energy spreading in strongly nonlinear lattices
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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 1 / 30 Motivation Study of nonlinear effects in disordered


  1. 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 1 / 30

  2. 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 2 / 30

  3. Strongly nonlinear lattices Usual nonlinear lattices � p 2 2 + ω 2 q 2 2 + κ ( q l +1 − q l ) 2 l l H = + U nl ( q l ) + V nl ( q l +1 − q l ) 2 Strongly nonlinear lattice � p 2 2 + ω 2 q 2 l l H = 2 + U nl ( q l ) + V nl ( q l +1 − q l ) 3 / 30

  4. 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 4 / 30

  5. 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 1 300 1 140 0.5 0.5 250 120 0 0 -0.5 -0.5 100 200 time time -1 -1 80 -1.5 150 -1.5 60 -2 -2 100 -2.5 -2.5 40 -3 -3 50 20 -3.5 -3.5 0 -4 0 -4 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 site site 5 / 30

  6. 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 6 / 30

  7. Guiding phenomenology Use Nonlinear Diffusion Equation (NDE) as a heuristic model ∂ρ ∂ t = D ∂ � ρ a ∂ρ � � , with ρ dx = E ∂ x ∂ x Self-similar solution � 1 / a ax 2 � 1 ρ ( x , t ) = E − [ D ( t − t 0 )] 1 / (2+ a ) 2( a + 2)[ D ( t − t 0 )] 2 / ( a +2) yields subdiffusion � 22 + a E a / (2+ a ) [ D ( t − t 0 )] 1 / (2+ a ) L = a 7 / 30

  8. One parameter scaling Reformulate � 22 + a E a / (2+ a ) ( D ( t − t 0 )) 1 / (2+ a ) L = a as scaling relaions: � 1 / (2+ a ) � − ( a +1) a ( w )+1 = − d log 1 d t � t − t 0 1 d t � E L E d L E ∼ d L ∼ E 2 d log w E L where w = E / L is the characteristic density, d t d L ≈ ∆ T 8 / 30

  9. Spreading in a homogeneously nonlinear lattice Fully self-similar lattice: rescaling energy ⇔ rescaling time p 2 q κ κ + β ( q k +1 − q k ) κ � 2 + W ω 2 k k H = k κ k From the rescaling of energy and time it follows a = κ − 2 2 κ 2 κ t ∼ E ⇒ ⇒ L ∼ ( t − t 0 ) 2 − κ 5 κ − 2 2 κ For the case κ = 4 we have L ∼ ( t − t 0 ) 4 / 9 ∆ T ∼ L 5 / 4 9 / 30

  10. Spreading in a lattice of nonlinearly coupled linear oscillators p 2 k + ω 2 k q 2 + ( q k +1 − q k ) 4 � k H = 2 4 k 9 10 8 8 a ( w ) 6 7 8 log 10 ∆ T / E 4 6 log 10 ∆ T 6 2 5 0 0 1 2 4 − log 10 w 4 E=0.2 E=0.35 3 E=0.5 E=1 2 2 E=2 E=4 1 E=8 0 E=16 0 E=32 E=64 -1 -2 1.2 1.6 2 2.4 2.8 3.2 -0.4 0 0.4 0.8 1.2 1.6 2 2.4 log 10 L log 10 L / E 10 / 30

  11. Spreading in a lattice of nonlinearly coupled linear oscillators p 2 k + ω 2 k q 2 + ( q k +1 − q k ) 6 � k H = 2 6 k 8 6 8 a ( w ) 6 6 5 log 10 ∆ T / E 4 log 10 ∆ T 2 4 4 0 0.8 1.2 1.6 2 − log 10 w 3 2 2 E=0.2 E=0.5 E=1 0 1 E=2 E=5 E=10 0 -2 1.2 1.6 2 2.4 2.8 0 0.4 0.8 1.2 1.6 2 2.4 log 10 L log 10 L / E 11 / 30

  12. Nonlinearly coupled nonlinear oscillators p 2 q 4 4 + ( q k +1 − q k ) 8 � 2 + ω 2 k k H = k 8 k 7 7 6.5 4 log 10 ∆ T / E 0 . 7 6 a +2 − γ 6 log 10 ∆ T / E 3 5.5 γ 5 2 5 1 2 2.5 3 − log 10 w 4.5 4 4 3 3.5 E=0.02 E=0.03 3 E=0.05 2 E=0.10 2.5 E=0.20 E=0.50 2 1 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 1.2 1.6 2 2.4 2.8 3.2 log 10 L / E log 10 L / E Different scaling: ∆ T / E 0 . 7 = F ( L / E ) 12 / 30

  13. Fractional nonlinear diffusion equation ∂ γ ρ ∂ t γ = D ∂ � ρ a ∂ρ � � , with ρ dx = E ∂ x ∂ x yields � L � a +2 − γ E 1 − 2 /γ dt γ dL ∼ E 13 / 30

  14. Nonlinearly coupled nonlinear oscillators p 2 q 4 4 + ( q k +1 − q k ) 6 � 2 + ω 2 k k H = k 6 k 6.5 9 6 log 10 ∆ T / E 0 . 85 8 5.5 log 10 ∆ T 7 5 4.5 6 4 E=0.001 5 E=0.002 3.5 E=0.003 E=0.005 4 E=0.01 3 E=0.02 E=0.05 2.5 3 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 2.8 3.2 3.6 4 4.4 4.8 log 10 L log 10 L / E 14 / 30

  15. 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 15 / 30

  16. 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. 16 / 30

  17. 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 oscillators with elastic collisions 17 / 30

  18. Ding-Dong dynamics Hamiltonian and collision condition p 2 k + q 2 � k H = when q k − q k +1 = 1 then p k → p k +1 , p k +1 → p k 2 k 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 18 / 30

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