ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Ivanchenko M. V. q-­breathers: from the fermi–pasta–ulam paradox to anomalous conductivity. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 1, pp. 73-85. DOI: 10.18500/0869-6632-2011-19-1-73-85

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 244)
Article type: 
530.182, 534.1, 534.015

q-­breathers: from the fermi–pasta–ulam paradox to anomalous conductivity

Ivanchenko Mihail Vasilevich, Lobachevsky State University of Nizhny Novgorod

The paper reviews the modern problems of nonlinear physics, where q-breathers theory finds its applications.

  1. Fermi E, Pasta J, and Ulam S. Los Alamos Report LA-1940; 1955. Also in: Segre E, editor. Collected Papers of Enrico Fermi. Vol. 2. University of Chicago Press; 1965. P. 978; Mattis DC, editor. Many-Body Problems. Singapore: World Scientific; 1993.
  2. Izrailev FM, and Chirikov BV. Statistical properties of a non-linear string. Sov. Phys. Dokl. 1966;11:30–32.
  3. Zabusky NJ, and Kruskal MD. Interaction of «solitons» in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 1965;15(6):240–243. DOI: 10.1103/PhysRevLett.15.240.
  4. Ford J. The Fermi–Pasta–Ulam problem: Paradox turns discovery. Phys. Rep. 1992;213(5):271–310. DOI: 10.1016/0370-1573(92)90116-H.
  5. Campbell DK. The Fermi–Pasta–Ulam problem – The first fifty years. Chaos. 2005;15(1):015101. DOI: 10.1063/1.1889345.
  6. Berman GP and Izrailev FM. The Fermi–Pasta–Ulam problem: Fifty years of progress. Chaos. 2005;15(1):015104. DOI: 10.1063/1.1855036.
  7. De Luca J, Lichtenberg AJ, and Lieberman MA. Time scale to ergodicity in the Fermi–Pasta–Ulam system. Chaos. 1995;5(1):283–297. DOI: 10.1063/1.166143.
  8. Shepelyansky DL. Low-energy chaos in the Fermi–Pasta–Ulam problem. Nonlinearity. 1997;10(5):1331–1338. DOI: 10.1088/0951-7715/10/5/017.
  9. Bocchierri P, Scotti A, Bearzi B, and Loigner A. Anharmonic chain with Lennard-Jones interaction. Phys. Rev. A. 1970;2(5):2013–2019. DOI: 10.1103/PhysRevA.2.2013.
  10. Galgani L, and Scotti A. Planck-like distributions in classical nonlinear mechanics. Phys. Rev. Lett. 1972;28(18):1173–1176. DOI: 10.1103/PhysRevLett.28.1173.
  11. Patrascioiu A. Blackbody radiation law: quantum or classical explanation? Phys. Rev. Lett. 1983;50(24):1879–1882. DOI: 10.1103/PhysRevLett.50.1879.
  12. Kantz H. Vanishing stability thresholds in the thermodynamic limit of nonintegrable conservative systems. Physica D. 1989;39(2–3):322–335. DOI: 10.1016/0167-2789(89)90014-6.
  13. Kantz H, Livi R, and Ruffo S. Equipartition thresholds in chains of anharmonic oscillators. J. Stat. Phys. 1994;76(1–2):627–643.
  14. Casetti L, Cerruti-Sola M, Pettini M, and Cohen EGD. The Fermi–Pasta–Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems. Phys. Rev. E. 1997;55(6):6566–6574. DOI: 10.1103/PhysRevE.55.6566.
  15. Flach S, Ivanchenko MV, and Kanakov OI. q-Breathers and the Fermi–Pasta–Ulam problem. Phys. Rev. Lett. 2005;95(6):064102. DOI: 10.1103/PhysRevLett.95.064102.
  16. Flach S, Ivanchenko MV, and Kanakov OI. q-breathers in Fermi–Pasta–Ulam chains: Existence, localization, and stability. Phys. Rev. E. 2006;73(3):036618. DOI: 10.1103/PhysRevE.73.036618.
  17. Fermi E. Evidence that a mechanic normal system is generally quasi-ergodic. Phys. Z. 1923;24:261.
  18. Tuck JL. Los Alamos Report. No. LA-3990; 1968.
  19. Izrailev FM, Khasamutdinov AI, and Chirikov BV. Numerical experiments on the statistical behaviour of dynamical systems with a few degrees of freedom. Comput. Phys. Commun. 1973;5(1):11–16. DOI: 10.1016/0010-4655(73)90003-9.
  20. Ullmann K, Lichtenberg AJ, and Corso G. Energy equipartition starting from high-frequency modes in the Fermi–Pasta–Ulam beta oscillator chain. Phys. Rev. E. 2000;61(3):2471–2477. DOI: 10.1103/PhysRevE.61.2471.
  21. De Luca J, Lichtenberg A. Transitions and time scales to equipartition in oscillator chains: Low-frequency initial conditions. Phys. Rev. E. 2002;66(2):026206. DOI: 10.1103/physreve.66.026206.
  22. Berchialla L, Giorgilli A, and Paleari S. Exponentially long times to equipartition in the thermodynamic limit. Physics Letters A. 2004;321(3):167–172. DOI: 10.1016/j.physleta.2003.11.052.
  23. Benettin G, Livi R, Ponno A. The Fermi–Pasta–Ulam problem: Scaling laws vs. initial conditions. J. Stat. Phys. 2009;135(5):873–893. DOI: 10.1007/s10955-008-9660-6.
  24. Giorgilli A, Paleari S, Penati T. Local chaotic behavior in the Fermi–Pasta–Ulam system. Discr. Cont. Dyn. Sys. B. 2005;5(4):991–1004. DOI: 10.3934/dcdsb.2005.5.991.
  25. Benettin G. Time scale for energy equipartition in a two-dimensional FPU model. Chaos. 2005;15(1):015108. DOI: 10.1063/1.1854278.
  26. Benettin G and Gradenigo G. A study of the Fermi– Pasta–Ulam problem in dimension two. Chaos. 2008;18(1):013112. DOI: 10.1063/1.2838458.
  27. Flach S and Willis C.R. Discrete breathers. Phys. Rep. 1998;295(5):181–264. DOI: 10.1016/S0370-1573(97)00068-9.
  28. Lyapunov MA. The General Problem of Stability of Motion. London: Taylor & Francis; 1992. 545 p.
  29. Ivanchenko MV et al. q-Breathers in finite two- and three-dimensional nonlinear acoustic lattices. Phys. Rev. Lett. 2006;97(2):025505. DOI: 10.1103/PhysRevLett.97.025505.
  30. Mishagin KG et al. q-breathers is discrete nonlinear Schroedinger lattices. New J. Phys. 2008;10:073034. DOI: 10.1088/1367-2630/10/7/073034.
  31. Nguenang JP, Pinto RA, Flach S. Quantum q-breathers in a finite Bose–Hubbard chain: The case of two interacting bosons. Phys. Rev. B. 2007;75(21):214303. DOI: 10.1103/PhysRevB.75.214303.
  32. Ivanchenko MV. q-Breathers and thermalization in acoustic chains with arbitrary nonlinearity Index. JETP Letters. 2010;92(6):365–369. DOI: 10.1134/S0021364010180013.
  33. Christodoulidi H, Efthymiopoulos C, and Bountis T. Energy localization on q-tori, long-term stability, and the interpretation of Fermi–Pasta–Ulam recurrences. Phys. Rev. E. 2010:81(1):016210. DOI: 10.1103/PhysRevE.81.016210.
  34. Penati T, Flach S. Tail resonances of Fermi–Pasta–Ulam q-breathers and their impact on the pathway to equipartition. Chaos. 2007;17(2):023102. DOI: 10.1063/1.2645141.
  35. Ivanchenko MV. q-Breathers in finite lattices: nonlinearity and weak disorder. Phys. Rev. Lett. 2009;102(17):175507. DOI: 10.1103/PhysRevLett.102.175507.
  36. Ivanchenko MV. q-Breathers in discrete nonlinear Schroedinger arrays with weak disorder. JETP Letters. 2009;89(3):150–155. DOI: 10.1134/S0021364009030114.
  37. Matsuda H, Ishii K. Localization of normal modes and energy transport in disordered harmonic chain. Suppl. Prog. Theor. Phys. 1970;45:56–86. DOI: 10.1143/PTPS.45.56.
  38. Lepri S, Livi R, and Politi A. Thermal conduction in classical low-dimensional lattices. Phys. Rep. 2003;377(1):1–80. DOI: 10.1016/S0370-1573(02)00558-6.
  39. Dhar A. Heat transport in low-dimensional systems. Adv. Phys. 2008;57(5):457–537. DOI: 10.1080/00018730802538522.
  40. Chang CW et al. Breakdown of Fourier’s law in nanotube thermal conductors. Phys. Rev. Lett. 2008;101(7):075903. DOI: 10.1103/PhysRevLett.101.075903.
  41. Ivanchenko MV and Flach S. Anomalous conductivity: impact of nonlinearity and disorder. arXiv:1009.3447v1. arXiv Preprint; 2010.
Short text (in English):
(downloads: 72)