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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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