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 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.