Chopping Time of the FPU alpha-Model

We study, both numerically and analytically, the time needed to observe the breaking of an FPU \u3b1-chain in two or more pieces, starting from an unbroken configuration at a given temperature. It is found that such a \u201cchopping\u201d time is given by a formula that, at low temperatures, is of the Arrhenius-Kramers form, so that the chain does not break up on an observable time-scale. The result explains why the study of the FPU problem is meaningful also in the ill-posed case of the \u3b1-model

The Fermi-Pasta-Ulam problem and its underlying integrable dynamics: an approach through Lyapunov Exponents

FPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include the usual $\beta$-model, perturbations of Toda include the usual $\alpha+\beta$ model. In this paper we explore and compare two families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each model, the maximal Lyapunov exponent $\chi$. We study the asymptotics of $\chi$ for large $N$ (the number of particles) and small $\epsilon$ (the specific energy $E/N$), and find, for all models, asymptotic power laws $\chi\simeq C\epsilon^a$, $C$ and $a$ depending on the model. The asymptotics turns out to be, in general, rather slow, and producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of $\chi$ introduced by Casetti, Livi and Pettini, originally formulated for the $\beta$-model. With great evidence the theory extends successfully to all models of the linear hierarchy, but not to models close to Toda

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi\u2013Pasta\u2013Ulam\u2013Tsingou (FPUT) chain composed by N 6b 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \u3b2- 1. Given a fixed 1 64 m 6a N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order \u3b2, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics

Mean Field Derivation of DNLS from the Bose–Hubbard Model

We prove that the flow of the discrete nonlinear Schrödinger equation (DNLS) is the mean field limit of the quantum dynamics of the Bose–Hubbard model for N interacting particles. In particular, we show that the Wick symbol of the annihilation operators evolved in the Heisenberg picture converges, as N becomes large, to the solution of the DNLS. A quantitative Lp-estimate, for any p≥ 1 , is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states

Resonance, metastability and blow up in FPU

We consider the FPU model with nonlinearity starting with terms of order n 65 3. We compute the resonant normal form in the region where only one low-frequency mode is excited and deduce rigorous results on the correspondence between the dynamics of the normal form and that of the complete system. As n varies, we give a criterion in order to deduce whether the FPU phenomenon (formation of a metastable packet of modes) is present or not. The criterion is that, if the normal form equation has smooth solutions then the FPU phenomenon is present, while it is absent if the solutions of the normal form equations have blow up in a finite time. In particular the phenomenon should be present for n 64 5 and absent for n 65 7

On metastability in FPU

We present an analytical study of the Fermi--Pasta--Ulam (FPU) $\alpha$--model with periodic boundary conditions. We analyze the dynamics corresponding to initial data with one low frequency Fourier mode excited. We show that, correspondingly, a pair of KdV equations constitute the resonant normal form of the system. We also use such a normal form in order to prove the existence of a metastability phenomenon. More precisely, we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes. Subsequently, the distribution remains unchanged up to the time scales of validity of our approximation. The phenomenon is controlled by the specific energy

Linear behavior in Covid19 epidemic as an effect of lockdown

We propose a mechanism explaining the approximately linear growth of Covid19 world total cases as well as the slow linear decrease of the daily new cases (and daily deaths) observed (in average) in USA and Italy. In our explanation, we regard a given population (the whole world or a single nation) as composed by many sub-clusters which, after lockdown, evolve essentially independently. The interaction is modeled by the fact that the outbreak time of the epidemic in a sub-cluster is a random variable with probability density slowly varying in time. The explanation is independent of the law according to which the epidemic evolves in the single sub cluster

The Fermi-Pasta-Ulam Problem

A review is given of the Fermi--Pasta--Ulam problem. Its foundational relevance in connection with the relations between classical and quantum mechanics is pointed out, and the status of the numerical and analytical results is discussed