Abstract
We study the original αFermi–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 αFPU equation of motion, we find that the first nontrivial resonances correspond to sixwave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for smallamplitude 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 (flipover) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.
Original language  English 

Pages (fromto)  4208–4213 
Number of pages  6 
Journal  Proceedings of the National Academy of Sciences of the United States of America (PNAS) 
Volume  112 
Issue number  14 
DOIs  
Publication status  Published  7 Apr 2015 
Keywords
 αFermi–Pasta–Ulam chain
 thermalization
 wave–wave interactions
 FPU recurrence
 resonant interactions
Profiles

Davide Proment
 School of Mathematics  Associate Professor in Applied Mathematics
 Centre for Photonics and Quantum Science  Member
 Quantum Fluids  Member
Person: Research Group Member, Academic, Teaching & Research