title: Route to thermalization in the α-Fermi-Pasta-Ulam system.
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/ϵ(8), 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.
authors: Dipartimento di Fisica, Università degli Studi di Torino and Istituto Nazionale di Fisica Nucleare-Sezione di Torino, 10125 Turin, Italy;
authors: Miguel
authors: M
authors: Onorato
authors: Dipartimento di Fisica, Università degli Studi di Torino and Istituto Nazionale di Fisica Nucleare-Sezione di Torino, 10125 Turin, Italy;
authors: Lara
authors: L
authors: Vozella
authors: School of Mathematics, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom; and.
authors: Davide
authors: http://orcid.org/0000-0002-9472-0097
authors: D
authors: Proment
authors: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 lvovy@rpi.edu.
authors: Yuri V
authors: YV
authors: Lvov
keywords: FPU recurrence; resonant interactions; thermalization; wave–wave interactions; α-Fermi–Pasta–Ulam chain