Energy Spreading in Strongly Nonlinear Lattices M. Mulansky, S. Roy - - PowerPoint PPT Presentation

Energy Spreading in Strongly Nonlinear Lattices

• M. Mulansky, S. Roy and A. Pikovsky

Institut for Physics and Astronomy, University of Potsdam, Germany

Florence, May 26, 2014

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Motivation

◮ Study of nonlinear effects in disordered lattices ◮ Linear lattices: Anderson localization ⇒ no propagation ◮ Nonlinear lattices: Weak subdiffusive spreading due to chaos ◮ Problems: Linear modes are only exponentially localized, no

clear picture of spreading

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Strongly nonlinear lattices

Usual nonlinear lattices H = p2

l

2 + ω2 q2

l

2 + κ(ql+1 − ql)2 2 + Unl(ql) + Vnl(ql+1 − ql) Strongly nonlinear lattice H = p2

l

2 + ω2 q2

l

2 + Unl(ql) + Vnl(ql+1 − ql)

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Sonic vacuum

◮ No phonons, no linear propagating waves and modes ◮ Localization length =1 (minimal possible) ◮ Only propagating waves are nonlinear ones – typically

compactons

◮ At finite energy density: typically strongly chaotic/turbulent

states

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Setup I: Spreading of a localized wave packet in 1-d lattices [with Mario Mulansky, New J. Phys. (2013)]

Strong compactness of the spreading field: Here ”Anderson modes” are one site oscillators ⇒ no exponential tails, the packet width L is well-defined at each moment of time Disorder to prevent ballistic quasi-compactons Regular lattice Disordered lattice

10 20 30 40 50 60 70 80 90 100 20 40 60 80 100 120 140

• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1

time site

10 20 30 40 50 60 70 80 90 100 50 100 150 200 250 300

• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1

time site

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How to average

Traditionally width at fixed time : log L(t), but due to large fluctuations one averages here propagation speed at different densities With sharp edges the averaging of propagation time at fixed width, i.e. at fixed density, is possible: log ∆T = log(T(L + 1) − T(L)) Goal: to describe ∆T(L, E) for different total energies E

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Guiding phenomenology

Use Nonlinear Diffusion Equation (NDE) as a heuristic model ∂ρ ∂t = D ∂ ∂x

• ρa ∂ρ

∂x

• ,

with

• ρ dx = E

Self-similar solution ρ(x, t) = 1 [D(t − t0)]1/(2+a)

• E −

ax2 2(a + 2)[D(t − t0)]2/(a+2) 1/a yields subdiffusion L =

• 22 + a

a E a/(2+a)[D(t − t0)]1/(2+a)

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One parameter scaling

Reformulate L =

• 22 + a

a E a/(2+a)(D(t − t0))1/(2+a) as scaling relaions: L E ∼ t − t0 E 2 1/(2+a) 1 E dt dL ∼ E L −(a+1) a(w)+1 = −d log 1

E dt dL

d log w where w = E/L is the characteristic density, dt

dL ≈ ∆T

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Spreading in a homogeneously nonlinear lattice

Fully self-similar lattice: rescaling energy ⇔ rescaling time H =

• k

p2

k

2 + W ω2

k

qκ

k

κ + β (qk+1 − qk)κ κ From the rescaling of energy and time it follows t ∼ E

2κ 2−κ

⇒ a = κ − 2 2κ ⇒ L ∼ (t − t0)

2κ 5κ−2

For the case κ = 4 we have L ∼ (t − t0)4/9 ∆T ∼ L5/4

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Spreading in a lattice of nonlinearly coupled linear

• scillators

H =

• k

p2

k + ω2 kq2 k

2 + (qk+1 − qk)4 4

• 1

1 2 3 4 5 6 7 8 9 1.2 1.6 2 2.4 2.8 3.2 E=0.2 E=0.35 E=0.5 E=1 E=2 E=4 E=8 E=16 E=32 E=64

log10 L log10 ∆T

• 2

2 4 6 8 10

• 0.4

0.4 0.8 1.2 1.6 2 2.4 2 4 6 8 1 2

log10 L/E log10 ∆T/E − log10 w a(w)

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Spreading in a lattice of nonlinearly coupled linear

• scillators

H =

• k

p2

k + ω2 kq2 k

2 + (qk+1 − qk)6 6

1 2 3 4 5 6 1.2 1.6 2 2.4 2.8 E=0.2 E=0.5 E=1 E=2 E=5 E=10

log10 L log10 ∆T

• 2

2 4 6 8 0.4 0.8 1.2 1.6 2 2.4 2 4 6 8 0.8 1.2 1.6 2

log10 L/E log10 ∆T/E − log10 w a(w)

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Nonlinearly coupled nonlinear oscillators

H =

• k

p2

k

2 + ω2

k

q4

k

4 + (qk+1 − qk)8 8

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 E=0.02 E=0.03 E=0.05 E=0.10 E=0.20 E=0.50

log10 L/E log10 ∆T/E

1 2 3 4 5 6 7 1.2 1.6 2 2.4 2.8 3.2 1 2 3 4 2 2.5 3

log10 L/E log10 ∆T/E 0.7 − log10 w

a+2−γ γ

Different scaling: ∆T/E 0.7 = F(L/E)

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Fractional nonlinear diffusion equation

∂γρ ∂tγ = D ∂ ∂x

• ρa ∂ρ

∂x

• ,

with

• ρ dx = E

yields E 1−2/γ dt dL ∼ L E a+2−γ

γ 13 / 30

Nonlinearly coupled nonlinear oscillators

H =

• k

p2

k

2 + ω2

k

q4

k

4 + (qk+1 − qk)6 6

2.5 3 3.5 4 4.5 5 5.5 6 6.5 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 E=0.001 E=0.002 E=0.003 E=0.005 E=0.01 E=0.02 E=0.05

log10 L log10 ∆T

3 4 5 6 7 8 9 2.8 3.2 3.6 4 4.4 4.8

log10 L/E log10 ∆T/E 0.85

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Conclusions for 1-dimensional wavepacket spreading

◮ Nonlinearly coupled linear oscillators:

NDE scaling works, slowing down of spreading

◮ Nonlinearly coupled nonlinear oscillators:

FracNDE scaling works, good power-law

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Relation to chaos properties [M. Mulansky, Chaos (2014)]

Probability to observe chaos in a finite lattice in dependence on length and density Nonlinear local osc. Linear local osc.

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Toy model: Ding-Dong lattice [with S. Roy, CHAOS, v. 22, n. 2, 026118 (2012)]

This is a strongly nonlinear lattice that is easy to model numerically

• Ding-Dong model (Prosen, Robnik, 92) is a chain of linear
• scillators with elastic collisions

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Ding-Dong dynamics

Hamiltonian and collision condition H =

• k

p2

k + q2 k

2 when qk−qk+1 = 1 then pk → pk+1, pk+1 → pk Effective calculation of the collision times – simulation on very long times pissible Strongly nonlinear lattice: no linear waves, no phonons, all propagating perturbations are nonlinear

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Compactons in a homogeneous Ding-Dong lattice

• 5

5 10 15 20 10 20 30 40 50 60

(a) time k + qk

• 5

5 10 15 10 20 30 40 50 60

(b) time k + qk

5 10 15 20 20 40 60 80 100 120 140 160

(c) time k + qk

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Spreading in a homogeneous lattice

From random initial conditions: chaos, breathers, and (almost)compactons appear

500 1000 1500 2000

• 20

20 40 60 80 100

k + qk time

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Examples of chaos and breathers

2400 2500 2600 2700 2800

• 21
• 20
• 19
• 18
• 17
• 16

k + qk time

2000 2100 2200 7 8 9 10 11 12

k + qk time

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Spreading in a disordered lattice

Disorder in distances or masses destroys compactons Spreading effectively stops: no spreading events for time interval 1010, a few chaotic spots appear

20 40 60 80 100 120 100 101 102 103 104 105 106 107 108 109 1010

time

width of excit region L(t)

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Chaotic spot at a boundary of spreading range

20 40 60 80 100

• 50
• 48
• 46
• 44
• 42
• 40

k + qk time

20 40 60 80 100

• 45
• 44
• 43
• 42
• 41
• 40
• 39

k + qk time

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Spreading in 2-dimensional lattices [M. Mulansky, A.P., Phys. Rev. E, 2012]

Hamiltonian H =

i,k Ei,k with

Ei,k = p2

i,k

2 + W ω2

i,k

κ |qi,k|κ+ + β 2λ(|qi+1,k − qi,k|λ + |qi−1,k − qi,k|λ + |qi,k+1 − qi,k|λ + |qi,k−1 − qi,k|λ) .

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Sharply localized field - no compactons even without disorder!

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2-dimensional Nonlinear Diffusion Equation

∂ρ ∂t = ∇ (ρa∇ρ) , with

• ρ d2

r = E . has a solution with growing radius R2 = 4B a + 1 a · (t − t0)1/(a+1) and B = E 4π

• a

a+1

. The scaling prediction: ∆n2 E ∼ t − t0 E ν ν = 1 a + 1 ,

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Spreading in regular 2-dimensional lattices

Linear oscillators, coupled via nonlinearity power 4 nonlinearity power 6

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A simple resonance model of spreading

An initially non-excited (linear) site is excited by a neighbor

• scillating with amplitude ǫ and frequency Ω = 1 + aǫλ−2 (this

shift of frequency follows from the nonlinear coupling term). H1 = p2 + q2 2 + |q − ǫ sin Ωt|λ λ . The resonant averaged Hamiltonian H1 = −aǫλ−2I + ǫλF( √ 2Iǫ−2 cos θ, √ 2Iǫ−2 sin θ) , can be rescaled by I → ǫ2I, t → ǫλ−2t which yields ∆n2 ∼ t2/λ in accordance with numerics

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Spreading in disordered 2-dimensional lattices

Linear oscillators, coupled via nonlinearity power 4 nonlinearity power 6

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Conclusions for 2-dimensional wavepacket spreading

◮ Nonlinearly coupled linear oscillators:

NDE ∂ρ

∂t = ∇ (ρa∇ρ) scaling works, good power laws ◮ Regular lattices:

effective powers in NDE a4,r ≈ 1, a6,r ≈ 2 ⇒ a = λ

2 − 1 ◮ Irregular lattices:

effective powers in NDE a4,d ≈ 3, a6,d ≈ 5 ⇒ a = λ − 1

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