The Metastable State of Fermi–Pasta–Ulam–Tsingou Models

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan–Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a “metastable” state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi–Pasta–Ulam–Tsingou (FPUT) models. We treat both the [Formula: see text]-FPUT and [Formula: see text]-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure—the spectral entropy ([Formula: see text])—and show that this measure has the power to quantify the distance from equipartition. For the [Formula: see text]-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state [Formula: see text] in the [Formula: see text]-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the [Formula: see text]- [Formula: see text] plane. Applying this procedure gives us a power-law scaling for [Formula: see text] , with the important result that the power laws for different system sizes collapse down to the same exponent as [Formula: see text]. We examine the energy spectrum [Formula: see text] over time in the [Formula: see text]-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the “wave turbulence” theory. We next apply a similar approach to the [Formula: see text]-FPUT model. Here, we explore in particular the different behavior for the two different signs of [Formula: see text]. Finally, we describe a procedure for calculating [Formula: see text] in the [Formula: see text]-FPUT model, a very different task than for the [Formula: see text]-FPUT model, because the [Formula: see text]-FPUT model is not a truncation of an integrable nonlinear model.