Scaling of thermalization time with system size in weakly nonlinear FPUT chains under frequency-weighted initial energies

Determine the asymptotic scaling with the number of particles N of the relaxation (thermalization) time required to reach equipartition of normal-mode energies in the modified Fermi–Pasta–Ulam–Tsingou chain with a quartic nonlinearity and a uniform harmonic restoring term, when initialized with normal-mode energies proportional to their frequencies E_k^nm(0) ∝ ω_k. The goal is to identify the functional dependence on N of the thermalization time in this weakly nonlinear regime; preliminary single-realization numerics exclude an exponential dependence but do not specify the scaling law.

Background

The paper studies thermalization in high-dimensional oscillator chains, comparing integrable (harmonic) and weakly nonlinear (chaotic) regimes. In the nonlinear case they consider a modified FPUT Hamiltonian with an added uniform on-site harmonic term and focus on initial conditions where each normal mode k starts with energy proportional to its frequency, E_k^nm(0) = (ω_k E_0)/Σ_j ω_j.

They examine how time-averaged observables and mode-energy equipartition evolve and, in particular, the timescale required to reach thermalization. For classical FPUT initializations where only low-frequency modes are excited, earlier studies found power-law dependences on N for the relaxation time. In the present, different initialization, their simulations show that the system eventually thermalizes but do not reveal a clear scaling with N for the thermalization time, except that an exponential dependence can be ruled out.