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

TL;DR

The paper addresses how the quartic FPUT chain behaves across all energy scales in a non-perturbative regime. It constructs a mean-field surrogate $\mathcal{H}$ that is close to the true Hamiltonian $H$ in measure, yielding $h_N=(H-\mathcal{H})/N$ that vanishes as $N\to\infty$, and shows that the dynamics reduces to $N-1$ independent effective normal modes with renormalized frequencies $\Omega_k=\omega_k\sqrt{1+\gamma(\varepsilon)}$. Analytically, it provides closed-form limits for $\alpha(\beta)$ and $\varepsilon(\beta)$, derives $\gamma(\varepsilon)$, and demonstrates that each mode obeys a damped-Langevin equation with white-noise forcing, aligning with hydrodynamic scaling. Numerically, the predicted spectra and energy-relaxation times match molecular-dynamics data, supporting an emergent effective integrability at all energies and offering a unified, non-perturbative framework for chaotic many-body dynamics with broad applicability to even potentials.

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 $Ω_k = ω_k\sqrt{1+γ(\varepsilon)}$, $ω_k$ being the frequency of the $k$-th normal mode of the linearized system, whereas $γ(\varepsilon)$ is an explicit function of the specific energy $\varepsilon$ 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.

Figures (3)

• Figure 1: Renormalizing coefficient $\gamma(\varepsilon)$, as a function of the specific energy. The blue solid curve represents the (canonical) ensemble estimate on the r.h.s. of Eq. \ref{['gamma']}. Red squares are computed considering empirical averages in molecular dynamics simulations of model \ref{['H']}, according to Eq. \ref{['gammaemp']}. We use a symplectic integration algortihm (Verlet), with a time-step chosen, for each value of $\varepsilon$, in such a way that the relative fluctuations on the total energy due to numerical effects are $\sim O(10^{-5})$. Parameters: $N=100$, $g=1$.
• Figure 2: Power density spectra of the normal modes $Q_k$'s. In panel (a) different values of $k$ are considered, at specific energy $\varepsilon=1$. Frequencies are rescaled by $\Omega_k$: as expected, the peaks of the spectra are found in 1. Panel (b) shows instead the behaviour of the spectrum for $k=34$ at different values of the energy. The dashed lines are obtained by fitting the functional form \ref{['SkomL2']} (with $c_0$ set to the phenomenological value $0.4$ and one free parameter for normalization). Details on the numerical simulations as in Fig. \ref{['fig:gamma']}.
• Figure 3: Mean square deviation of the spectral energies, for different values of $k$. Each curve has been obtained by averaging over 500 branches of a single trajectory of the system. Each branch spans a time window of 8 periods of the slowest normal mode. Here $\varepsilon=1$. The dashed line, proportional to $t$, is a guide for the eyes. Further details on the numerical simulations as in Fig. \ref{['fig:gamma']}.