A mean-field theory of effective normal modes in the Fermi-Pasta-Ulam-Tsingou model

Abstract

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 $\mathcal{H}$ such that the difference $h_N=(H-\mathcal{H})/N$, considered as a random variable with respect to the Gibbs measure, tends to zero as $N\to \infty$, in probabilistic sense. The dynamics of the mean-field Hamiltonian $\mathcal{H}$ consists of $N$ independent oscillation modes with renormalized frequencies ${\Omega }_k={\omega }_k\sqrt{1+\gamma (\epsilon )}$, ${\omega }_k$ being the frequency of the $k$-th normal mode of the linearized system, whereas $\gamma (\epsilon )$ is an explicit function of the specific energy $\epsilon$ 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.

30 pages, 3 figures