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Universal route to thermalization in weakly-nonlinear one-dimensional chains

1 Dipartimento di Fisica, Università di Torino, via Pietro Giuria 1, 10125, Torino, Italy
2 Sorbonne Université, CNRS, Institut Jean Le Rond D’Alembert, F-75005 Paris, France
3 School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
4 Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180
5 INFN, Sezione di Torino, Via P. Giuria, 1-Torino, 10125, Italy

$^\dagger$This contribution is part of the Special Issue: Hamiltonian Lattice Dynamics
  Guest Editors: Simone Paleari; Tiziano Penati
  Link: https://www.aimspress.com/newsinfo/1165.html

Special Issues: Hamiltonian Lattice Dynamics

Assuming that resonances play a major role in the transfer of energy among the Fourier modes, we apply the Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider α and β Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We consider both the thermodynamic limit and the discrete regime and we conjecture that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity, at least for a range of values of the nonlinear parameter. We perform state of the art numerical simulations and show that the results are mostly consistent with theoretical predictions. Some observed discrepancies are discussed. We suggest that the route to thermalization, based on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.
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Keywords FPUT; Klein-Gordon; nonlinear interactions; Wave Turbulence; thermalization

Citation: Lorenzo Pistone, Sergio Chibbaro, Miguel D. Bustamante, Yuri V. Lvov, Miguel Onorato. Universal route to thermalization in weakly-nonlinear one-dimensional chains. Mathematics in Engineering, 2019, 1(4): 672-698. doi: 10.3934/mine.2019.4.672

References

  • 1. Benettin G, Christodoulidi H, Ponno A (2013) The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J Stat Phys 152: 195–212.    
  • 2. Benettin G, Livi R, Ponno A (2009) The Fermi-Pasta-Ulam problem: Scaling laws vs. initial conditions. J Stat Phys 135: 873–893.
  • 3. Benettin G, Ponno A (2011) Time-scales to equipartition in the Fermi-Pasta-Ulam problem: Finite-size effects and thermodynamic limit. J Stat Phys 144: 793–812.    
  • 4. Berchialla L, Giorgilli A, Paleari S (2004) Exponentially long times to equipartition in the thermodynamic limit. Phys lett A 321: 167–172.    
  • 5. Bustamante MD, Hutchinson K, Lvov YV, et al. (2019) Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system. Commun Nonlinear Sci 73: 437–471.    
  • 6. Bustamante MD, Kartashova E (2011) Resonance clustering in wave turbulent regimes: Integrable dynamics. Commun Comput Phys 10: 1211–1240.    
  • 7. Carati A, Ponno A (2018) Chopping time of the FPU α-model. J Stat Phys 170: 883–894.    
  • 8. Carati A, Galgani L, Giorgilli A, et al. (2007) Fermi-Pasta-Ulam phenomenon for generic initial data. Phys Rev E 76: 022104.
  • 9. Chibbaro S, Dematteis G, Josserand C, et al. (2017) Wave-turbulence theory of four-wave nonlinear interactions. Phys Rev E 96: 021101.    
  • 10. Chibbaro S, Dematteis G, Rondoni L (2018) 4-wave dynamics in kinetic wave turbulence. Phys D 362: 24–59.    
  • 11. Chirikov BV (1979) A universal instability of many-dimensional oscillator systems. Phys Rep 52: 263–379.    
  • 12. Choi Y, Lvov YV, Nazarenko S (2004) Probability densities and preservation of randomness in wave turbulence. Phys Lett A 332: 230–238.    
  • 13. Choi Y, Lvov YV, Nazarenko S (2005) Joint statistics of amplitudes and phases in wave turbulence. Phys D 201: 121–149.    
  • 14. Düring G, Josserand C, Rica S (2017) Wave turbulence theory of elastic plates. Phys D 347: 42–73.    
  • 15. Dyachenko AI, Lvov YV, Zakharov VE (1995) Five-wave interaction on the surface of deep fluid. Phys D 87: 233–261.    
  • 16. Eyink GL, Shi YK (2012) Kinetic wave turbulence. Phys D 241: 1487–1511.    
  • 17. Falkovich G, Lvov VS, Zakharov VE (1992) Kolmogorov Spectra of Turbulence. Berlin: Springer.
  • 18. Fermi E, Pasta J, Ulam S (1955) Studies of the nonlinear problems. Technical report I, Los Alamos Scientific Lab Report No. LA-1940.
  • 19. Ford J (1961) Equipartition of energy for nonlinear systems. J Math Phys 2: 387–393.    
  • 20. Fu WC, Zhang Y, Zhao H (2018) Universality of energy equipartition in one-dimensional lattices. arXiv preprint arXiv:1811.05697.
  • 21. Fu WC, Zhang Y, Zhao H (2019) Universal law of thermalization for one-dimensional perturbed toda lattices. arXiv preprint arXiv:1901.04245.
  • 22. Fucito F, Marchesoni F, Marinari E, et al. (1982) Approach to equilibrium in a chain of nonlinear oscillators. J Phys 43: 707–713.    
  • 23. Gallavotti G (2008) The Fermi-Pasta-Ulam Problem: A Status Report. Springer.
  • 24. Henrici A, Kappeler T (2008) Results on normal forms for FPU chains. Commun Math Phys 278: 145–177.    
  • 25. Arnold VI (1963) Small denominators and problems of stability of motion in classical and celestial mechanics. Russ Math Surv 18: 85–191.
  • 26. Izrailev FM, Chirikov BV (1966) Statistical properties of a nonlinear string. Sov Phys Dokl 11: 30–32.
  • 27. Janssen P (2004) The Interaction of Ocean Waves and Wind. Cambridge: Cambridge University Press.
  • 28. Moser JK (1962) On invariant curves of area-preserving mappings of an annulus. Nachr Akad Wiss Göttingen Math Phys kl 166: 1–20.
  • 29. Kartashova E (2007) Exact and quasiresonances in discrete water wave turbulence. Phys Rev Lett 98: 214502.    
  • 30. Khinchin A (1949) Mathematical Foundations of Statistical Mechanics. Courier Corporation.
  • 31. Kolmogorov AN (1954) On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl Akad Nauk SSR 98: 527–530.
  • 32. Krasitskii VP (1994) On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J Fluid Mech 272: 1–20.    
  • 33. Landau LD, Lifshitz EM, Pitaevskii LP (1980) Statistical Physics, Part I. Oxford: Pergamon.
  • 34. Laurie J, Bortolozzo U, Nazarenko S, et al. (2012) One-dimensional optical wave turbulence: Experiment and theory. Phys Rep 514: 121–175.    
  • 35. Lebowitz JL (1993) Boltzmann's entropy and time's arrow. Phys Today 46: 32–32.
  • 36. Lvov YV, Onorato M (2018) Double scaling in the relaxation time in the β-fermi-pasta-ulam-tsingou model. Phys Rev Lett 120: 144301.    
  • 37. Matkowski J (2011) Subadditive periodic functions. Opusc Math 31: 75–96.    
  • 38. Nazarenko S (2011) Wave Turbulence. Springer Science Business Media.
  • 39. Newell AC (1968) System of random gravity waves. Rev Geophys 6: 1–31.    
  • 40. Newell AC, Rumpf B (2011) Wave turbulence. Annu Rev Fluid Mech 43: 59–78.    
  • 41. Onorato M, Vozella L, Proment D, et al. (2015) Route to thermalization in the α-Fermi-Pasta-Ulam system. Proc Natl Acad Sci 112: 4208–4213.    
  • 42. Pistone L, Onorato M, Chibbaro S (2018) Thermalization in the discrete nonlinear klein-gordon chain in the wave-turbulence framework. Europhys Lett 121: 44003.    
  • 43. Rink B (2006) Proof of Nishida's conjecture on anharmonic lattices. Commun Math Phys 261: 613–627.    
  • 44. Spohn H (2006) The phonon boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J Stat Phys 124: 1041–1104.    
  • 45. Yoshida H (1990) Construction of higher order symplectic integrators. Phys Lett A 150: 262–268.    
  • 46. Zabusky NJ, Kruskal MD (1965) Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15: 240–243.
  • 47. Zakharov VE, Schulman EI (1988) On additional motion invariants of classical Hamiltonian wave systems. Phys D 29: 283–320.    
  • 48. Zakharov VE, Schulman EI (1991) Integrability of nonlinear systems and perturbation theory. In: What Is Integrability? Springer Series in Nonlinear Dynamics. Berlin: Springer.
  • 49. Zakharov VE, Odesskii AV, Onorato M, et al. (2012) Integrable equations and classical s-matrix. arXiv preprint arXiv:1204.2793.

 

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