arXiv:1402.1603v1 [nlin.CD] 7 Feb 2014 On the thermalization of the α-Fermi-Pasta-Ulam system M. Onorato 1,2 , L. Vozella 1 , D. Proment 3 , and Y. V. Lvov 4 1 Dip. di Fisica, Universit` a di Torino, Via P. Giuria, 1 - Torino, 10125, Italy; 2 INFN, Sezione di Torino, Via P. Giuria, 1 - Torino, 10125, Italy; 3 School of Mathematics, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom 4 Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180, USA; (Dated: February 10, 2014) We study theoretically 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. We show that the route to thermalization consists of three stages. The first one is associated with non-resonant three-wave interactions. At this short time scale, the dynamics is reversible; this stage coincides with the observation of recurrent phenomena in numerical simulations of the α-FPU. On a larger time scale, exact four-wave resonant interactions start to take place; however, we find that all quartets are isolated, preventing a full mixing of energy in the spectrum and thermalization. The last stage corresponds to six-wave resonant interactions. Those are responsible for the energy equipartition recently observed in numerical simulations. A key role in our finding is played by the Umklapp (flip over) resonant interactions, typical of discrete systems. PACS numbers: Valid PACS appear here The Fermi Pasta Ulam (FPU) chains is a mathemati- cal model introduced in the fifties to study the thermal equipartition in crystals [1]. The model consists of N identical masses connected by a nonlinear spring; the elastic force can be expressed as a power series in the spring deformation Δx: F = −γ Δx + αΔx 2 + βΔx 3 + ..., (1) where γ,α and β are elastic, spring dependent, con- stants. The α-FPU chain, the system studied herein, corresponds to the case of α = 0 and β = 0. Fermi, Pasta and Ulam integrated numerically the equations of motion and conjectured that, after many iterations, the system would exhibit a thermalization, i.e. a state in which the influence of the initial modes disappears and the system becomes random, with all modes ex- cited more or less equally (equipartition of energy). Con- trary to their expectations, the system exhibited a very complicated quasi-periodic behavior. This phenomenon has been named “FPU recurrence”. The discovery has spurred many great mathematical and physical discover- ies such as integrability [2] and soliton physics [3]. More recently, very long numerical simulations have shown a clear evidence of the phenomenon of equiparti- tion (see for example [4] and references therein). Yet, de- spite substantial progresses on the subject (see for exam- ple recent reviews [5–9]), to our knowledge, no complete understanding of the original problem has been achieved so far and the numerical results of the original α-FPU sys- tem remain largely unexplained from a theoretical point of view; more precisely, the physical mechanism respon- sible for a first metastable state [4] and the observation of equipartition for very large times have not been un- derstood. In this Letter, we present an approach to the FPU problem based on the nonlinear interaction of weakly nonlinear dispersive waves. Our main assumption is that the irreversible transfer of energy in the spectrum in a weakly nonlinear system is achieved by exact resonant wave-wave interactions. Such resonant interactions are the bases for the so called wave turbulence theory [10, 11] and are responsible for the phenomenon of thermaliza- tion. We will show that in the α-FPU system six-wave resonant interactions are responsible for an efficient irre- versible transfer of energy in the spectrum. The equation of motion for a chain of N identical par- ticles of mass m, subject to a force of the type in (1) with α = 0 and β = 0, has the following form: m¨ q j =(q j+1 + q j−1 − 2q j )(γ + α(q j+1 − q j−1 )) , (2) with j =0, 1, .., N − 1. q j (t) is the displacement of the particle j from the equilibrium position. We consider pe- riodic boundary conditions, i.e. q N = q 0 . Our approach is developed in Fourier space and the following definitions of the direct and inverse Discrete Fourier Transform are adopted: Q k = 1 N N−1  j=0 q j e −i2πkj/N ,q j = N/2  k=−N/2+1 Q k e i2πjk/N , (3)