Route to thermalization in the α-Fermi–Pasta–Ulam system

Miguel Onorato; Lara Vozella; Davide Proment; Yuri V. Lvov

doi:10.1073/pnas.1404397112

New Research In

Edited* by David W. McLaughlin, New York University, New York, NY, and approved February 25, 2015 (received for review March 7, 2014)

Significance

Despite the fact that more than 60 years have passed, the α-Fermi–Pasta–Ulam (FPU) system has not yet been fully understood. Their seminal work stimulated many interdisciplinary research topics in mathematics and physics like integrable systems, soliton theory, ergodic theory, and chaos. In this article, we theoretically investigate the original problem by applying the wave–wave interaction theory. By using this mathematical approach, we are able to explain why the emergence of equipartition requires very long times (inaccessible when the original numerical experiments were performed but nowadays recently observed using computer power). Our approach is general and can be used to attack other problems of weakly nonlinear dispersive waves.

Abstract

We study the original α-Fermi–Pasta–Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave–wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ⁸, where ϵ is the small parameter in the system. The wave–wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

Footnotes

↵¹To whom correspondence should be addressed. Email: lvovy{at}rpi.edu.

Author contributions: M.O., L.V., D.P., and Y.V.L. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.
The authors declare no conflict of interest.
↵*This Direct Submission article had a prearranged editor.