Antonio Ponno

Collective turns in starling flocks propagate linearly with negligible attenuation, indicating the existence of an underdamped sector in the dispersion relation. According to linear response theory, beside granting linear propagation of the phase perturbations, the real part of the frequency should also yield a spin-wave form of the unperturbed correlation function. However, new high-resolution experiments on real flocks show that underdamped traveling waves coexist with an overdamped Lorentzian correlation. Theory and experiments are reconciled once we add to the dynamics a Fermi-Pasta-Ulam-Tsingou term.

We present a non-perturbative, mean-field theory for the Fermi–Pasta–Ulam–Tsingou model with quartic interaction, capturing the quasiperiodic features shown by the system at all energies in the thermodynamic limit. Starting from the true Hamiltonian H of the system with N degrees of freedom, we introduce a mean-field Hamiltonian H such that the difference h N = ( H − H ) / N , considered as a random variable with respect to the Gibbs measure, tends to zero as N → ∞ , in probabilistic sense. The dynamics of the mean-field Hamiltonian H consists of N independent oscillation modes with renormalized frequencies Ω k = ω k 1 + γ ( ε ) , ω k being the frequency of the k th normal mode of the linearized system, whereas γ ( ε ) is an explicit function of the specific energy ε of the system. Analytical predictions drawn from the effective Langevin equations ruling the dynamics of such oscillation modes are successfully compared with the numerical data from the original Hamiltonian dynamics. Such a simple decomposition of the true dynamics into N effective normal modes holds at all energy scales, i.e. from the quasi-integrable regime to the strongly chaotic one.

We investigate the dynamics of the Fermi-Pasta-Ulam-Tsingou chain with long-wavelength random initial data. When the energy per particle is small, thermal equilibrium is not reached on a fast timescale, and the system enters prethermalization. The formation of the prethermal state is characterized by the development of a Burgers-type shock and the onset of a turbulentlike spectrum with a time dependent exponent ζ ( t ) in the inertial range. We perform a significant step forward by demonstrating that these features are robust under generic long-wavelength random initial conditions. By employing advanced probabilistic techniques inspired by the works of Dudley and Talagrand, we derive a sharp asymptotic expression for the average shock time in the thermodynamic limit. For large p , this time scales as ( p log p ) − 1 , where p is the number of excited modes, proving that it is an intensive quantity up to a logarithmic correction in the size of the system.

We consider a Klein–Gordon-Wave system, describing the evolution of a massive field and a massless one interacting through a Yukawa-like coupling, and we explicitly derive its Hamiltonian normal form to first and second order. To the first-order approximation, the normal form results in a Schrödinger-Wave (SW) system, which reduces to the Schrödinger–Poisson (SP) one in the singular limit of vanishing perturbative parameter. The second-order approximation provides the successive corrections to the SW system, and is presented in order to show that higher-order approximations to all orders can be obtained by iterating our constructive procedure. The normal form technique adopted here formally extends the standard Birkhoff normal form procedure for harmonic oscillators to include a set of free particles in the unperturbed problem. The mathematical result obtained here might explain, for example, the ‘cooling’ process of ultra-light dark matter, the approximate validity of the SP system describing its dynamics and the long term conservation of the total dark matter mass.

We describe the point spectra of some dissipative version of the celebrated ?Rayleigh loaded string?, an elastic string of finite length carrying a number n ? 1 of equally spaced, equal point masses, which is a basic model that exhibits a band structure and appears in many applied areas. We consider the case in which the dissipation is due to a viscous damping due to the interaction string-environment, a standard model for internal visco-elastic dissipation (the Kelvin-Voigt model), and their combined presence. We show that the point spectrum of each of these damped versions of the Rayleigh loaded string is a continuous deformation of the point spectrum of the unloaded elastic strings with that damping and that presents a band structure similar to that of the undamped case. We also provide explicit analytical expressions for the eigenfunctions, for any value of n.

We describe the point spectra of some dissipative version of the celebrated "Rayleigh loaded string", an elastic string of finite length carrying a number n ≥ 1 of equally spaced, equal point masses, which is a basic model that exhibits a band structure and appears in many applied areas. We consider the case in which the dissipation is due to a viscous damping due to the interaction string-environment, a standard model for internal visco-elastic dissipation (the Kelvin-Voigt model), and the combined presence of them. We show that the point spectrum of each of these damped versions of the Rayleigh loaded string is a continuous deformation of the point spectrum of the unloaded elastic strings with that damping and that presents a band structure similar to that of the undamped case. We also provide explicit analytical expressions for the eigenfunctions, for any value of n.