Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou problem revisited: an approach using ideas from wave turbulence

Abstract: The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear $\alpha-$FPUT system in more detail. We aim to reconstruct the quasiperiodic behaviour observed in the original paper from the canonical transformation used to remove the three wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work we show that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the $\alpha-$FPUT system, and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the $\alpha-$FPUT problem. We argue that certain aspects of the $\alpha-$FPUT system such as presence of the stochasticity threshold, thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.