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The Fermi-Pasta-Ulam Problem and Its Underlying Integrable Dynamics

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Abstract

This paper is devoted to a numerical study of the familiar α+β FPU model. Precisely, we here discuss, revisit and combine together two main ideas on the subject: (i) In the system, at small specific energy ε=E/N, two well separated time-scales are present: in the former one a kind of metastable state is produced, while in the second much larger one, such an intermediate state evolves and reaches statistical equilibrium. (ii) FPU should be interpreted as a perturbed Toda model, rather than (as is typical) as a linear model perturbed by nonlinear terms. In the view we here present and support, the former time scale is the one in which FPU is essentially integrable, its dynamics being almost indistinguishable from the Toda dynamics: the Toda actions stay constant for FPU too (while the usual linear normal modes do not), the angles fill their almost invariant torus, and nothing else happens. The second time scale is instead the one in which the Toda actions significantly evolve, and statistical equilibrium is possible. We study both FPU-like initial states, in which only a few degrees of freedom are excited, and generic initial states extracted randomly from an (approximated) microcanonical distribution. The study is based on a close comparison between the behavior of FPU and Toda in various situations. The main technical novelty is the study of the correlation functions of the Toda constants of motion in the FPU dynamics; such a study allows us to provide a good definition of the equilibrium time τ, i.e. of the second time scale, for generic initial data. Our investigation shows that τ is stable in the thermodynamic limit, i.e. the limit of large N at fixed ε, and that by reducing ε (ideally, the temperature), τ approximately grows following a power law τ∼ε −a, with a=5/2.

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Notes

  1. From Ref. [1]: … It is, therefore, very hard to observe the rate of ‘thermalization’ or mixing in our problem, and this was the initial purpose of the calculation.

  2. The limit is delicate, and approaching it at low ε requires larger and larger N, see [19]; for example, for ε=10−4 the correct asymptotics appears for, say, N=1023 or larger, while N=255 is sufficient at ε=10−3. The two limits N→∞ and ε→0 do not commute at all.

  3. Of course the divergence stops for β so close to β T , that the terms of order five, different in FPU and in Toda, become dominant; see [19].

  4. Odd powers of L do provide the additional constants of motion of the larger periodic model, but identically vanish for motions satisfying (9).

  5. Here and in the following, smaller values of ε would be desirable, but times then get much larger and the numerical work goes behind our possibilities.

  6. It is not easy to compare the speed of MANIAC I used by FPU with the speed of modern CPUs. A prudent estimate gives a ratio of the order 105; this means that each panel of Fig. 1 would require, on MANIAC I, about 105 CPU years.

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Acknowledgements

We feel indebted to all our colleagues of Milano, in particular Luigi Galgani, Antonio Giorgilli, Dario Bambusi, Andrea Carati, Simone Paleari and Tiziano Penati, for (years of) intense discussions; their criticism has been particularly helpful. The results contained in the paper have been preliminarily shown to Giovanni Gallavotti and Giorgio Parisi (Rome), and more recently to the research group around Jean-Pierre Eckmann (Genéve); the comments we received have been quite useful to improve the exposition of our results. This research has been supported by the University of Padova, also covering the post-doc position of H.C.

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  1. H. Christodoulidi

    Present address: Department of Mathematics, University of Patras, 26500, Patras, Greece

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  1. Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy

    G. Benettin, H. Christodoulidi & A. Ponno

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  1. G. Benettin
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Benettin, G., Christodoulidi, H. & Ponno, A. The Fermi-Pasta-Ulam Problem and Its Underlying Integrable Dynamics. J Stat Phys 152, 195–212 (2013). https://doi.org/10.1007/s10955-013-0760-6

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