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 � 1 N N N j + κ � + α ( q j − q j +1 ) 3 + β � 2 mp 2 2( q j − q j +1 ) 2 � � ( q j − q j +1 ) 4 + ... H = 3 4 j =1 j =1 j =1 MANIAC I Mary Tsingou-Menzel Enrico Fermi (1901-1954) Stanislaw Ulam John Pasta (1909-1984) (1952-1957) (1928- ) (1918-1984)

The result expected by Fermi and collaborators Equipartition of linear energy in Fourier space for large times N − 1 N − 1 Q k = 1 N , P k = 1 q j e − i 2 πkj p j e − i 2 πkj � � N , N N j =0 j =0 then E k = | P k | 2 + ω 2 k | Q k | 2 = const with � � πk �� � � ω k = 2 � sin � � 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: ∂x + ∂ 3 η ∂η ∂t + η ∂η ∂x 3 = 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) 10 (b) 15m elevation [cm] (a) 1st harmonic 2nd harmonic 5 1 3rd harmonic 0 Fourier amplitudes (a.u.) 65 75 85 95 10 (c) 65m elevation [cm] 5 recurrence 0 0.5 90 100 110 120 10 (d) 75m elevation [cm] 5 0 0 5 15 25 35 45 55 65 75 100 110 120 130 distance z time [s]

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 ( q j +1 − q j ) 2 − ( q j − 1 − q j ) 2 � � q j = ( q j +1 + q j − 1 − 2 q j ) + α ¨ β -FPUT ( q j +1 − q j ) 3 − ( q j − 1 − q j ) 3 � � q j = ( q j +1 + q j − 1 − 2 q j ) + β ¨ Discrete Nonlinear Klein Gordon (DNKG) q j = ( q j +1 + q j − 1 − 2 q j ) − q j − gq 3 ¨ j , Toda Lattice q j = 1 ¨ 2 α (exp[2 α ( q j +1 − q j )] − exp[2 α ( q j − q j − 1 )])

The linear and the weakly nonlinear regime Linear regime For α -FPUT, β -FPUT, Toda ω k = 2 | sin ( kπ/N ) | For DNKG � 1 + 4 sin ( kπ/N ) 2 ω k = Weakly nonlinear regime β ∼ g ∼ α 2 ∼ ǫ

Normal modes Assuming periodic boundary conditions, we introduce the wave action variable 1 √ 2 ω k a k = ( ω k Q k + iP k ) , with P k = ˙ Q k and ω k = 2 | sin( πk/N ) | Because of the absence of three wave interactions, i.e.: k 1 ± k 2 ± k 3 = 0 ω 1 ± ω 2 ± ω 3 � = 0 quadratic nonlinearity can be removed from α -FPUT and Toda.

Same (approximate) Hamiltonian for all 4 models N − 1 H ω k | a k | 2 + ǫ T (1) � � � 1 , 2 , 3 , 4 ( a ∗ N = 1 a 2 a 3 a 4 + c.c. ) δ 1 − 2 − 3 − 4 + k =0 k 1 ,k 2 ,k 3 ,k 4 + 1 2 a 3 a 4 δ 1+2 − 3 − 4 + 1 2 T (2) 4 T (3) 1 , 2 , 3 , 4 a ∗ 1 a ∗ � 1 , 2 , 3 , 4 ( a 1 a 2 a 3 a 4 + c.c. ) δ 1+2+3+4 with δ 1 ± 2 ± 3 ± 4 = δ ( k 1 ± k 2 ± k 3 ± k 4 ) , a i = a ( k i , t ) , T 1 , 2 , 3 , 4 = T ( k 1 , k 2 , k 3 , k 4 ) ǫ ∼ β ∼ g ∼ α 2 Starting point for statistical theory (see Nazarenko 2011)

The thermodynamic limit L N → ∞ , L → ∞ with N = ∆ x = const Then the dispersion relations become: � 1 + 4 sin( κ/ 2) 2 ω κ = 2 | sin( κ/ 2) | , ω κ = with κ ∈ R . The following 4-wave resonant interactions are satisfied: κ 1 + κ 2 − κ 3 − κ 4 = 0 ω 1 + ω 2 − ω 3 − ω 4 = 0 Standard Wave Turbulence can be developed

The Wave Kinetic Equation Look for an evolution equation for the correlator < a ( κ i , t ) a ( κ j , t ) ∗ > = n i δ ( κ i − κ j ) with n i = n ( κ i , t ) Assume random initial phases and amplitudes ∂n ( κ 1 , t ) = J ( κ 1 , t ) ∂t � 1 � 2 π + 1 − 1 − 1 � J ( κ 1 , t ) = ǫ 2 T 2 1 , 2 , 3 , 4 n 1 n 2 n 3 n 4 δ (∆ κ ) δ (∆ ω ) dκ 2 , 3 n 1 n 2 n 3 n 4 0 δ (∆ κ ) = κ 1 + κ 2 − κ 3 − κ 4 δ (∆ ω ) = ω ( κ 1 ) + ω ( κ 2 ) − ω ( κ 3 ) − ω ( κ 4 )

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

Small N regime ω k = 2 | sin( πk/N ) | with k ∈ Z k 1 ± k 2 ± k 3 ± k 4 = 0 (mod N ) ω 1 ± ω 2 ± ω 3 ± ω 4 = 0 It can be shown that only the following interactions are possible (of Umklapp type): k 1 + k 2 − k 3 − k 4 = 0 (mod N ) ω 1 + ω 2 − ω 3 − ω 4 = 0

Umklapp (flip-over) scattering Normal process (N-process) and Umklapp process (U-process). Example of an Umklapp scattering with N = 32 ( k max = 16 ), k 1 = 7 , k 2 = 9 , k 3 = − 7 , k 4 = 23 → outside the Brillouin zone, therefore the wave-number is flip-over k ′ 4 = k 4 − 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 k 1 = k 3 , k 2 = k 4 , or k 1 = k 4 , k 2 = k 3 Nontrivial solutions : � � k 1 , N 2 − k 1 ; N − k 1 , N { k 1 , k 2 ; k 3 , k 4 } = 2 + k 1 with k 1 = 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 N − 1 H ω k | a k | 2 + ǫ T (1) � � 1 , 2 , 3 , 4 ( a ∗ � N = 1 a 2 a 3 a 4 + c.c. ) δ 1 − 2 − 3 − 4 + k =0 k 1 ,k 2 ,k 3 ,k 4 + 1 2 a 3 a 4 δ 1+2 − 3 − 4 + 1 2 T (2) 4 T (3) 1 , 2 , 3 , 4 a ∗ 1 a ∗ � 1 , 2 , 3 , 4 ( a 1 a 2 a 3 a 4 + c.c. ) δ 1+2+3+4 Eliminate the non-resonant terms from the Hamiltonian using a near-identity (canonical) transformation from { ia, a ∗ } to { ib, b ∗ } � ( B (1) 1 , 2 , 3 , 4 b 2 b 3 b 4 δ 1 − 2 − 3 − 4 + B (2) 1 , 2 , 3 , 4 b ∗ a 1 = b 1 + ǫ 2 b 3 b 4 δ 1+2 − 3 − 4 + k 2 ,k 3 ,k 4 + B (3) 3 b 4 δ 1+2+3 − 4 + B (4) 1 , 2 , 3 , 4 b ∗ 2 b ∗ 1 , 2 , 3 , 4 b ∗ 2 b ∗ 3 b ∗ 4 δ 1+2+3+4 ) + O ( ǫ 2 ) with B 1 , 2 , 3 , 4 ≃ T 1 , 2 , 3 , 4 / ( ω 1 ± ω 2 ± ω 3 ± ω 4 ) .

Removing non-resonant four-wave interactions: the appearance six-wave interactions in the β -FPUT check for exact resonances at higher order idb 1 � T 1 , 2 , 3 , 4 b ∗ dt = ω 1 b 1 + ǫ 2 b 3 b 4 δ 1+2 − 3 − 4 + k 2 ,k 3 ,k 4 + ǫ 2 � W 1 , 2 , 3 , 4 , 5 , 6 b ∗ 2 b ∗ 3 b 4 b 5 b 6 δ 1+2+3 − 4 − 5 − 6 Resonant conditions: k 1 + k 2 + k 3 − k 4 − k 5 − k 6 = 0 (mod N ) ω 1 + ω 2 + ω 3 − ω 4 − ω 5 − ω 6 = 0 Non-isolated solutions exist for integer values of k with arbitrary N . dn 1 dt ∼ ǫ 4 ....

Estimation of the equipartition time scale for incoherent waves Look for the evolution equation of < b ( k i , t ) b ( k j , t ) ∗ > = n ( k i , t ) δ i − j dn 1 dt ∼ ǫ 2 < b ∗ 1 b ∗ 2 b ∗ 3 b 4 b 5 b 6 > d < b ∗ 1 b ∗ 2 b ∗ 3 b 4 b 5 b 6 > ∼ ǫ 2 < b ∗ 1 b ∗ 2 b ∗ 3 b ∗ 4 b 5 b 6 b 7 b 8 > dt therefore dn 1 dt ∼ ǫ 4 .... and the time of equipartition scales as t eq ∼ 1 /ǫ 4

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 f k = N − 1 � ω k �| a k | 2 � , � ω k �| a k | 2 � s ( t ) = f k log f k with E tot = E tot k k

Scaling in time

Collapse of entropy curves

Example of equipartition: β -FPUT, N =32, ǫ = 7 . 05 × 10 − 2