Abstract: The so-called Fermi-Pasta-Ulam-Tsingou (FPUT) problem, in large, concerns the problem of the approach of a Hamiltonian system to the state of equilibrium predicted by statistical mechanics. More precisely, the question is whether an initial measure evolves, through the dynamics, and in some specific sense, into the micro-canonical one, and on which time-scale(s). Still in other words, the problem consists in characterizing the mixing properties of the Hamiltonian system at hand.
Starting with the seminal paper of FPUT, and due to tradition and to the sake of simplicity, most of the studies in the literature concern the dynamics of classical, identical, interacting particle chains (one-dimensional systems), that for smooth interaction potentials are equivalent to perturbations of harmonic oscillators. The original, negative result obtained by FPUT, the lack of any trend to equilibrium observed within the available computation time, was named, after them, the FPUT paradox.
The aim of the present communication is to show that there is no paradox at all. Depending on energy, number of particles and initial conditions, the Hamiltonian system examined can usually be regarded as some small perturbation of a suitable integrable system close to it. The observed dynamics of the system resembles then, for a while, that of its closest integrable approximation, to then detach from it on a possibly long time-scale, to eventually reach the expected equilibrium. The whole process is referred to, in jargon, as "thermalization", the paradox being resolved by the existence of a nontrivial, short term, close to integrable "pre-thermal" stage. Some quantitative results on particle chains will be reviewed, with a particular emphasis on the short term dynamics for long wave-length, low energy initial conditions, where Burgers turbulence plays an important role.

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