5b6 >

d < b∗

1b∗ 2b∗ 3b4b5b6 >

dt ∼ ǫ2 < b∗

1b∗ 2b∗ 3b∗ 4b5b6b7b8 >

therefore dn1 dt ∼ ǫ4.... and the time of equipartition scales as teq ∼ 1/ǫ4

Numerical simulations (Symplectic integrator (H. Yoshida, 1990 Phys. Lett. A) )

Example of Umklapp resonance

Numerical simulations

Example of thermalization for α-FPUT with N=32, ǫ = 7.3×10−2 (1000 realizations)

Entropy

s(t) =

• k

fk log fk with fk = N − 1 Etot ωk|ak|2, Etot =

• k

ωk|ak|2

Scaling in time

Collapse of entropy curves

Example of equipartition: β-FPUT, N=32, ǫ = 7.05 × 10−2

Entropy: β-FPUT, ǫ = 7.05 × 10−2, N = 32

s(t) =

• k

fk log fk with fk = N − 1 H0 ωk|ak|2, H0 =

• k

ωk|ak|2

10-4 10-3 10-2 10-1 100 101 10-1 100 101 102 103 104 105 s(t) t

Equipartition time as a function of ǫ

102 103 104 105 106 107 108 0.01 0.1 1 teq

ε

ε-4

DNKG equation: from discrete to the Large Box Limit

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• ■

■ ■ ■ ■ ■ ■ ■■ ■ ■■■■■■■■ ■ ■■■■■■ ■■■ ■■ ■ ■■■■■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲ ▲▲ ▲ ▲▲▲▲▲▲▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○○ ○ ○ ○ ○○○○○ ○ ○ □ □ □ □ □ □ □□ □□□□□□ □□□ □□□□ □ □ □ □□□□□ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ 32 64 96 128 512 1024 31 15104

5.×10-4 0.001 0.005 0.010

ϵ

1000 104 105 106 107 108

Teq

Figure: The scaling of Teq on ǫ for multiple values of N, with m = 1 and E = 0.1N/32. Scaling laws ǫ−2 and ǫ−4 in red dotted and black dash-dotted lines for reference.

Equipartition time as a function of ǫ: α − FPUT N=64

102 103 104 105 106 107 108 109 1010 10-4 10-3 Teq

ε

α-FPUT N=64 ε-3.84

Equipartition time as a function of ǫ: β − FPUT N=64

104 105 106 107 108 109 1010 10-3 10-2 Teq

ε

β-FPUT N=64 ε-3.72

Equipartition time as a function of ǫ: NLKG N=64

103 104 105 106 107 108 0.001 0.01 Teq

ε

DNKG N=64 ε-3.79

Equipartition time as a function of ǫ: α − FPUT N=1024

105 106 107 108 10-4 10-3 Teq

ε

α-FPUT N=1024 ε-2.49

Equipartition time as a function of ǫ: β − FPUT N=1024

104 105 106 107 0.01 Teq

ε

β-FPUT N=1024 ε-2.00

Equipartition time as a function of ǫ: NLKG N=1024

102 103 104 105 106 10-2 Teq

ε

DNKG N=1024 ε-1.97