Abstract
We give a review of the Fermi–Pasta–Ulam (FPU) problem from the perspective of its possible impact on the foundations of physics, concerning the relations between classical and quantum mechanics. In the first part we point out that the problem should be looked upon in a wide sense (whether the equilibrium Gibbs measure is attained) rather than in the original restricted sense (whether energy equipartition is attained). The second part is devoted to some very recent results of ours for an FPU-like model of an ionic crystal, which has such a realistic character as to reproduce in an impressively good way the experimental infrared spectra. Since the existence of sharp spectral lines is usually considered to be a characteristic quantum phenomenon, even unconceivable in a classical frame, this fact seems to support a thesis suggested by the original FPU result. Namely, that the relations between classical and quantum mechanics are much subtler than usually believed, and should perhaps be reconsidered under some new light.
Similar content being viewed by others
References
Campbell, D.K., Rosenau, Ph., and Zaslavsky, G. M., Introduction: The Fermi–Pasta–Ulam Problem — The First Fifty Years, Chaos, 2005, vol. 15, no. 1, 015101, 4 pp.
The Fermi–Pasta–Ulam Problem: A Status Report, G.Gallavotti (Ed.), Lect. Notes Phys., vol. 728, Berlin: Springer, 2008.
Schilpp, P. A., Albert Einstein: Philosopher-Scientist, 3rd ed., Library of Living Philosophers, vol. 7, Peru, Ill.: Open Court, 1998.
Carati, A. and Galgani, L., Progress along the Lines of the Einstein Classical Program: An Enquiry on the Necessity of Quantization in Light of the Modern Theory of Dynamical Systems: Notes (in an Extremely Preliminary Form) for a Course on the Foundations of Physics at the Milan University, https://doi.org/www.mat.unimi.it/users/carati/didattica/fondamenti/indice.pdf (2018).
Born, M., Atomic Physics, 8th ed., New York: Dover, 1989.
Carati, A., Galgani, L., and Giorgilli, A., The Fermi–Pasta–Ulam Problem As a Challenge for the Foundations of Physics, Chaos, 2005, vol. 15, no. 1, 015105, 19 pp.
Benettin, G., Carati, A., Galgani, L., and Giorgilli, A., The Fermi–Pasta–Ulam Problem and the Metastability Perspective, in The Fermi–Pasta–Ulam Problem: A Status Report, G.Gallavotti (Ed.), Lect. Notes Phys., vol. 728, Berlin: Springer, 2008, pp. 151–189.
Poincaré, H., Les méthodes nouvelles de la mécanique céleste: Vol. 1. Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotique, Paris: Gauthier-Villars, 1892, pp. 233–268.
Fermi, E., Pasta, J., and Ulam, S., Studies of Nonlinear Systems, in Collected Papers of Enrico Fermi: Vol. 2. United States, 1939–1945, E.Amaldi et al. (Eds.), Chicago, Ill.: Univ. of Chicago, 1965, pp. 978–993.
Fermi, E., Beweis, dass ein mechanisches Normalsystem im allgemeinen quasi-ergodisch ist, in Collected Papers of Enrico Fermi: Vol. 1. Italy, 1921–1938, E.Amaldi et al. (Eds.), Chicago, Ill.: Univ. of Chicago, 1962, pp. 79–86.
Zabusky, N. J., Fermi–Pasta–Ulam, Solitons and the Fabric of Nonlinear and Computational Science: History, Synergetics, and Visiometrics, Chaos, 2005, vol. 15, no. 1, 015102, 16 pp.
Zabusky, N. J. and Kruskal, M. D., Interaction of “Solitons” in a Collisionless Plasma and the Recurrence of Initial States, Phys. Rev. Lett., 1965, vol. 15, no. 6, pp. 240–243.
Gardner, C. S., Greene, J.M., Kruskal, M.D., and Miura, R.M., Method for Solving the Korteweg–deVries Equation, Phys. Rev. Lett., 1967, vol. 19, no. 19, pp. 1095–1097.
Berman, G.P. and Izrailev, F.M., The Fermi–Pasta–Ulam Problem: Fifty Years of Progress, Chaos, 2005, vol. 15, no. 1, 015104, 18 pp.
Chirikov, B.V., Resonance Processes in Magnetic Traps, J. Nucl. Energy: Part C, 1960, vol. 1, no. 4, pp. 253–260; see also: Soviet J. Atom. Energy, 1960, vol. 6, no. 6, pp. 464–470.
Izrailev, F.M. and Chirikov, B.V., Statistical Properties of a Nonlinear String, Sov. Phys. Dokl., 1966, vol. 11, no. 1, pp. 30–34; see also: Dokl. Akad. Nauk SSSR, 1966, vol.166, no. 1, pp. 57–59.
Izrailev, F. M., Khisamutdinov, A. I., and Chirikov, B. V., Numerical Experiments with a Chain of Coupled Anharmonic Oscillators, Los Alamos Technical Report LA-4440 (1970); see also: Report 252, Novosibirsk: Institute of Nuclear Physics, 1968.
Nekhoroshev, N. N., An Exponential Estimate of the Stability Time of Near-Integrable Hamiltonian Systems, Russian Math. Surveys, 1977, vol. 32, no. 6, pp. 1–65; see also: Uspekhi Mat. Nauk, 1977, vol. 32, no. 6(198), pp. 5–66.
Benettin, G., Galgani, L., and Giorgilli, A., A Proof of Nekhoroshev’s Theorem for the Stability Times in Nearly Integrable Hamiltonian Systems, Celestial Mech., 1985, vol. 37, no. 1, pp. 1–25.
Bambusi, D. D. and Giorgilli, A., Exponential Stability of States Close to Resonance in Infinite Dimensional Hamiltonian Systems, J. Stat. Phys., 1993, vol. 71, nos. 3–4, pp. 569–606.
Bocchieri, P., Scotti, A., Bearzi, B., and Loinger, A., Anharmonic Chain with Lennard–Jones Interaction, Phys. Rev. A, 1970, vol. 2, no. 5, pp. 2013–2019.
Galgani, L. and Scotti, A., Planck-Like Distribution in Classical Nonlinear Mechanics, Phys. Rev. Lett., 1972, vol. 28, no. 18, pp. 1173–1176.
Cercignani, C., Galgani, L., and Scotti, A., Zero-Point Energy in Classical Nonlinear Mechanics, Phys. Lett. A, 1972, vol. 38, no. 6, pp. 403–404.
Galgani, L. and Scotti, A., Recent Progress in Classical Nonlinear Dynamics, Riv. Nuovo Cimento, 1972, vol. 2, no. 2, pp. 189–209.
Carati, A., An Averaging Theorem for Hamiltonian Dynamical Systems in the Thermodynamic Limit, J. Stat. Phys., 2007, vol. 128, no. 4, pp. 1057–1077.
Neishstadt, A. I., Averaging in Multifrequency Systems: 2, Sov. Phys. Dokl., 1976, vol. 21, no. 2, pp. 80–82; see also: Dokl. Akad. Nauk SSSR, 1976, vol. 226, no. 6, pp. 1295–1298.
De Roeck, W. and Huveneers, F., Asymptotic Localization of Energy in Non-Disordered Oscillator Chains, Comm. Pure Appl. Math., 2015, vol. 68, no. 9, pp. 1532–1568.
Maiocchi, A., Bambusi, D., and Carati, A., An Averaging Theorem for FPU in the Thermodynamic Limit, J. Stat. Phys., 2014, vol. 155, no. 2, pp. 300–322.
Carati, A. and Maiocchi, A., Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit, Comm. Math. Phys., 2012, vol. 314, no. 1, pp. 129–161.
Giorgilli, A., Paleari, S., and Penati, T., An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit, Ann. Henri Poincaré, 2015, vol. 16, no. 4, pp. 897–959.
Maiocchi, A., Freezing of the Optical Branch Energy in a Diatomic FPU Chain, arXiv:1808.09359 (2018).
Carati, A., Galgani, L., Giorgilli, A., and Paleari, S., Fermi–Pasta–Ulam Phenomenon for Generic Initial Data, Phys. Rev. E, 2007, vol. 76, no. 2, 022104, 4 pp.
Berchialla, L., Galgani, L., and Giorgilli, A., Localization of Energy in FPU Chains, Discrete Contin. Dyn. Syst., 2004, vol. 11, no. 4, pp. 855–866.
Benettin, G. and Ponno, A., Time-scales to Equipartition in the Fermi–Pasta–Ulam Problem: Finite-Size Effects and Thermodynamic Limit, J. Stat. Phys., 2011, vol. 144, no. 4, pp. 793–812.
Benettin, G., Christodoulidi, H., and Ponno, A., The Fermi–Pasta–Ulam Problem and Its Underlying Integrable Dynamics, J. Stat. Phys., 2013, vol. 152, no. 2, pp. 195–212.
Berchialla, L., Giorgilli, A., and Paleari, S., Exponentially Long Times to Equipartition in the Thermodynamic Limit, Phys. Lett. A, 2004, vol. 321, no. 3, pp. 167–172.
Paleari, S. and Penati, T., Equipartition Times in a Fermi–Pasta–Ulam System, Discrete Contin. Dyn. Syst., 2005, suppl., pp. 710–719.
Siegel, C. L., On the Integrals of Canonical Systems, Ann. of Math. (2), 1941, vol. 42, pp. 806–822.
Ferguson, W.E. Jr., Flaschka, H., and McLaughlin, D.W., Nonlinear Normal Modes for the Toda Chain, J. Comput. Phys., 1982, vol. 45, no. 2, pp. 157–209.
Rink, B., An Integrable Approximation for Fermi–Pasta–Ulam Lattice, in The Fermi–Pasta–Ulam Problem: A Status Report, G.Gallavotti (Ed.), Lect. Notes Phys., vol. 728, Berlin: Springer, 2008, pp. 283–301.
Giorgilli, A., Paleari, S., and Penati, T., Local Chaotic Behaviour in the Fermi–Pasta–Ulam System, Discrete Contin. Dyn. Syst. Ser. B, 2005, vol. 5, no. 4, pp. 991–1004.
Danieli, C., Campbell, D.K., and Flach, S., Intermittent Many-Body Dynamics at Equilibrium, Phys. Rev. E, 2017, vol. 95, no. 6, 060202, 5 pp.
Mithun, Th., Kati, Y., Danieli, C., and Flach, S., Weakly Nonergodic Dynamics in the Gross–Pitaevskii Lattice, Phys. Rev. Lett., 2018, vol. 120, no. 18, 184101, 6 pp.
Carati, A. and Galgani, L., On the Specific Heat of the Fermi–Pasta–Ulam Systems and Their Glassy Behavior, J. Stat. Phys., 1999, vol. 94, nos. 5–6, pp. 859–869.
Carati, A. and Galgani, L., Metastability in Specific-Heat Measurements: Simulations with the FPU model, Europhys. Lett., 2006, vol. 75, no. 4, pp. 528–534.
Carati, A., Galgani, L., and Pozzi, B., Lévy Flights in Landau–Teller Model of Molecular Collision, Phys. Rev. Lett., 2003, vol. 90, no. 1, 010601, 4 pp.
Kozlov, V.V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Regul. Chaotic Dyn., 2008, vol. 13, no. 3, pp. 141–154.
Kozlov, V.V., Thermal Equilibrium in the Sense of Gibbs and Poincaré, Izhevsk: R&C Dynamics, Institute of Computer Science, 2002 (Russian).
Kozlov, V.V., On Justification of Gibbs Distribution, Regul. Chaotic Dyn., 2002, vol. 7, no. 1, pp. 1–10.
Kozlov, V.V. and Treshchev, D. V., Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems, Theoret. and Math. Phys., 2003, vol. 134, no. 3, pp. 339–350; see also: Teoret. Mat. Fiz., 2003, vol. 134, no. 3, pp. 388–400.
Ponno, A., A Theorem on the Equilibrium Thermodynamics of Hamiltonian Systems, Phys. A, 2006, vol. 359, pp. 162–176.
Tsallis, C., Possible Generalization of Boltzmann–Gibbs Statistics, J. Stat. Phys., 1998, vol. 52, nos. 1–2, pp. 479–487.
Carati, A., On the Definition of Temperature Using Time-Averages, Phys. A, 2006, vol. 369, no. 2, pp. 417–431.
Fucito, E., Marchesoni, F., Marinari, E., Parisi, G., Peliti, L., Ruffo, S., and Vulpiani, A., Approach to Equilibrium in a Chain of Nonlinear Oscillators, J. Phys., 1982, vol. 43, no. 5, pp. 707–713.
Gangemi, F., Carati, A., Galgani, L., Gangemi, R., and Maiocchi, A., Agreement of Classical Kubo Theory with the Infrared Dispersion Curves n(ω) of Ionic Crystals, Europhys. Lett., 2015, vol. 110, no. 4, 47003, 11 pp.
Carati, A., Galgani, L., Maiocchi, A., Gangemi, F., and Maiocchi, R., Classical Infrared Spectra of Ionic Crystals and Their Relevance for Statistical Mechanics, Phys. A, 2018, vol. 506, pp. 1–10.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carati, A., Galgani, L., Maiocchi, A. et al. The FPU Problem as a Statistical-mechanical Counterpart of the KAM Problem, and Its Relevance for the Foundations of Physics. Regul. Chaot. Dyn. 23, 704–719 (2018). https://doi.org/10.1134/S1560354718060060
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354718060060